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  <title>kNOw self</title>
  <subtitle>Tsan-Kuang Lee&#39;s Blog</subtitle>
  <link href="https://blog.tklee.org/en/feed.xml" rel="self"/>
  <link href="https://blog.tklee.org/en/"/>
  <updated>2026-08-24T00:00:00Z</updated>
  <id>https://blog.tklee.org/en/</id>
  <author>
    <name>Tsan-Kuang &quot;TK&quot; Lee 李璨光</name>
  </author><entry>
    <title>About</title>
    <link href="https://blog.tklee.org/en/posts/0748-about-關於本站/"/>
    <updated>2026-08-24T07:48:15Z</updated>
    <id>https://blog.tklee.org/en/posts/0748-about-關於本站/</id>
    <content type="html"><![CDATA[
      <p></p>

<div class="post-teaser-box" data-lang="en">
<p>Welcome to Tsan-Kuang &quot;TK&quot; Lee's blog, <strong>kNOw self</strong>. You can read <a href="https://blog.tklee.org/posts/2026-08-17-2307-kNOw-self-%E8%AD%98%E7%84%A1%E5%90%BE/">my other post about the site's namesake and history</a>. This post is an overview of what this blog is about.</p>
<p>For the tutorial of this site's design, please see <a href="https://blog.tklee.org/posts/2026-08-22-2212-site-help-and-usage-guide-%E7%B6%B2%E7%AB%99%E4%BD%BF%E7%94%A8%E6%8C%87%E5%8D%97/">Help &amp; Usage Guide</a>.<br />
</p></div><p></p>
<h1>Site Content</h1>
<p>This is a repository of my writing since 1990. Each post is tagged according to <a href="https://blog.tklee.org/tags/">its topics</a>.</p>
<p>My main profession is software engineering; my side profession is music composition; I was immersed and trained in literature and linguistics in higher ed; later in my life I have also undergone profound transformation via meditation. Therefore, the topics I share the most are technology, music, cultural observations and commentary, and personal introspection.</p>
<p>In addition, years of writing and sharing have also prompted some <a href="https://blog.tklee.org/tags/meta-blog/">meta-thinking</a> regarding interactions across media, platforms, and communities.</p>
<h1>Keeping Up with the Times</h1>
<p>Early versions of the site reflected the internet ecology of their times, both in content and meta-content: what I wrote, what I wrote about, why I wrote, how I presented it, and how my writing interacted with the world—all of these have changed over time.</p>
<p>For example, in the infant era of the internet, much of what I wrote was to disseminate <a href="https://blog.tklee.org/posts/1990-01-01-2334-complete-list-of-mozart-s-works-%E8%8E%AB%E6%9C%AD%E7%89%B9%E4%BD%9C%E5%93%81%E7%B8%BD%E8%A1%A8/">information that was not easily accessible to the general public</a>. When the internet later became slightly more widespread, full-text search engines were still not yet mature, and online resources had to be manually compiled into <a href="https://en.wikipedia.org/wiki/Yahoo_Directory">directories</a>. During this period, my content also included <a href="https://blog.tklee.org/posts/1998-06-30-1305-%E8%8E%AB%E6%9C%AD%E7%89%B9%E9%80%A3%E7%B5%90%E5%8F%8A%E8%B3%87%E6%BA%90/">curation of useful web links</a>.</p>
<p>While the web technology kept evolving, I also aspired to contribute to user education in those early days. For example , I wrote about <a href="https://blog.tklee.org/posts/1999-10-14-0508-%E7%94%A8%E9%BB%9E%E5%BF%83-%E4%BF%9D%E8%AD%B7%E6%9C%8B%E5%8F%8B%E7%9A%84%E7%B6%B2%E8%B7%AF%E9%9A%B1%E7%A7%81/">blocking spam emails</a>, <a href="https://blog.tklee.org/posts/1999-10-27-0510-%E5%88%A4%E6%96%B7%E6%98%AF%E5%90%A6%E4%B8%ADpretty-park-exe%E7%97%85%E6%AF%90%E5%8F%8A%E8%A7%A3%E6%AF%90%E6%B3%95/">solutions for specific computer viruses</a>, my <a href="https://blog.tklee.org/posts/2003-01-20-0516-qa-%E5%9C%8B%E4%B8%AD%E9%9F%B3%E6%A8%82%E8%80%81%E5%B8%AB%E4%BE%86%E4%BF%A1%E8%A8%8E%E8%AB%96%E5%AD%B8%E7%94%9F%E6%8A%84%E8%A5%B2%E5%95%8F%E9%A1%8C/">discussion on plagiarism issues</a> with teachers, and my <a href="https://blog.tklee.org/posts/2002-11-01-1536-qa-%E8%8E%AB%E6%9C%AD%E7%89%B9%E6%AD%8C%E5%8A%87%E9%AD%94%E7%AC%9B%E7%9A%84%E8%A7%92%E8%89%B2-%E8%81%B2%E6%A8%82%E5%AE%B6%E5%88%86%E9%A1%9E-%E6%AD%8C%E5%8A%87%E5%88%86%E6%AE%B5%E6%B3%95-%E8%8A%B1%E8%85%94-%E5%A4%9C%E5%90%8E%E8%A9%A0%E5%98%86%E8%AA%BF/">exchanges</a> with students.</p>
<p>Once the full-text search engines became mature enough for the public to find raw information, the focus of my sharing shifted to more complex, integrated content that was harder to find online. For instance, <a href="https://blog.tklee.org/posts/2010-11-24-0850-%E6%97%A9%E6%9C%9F%E4%B8%AD%E6%96%87%E7%B7%A8%E7%A2%BC%E8%BD%89%E6%8F%9B/">solutions</a> for early Chinese character <a href="https://en.wikipedia.org/wiki/Mojibake">encoding issues</a>, <a href="https://blog.tklee.org/posts/2008-09-19-2103-modifiers-across-keyboards/">a driver I wrote for multiple keyboards</a>, how to allow two local users to <a href="https://blog.tklee.org/posts/2015-05-07-0243-x11-multi-seat-running-kde-and-gnome/">use the same computer independently and simultaneously</a>, or the sheet music <a href="https://blog.tklee.org/posts/2009-03-31-0425-practical-tips-for-score-part-preparation/">preparation tips</a> for composers.</p>
<p>Today, as AI can directly solve many problems, I am re-contemplating on what, how, and why I share.</p>
<h1>Site Design Philosophy: Preserving History</h1>
<p>These evolutions have been recorded by the site. It witnesses the changes of the world as well as records my personal growth.</p>
<p>This site preserves this history through a <strong>post-centric</strong> design—everything is a post with a date. For example, this &quot;About&quot; page you are reading is a dated post; if I write a new &quot;About&quot; page, the old post will still be preserved, and the menu will simply link to the new one. The &quot;Contact&quot; form is a post, and the &quot;Help &amp; Usage Guide&quot; is also a post. Posts like these are mostly tagged with the <a href="https://blog.tklee.org/tags/meta-blog/">Meta-Blog</a> topic.</p>
<p>Realistically, my preservation efforts can only cover my own site; many external resources referenced in my posts have long been gone. Fortunately, <a href="https://archive.org/">Archive.org</a> has preserved quite a lot (though not all) for nostalgic archaeologists to dig through.</p>
<p>Therefore, <strong>dead links are preserved as they are</strong>.</p>
<p>Older posts have gone through different editors, systems, platforms, and formats, so they inevitably suffer some loss of form; I am less insistent on preserving this aspect. I also try to correct typos in my own posts whenever possible.</p>
<p>However, the immature content and clumsy writing of my early days are preserved as part of my personal history.</p>
<h1>The Completion Project</h1>
<p>Over the past decade or so, social media has become the primary mode of public communication. I moved most of my intellectual exchanges to social media, resulting in a long hiatus on my own blog.</p>
<p>There are also some articles that I never published.</p>
<p>I intend to gradually migrate those, and keep their actual authoring dates. Such newly collected old articles will be marked &quot;✨ Recently Added&quot; and show up in a recent slot in the post list.</p>
<h1>Interaction and Moderation</h1>
<p>Early on, readers wrote directly to me, and I would re-post our exchanges. In later iterations of the site, readers could leave comments directly, which appeared instantly, making interaction much more convenient.</p>
<p>Over the years, however, this convenience was abused -- spams, scams, and malicious links for password theft filled my comment section. One countermeasure is to require readers to log in with real names before commenting, but such friction would discourage some readers from speaking freely.</p>
<p>Therefore, for this new generation of the system, I ask readers to use a contact form to leave comments. I will moderate the messages before making them public.</p>
<p>I hope such necessary evil is only needed in this transitinoal phase of human civilization when we are still learning to respect and care for one another.<br />
</p>

    ]]></content>
  </entry><entry>
    <title>Site Help &amp; Usage Guide</title>
    <link href="https://blog.tklee.org/en/posts/2212-site-help-and-usage-guide-網站使用指南/"/>
    <updated>2026-08-22T22:12:09Z</updated>
    <id>https://blog.tklee.org/en/posts/2212-site-help-and-usage-guide-網站使用指南/</id>
    <content type="html"><![CDATA[
      

<p>Welcome to my blog! Here are some tips for a better reading experience.</p>
<h2>❓ Help Page</h2>
<p>Clicking the question mark icon (?) in the upper-right corner of the page takes you to this help page. You can return to where you were using your browser's Back function.</p>
<h2>🌐 Language Settings</h2>
<p>Because I write in multiple languages, I ended up translating some of my own posts (they are marked &quot;translated by author&quot;), but often times I couldn't afford to do so. Now with modern technology, I don't have to.</p>
<p>Functionally, this blog is fully bilingual (English and Traditional Chinese), and supports real-time translation into many other languages. You can switch languages at any time using <svg xmlns="http://www.w3.org/2005/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" style="display: inline; vertical-align: -3px; margin-right: 2px;"><circle cx="12" cy="12" r="10"></circle><path d="M12 2a15.3 15.3 0 0 1 4 10 15.3 15.3 0 0 1-4 10 15.3 15.3 0 0 1-4-10 15.3 15.3 0 0 1 4-10z"></path><path d="M2 12h20"></path></svg> (the globe icon) at the top of the page. By the way, clicking on that icon will minimize the widget for a cleaner layout.</p>
<p>Your last choice of language will stick until you change again.</p>
<h2>☰ Menu</h2>
<p>The hamburger icon ☰ in the upper-left toggles the site menu, granting access to:</p>
<ul>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><circle cx="11" cy="11" r="8"></circle><line x1="21" y1="21" x2="16.65" y2="16.65"></line></svg> Search (powered by Google)</li>
<li><svg xmlns="http://www.w3.org/2005/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" style="display: inline; vertical-align: -3px; margin-right: 2px;"><circle cx="12" cy="12" r="10"></circle><path d="M12 2a15.3 15.3 0 0 1 4 10 15.3 15.3 0 0 1-4 10 15.3 15.3 0 0 1-4-10 15.3 15.3 0 0 1 4-10z"></path><path d="M2 12h20"></path></svg> Language toggle (see above)</li>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><path d="M14 2H6a2 2 0 0 0-2 2v16a2 2 0 0 0 2 2h12a2 2 0 0 0 2-2V8z"></path><polyline points="14 2 14 8 20 8"></polyline><line x1="16" y1="13" x2="8" y2="13"></line><line x1="16" y1="17" x2="8" y2="17"></line><polyline points="10 9 9 9 8 9"></polyline></svg> Posts (list of posts with summaries, ordered by the time they were added here. Articles written long ago but added recently are marked &quot;✨ Recently Added&quot;)</li>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><circle cx="12" cy="12" r="10"></circle><polyline points="12 6 12 12 16 14"></polyline></svg> Timeline (chronological list of all posts on a timeline, based on their authoring time)</li>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><path d="M20.59 13.41l-7.17 7.17a2 2 0 0 1-2.83 0L2 12V2h10l8.59 8.59a2 2 0 0 1 0 2.82z"></path><line x1="7" y1="7" x2="7.01" y2="7"></line></svg> Tags (find posts by topic)</li>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><circle cx="12" cy="12" r="10"></circle><line x1="12" y1="16" x2="12" y2="12"></line><line x1="12" y1="8" x2="12.01" y2="8"></line></svg> About (background and overview of this generation of the blog)</li>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><path d="M21 15a2 2 0 0 1-2 2H7l-4 4V5a2 2 0 0 1 2-2h14a2 2 0 0 1 2 2z"></path></svg> Contact</li>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><circle cx="12" cy="12" r="10"></circle><path d="M9.09 9a3 3 0 0 1 5.83 1c0 2-3 3-3 3"></path><line x1="12" y1="17" x2="12.01" y2="17"></line></svg> Help (this page)</li>
<li><svg xmlns="http://www.w3.org/2000/svg" width="16" height="16" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="menu-icon"><path d="M4 11a9 9 0 0 1 9 9"></path><path d="M4 4a16 16 0 0 1 16 16"></path><circle cx="5" cy="19" r="1"></circle></svg> Subscribe (RSS feed)</li>
</ul>
<h2>The Unique Design of This Site</h2>
<p>This site is a repository of my writing since 1990. In my post <a href="https://blog.tklee.org/posts/2026-08-24-0748-about-%E9%97%9C%E6%96%BC%E6%9C%AC%E7%AB%99/">about the site</a>, I explain its design philosophy, topics, history, and how outdated contents are handled.</p>


    ]]></content>
  </entry><entry>
    <title>Contact or Comment</title>
    <link href="https://blog.tklee.org/en/posts/1830-contact-comment-聯絡留言/"/>
    <updated>2026-08-22T18:30:00Z</updated>
    <id>https://blog.tklee.org/en/posts/1830-contact-comment-聯絡留言/</id>
    <content type="html"><![CDATA[
      

<p>To respond to one of my posts, please leave your comments there. For generic communication, please use the form below.</p>


    ]]></content>
  </entry><entry>
    <title>kNOw self</title>
    <link href="https://blog.tklee.org/en/posts/2307-kNOw-self-識無吾/"/>
    <updated>2026-08-17T23:07:00Z</updated>
    <id>https://blog.tklee.org/en/posts/2307-kNOw-self-識無吾/</id>
    <content type="html"><![CDATA[
      

<p>Whether you have followed my writing since 1990 or are just finding it now, welcome to my blog <em>kNOw self</em>.</p>
<p>The name of this incarnation carries a threefold meaning that has anchored my life's journey. First, it is a nod to the ancient Greek philosophy to &quot;<a href="https://en.wikipedia.org/wiki/Know_thyself">know thyself</a>.&quot; Second, it is an invitation for friends to know me better, so that we might build meaningful connections. Finally, it reflects the Buddhist concept of &quot;no self&quot; (<em><a href="https://en.wikipedia.org/wiki/Anatt%C4%81">anattā</a></em>). This is a reminder to embrace the fluid nature of the self, reduce my own ego, and share my journey with interconnected beings everywhere.</p>
<p>The website's favicon represents the silent consonant 'k' in the site name (<strong>kNOw self</strong>). Furthermore, the 'k' can be broken down into the letter 'I' on the left and two reflecting beams on the right, together symbolizing self-reflection in silence.</p>
<p>In the past decade, my blogging became sporadic. The current of social media had pulled me away from a dedicated, self-managed space. The convenience of exchanging fleeting thoughts in crowded digital squares replaced my long-form reflection. But I now realize the necessity of returning to a self-owned platform. I want a quiet place for me (and readers) to focus on the signal rather than the noise.</p>
<p>This return is accompanied by a technical reset. I have been publishing my writing since 1990, witnessing the entire evolution of our communication mediums. My words have lived through traditional paper media, email lists, newsletters, and BBS, before I rode the wave of Web 1.0, carving out corners in academic mainframe user spaces, <a href="https://en.wikipedia.org/wiki/GeoCities">GeoCities</a>, and pure hand-written HTML on self-hosted servers, eventually syndicating through <a href="https://en.wikipedia.org/wiki/RSS">RSS</a>. Then came Web 2.0—the era of interactive CMS platforms and countless blogging engines—before social media swallowed the web whole.</p>
<p>After wandering through all those generations of platforms, I have finally settled on something refreshingly foundational: static files, powered by a custom rendering engine to cleanly separate the content from the presentation. It is a return to the roots of the web, but with the <a href="https://blog.tklee.org/tags/meta-blog/">experience</a> of decades of software engineering and sharing my writing behind it.</p>
<p>The medium isn't the only thing that has evolved; the landscape of global communication has completely transformed. When I began, navigating the web as a multi-lingual writer was a very different experience. Today, <a href="https://en.wikipedia.org/wiki/Unicode">Unicode</a> is standard, the internet has elevated global English literacy, and machine translation is instantly accessible. My own primary audience has shifted from Chinese to English readers over the years. And my own English proficiency and cultural awareness have also grown a bit.</p>
<p>At the same time, the fundamental utility of writing on the internet has shifted. We now have smart search and AI-driven solutions that can synthesize answers in seconds. Readers' attention spans have shortened drastically, and multimedia is rapidly replacing text consumption. Why write long-form text on a personal site when AI can answer questions and video dominates our screens?</p>
<p>Because the purpose of this space isn't merely to distribute data; it is to share my deeply personal journey of self-observation, self-correction, and ultimately, self-liberation.</p>
<p>This blog is a living record of my growth across all facets of my life: as a software professional and a technology tinkerer; as a composer and a music enthusiast; as a multi-lingual writer navigating nuance; and, most importantly, as a layperson who simply wishes to be a better being tomorrow than I am today.</p>
<p><em>kNOw self</em> is a return to that core pursuit. I am glad to be back on track, and I am glad you are here with me.</p>


    ]]></content>
  </entry><entry>
    <title>Copyright and Licensing</title>
    <link href="https://blog.tklee.org/en/posts/2234-copyright-and-licensing/"/>
    <updated>Sun Aug 16 2026 22:34:00 GMT+0000 (Coordinated Universal Time)</updated>
    <id>https://blog.tklee.org/en/posts/2234-copyright-and-licensing/</id>
    <content type="html"><![CDATA[
      <p></p>

<p>This policy applies to all past, present, and future content published on this blog.</p>
<h2>General Content (Text, Images, and Video)</h2>
<p>This policy serves as the default license for any content on this blog where a specific copyright or license is not explicitly stated.</p>
<p>Unless otherwise noted, the text content, general photography and images, and original video content hosted on this blog are licensed under the <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0)</a>. You are free to share, copy, distribute, transmit, and adapt this content, provided you attribute the work to Tsan-Kuang &quot;TK&quot; Lee, do not use it for commercial purposes, and distribute your contributions under the same license.</p>
<p><strong>Third-Party Hosting:</strong> Any videos or media embedded from external sites (such as YouTube) are not covered by this default license and are instead governed by their own specific terms and licenses on those platforms.</p>
<p><strong>Exception:</strong> Any content that contains, depicts, or encodes my musical works (e.g. plain-text music notation like ABCjs, screenshots of sheet music, video performances of my music, etc.) is <strong>exempt</strong> from this license. These materials are treated as the musical works themselves and all rights are reserved.</p>
<h2>Code</h2>
<p>Unless otherwise mentioned in their posts or respective source code repositories, all code snippets shared on this blog are released under the <a href="https://opensource.org/licenses/MIT">MIT License</a>. You are free to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the code, provided that the original copyright notice and permission notice are included in all copies or substantial portions of the software. The code is provided &quot;AS IS&quot;, without warranty of any kind.</p>
<h2>Music, Notation, and Recording</h2>
<p>Please note that the Creative Commons license <strong>does not apply</strong> to any musical works, compositions, sheet music, musical notation, or recordings hosted on this site.</p>
<p>All musical works by me remain the exclusive property of Tsan-Kuang &quot;TK&quot; Lee, and <strong>all rights are reserved</strong>. Unless you obtain explicit <a href="https://blog.tklee.org/posts/2026-08-22-1830-contact-comment-%E8%81%AF%E7%B5%A1%E7%95%99%E8%A8%80/">authorization</a>, you may not reproduce, distribute, or perform these works.<br />
</p>

    ]]></content>
  </entry><entry>
    <title>Enemy</title>
    <link href="https://blog.tklee.org/en/posts/0716-敵人-enemy/"/>
    <updated>2023-10-11T07:16:00.003Z</updated>
    <id>https://blog.tklee.org/en/posts/0716-敵人-enemy/</id>
    <content type="html"><![CDATA[
      <p></p>

<p>A friend lost one of her family members to the recent war bombing. Others had to leave that disabled member behind.</p>
<p>Taiwan experienced a devastating earthquake in 1999, during which a lot of people also had to leave their loved ones behind in their narrow escape. On top of the loss, they also have to carry the survivor guilt.</p>
<p>Facing natural disasters is part of life and we do not have other choices but to face them, but wars happen because somebody prioritizes their interest over other people's lives. The pain brought by the war is therefore on another level.</p>
<p>Many people's birthday wishes include world peace. Who are we hoping to grant these wishes? I am unaware of any deity who specializes in granting birthday wishes.</p>
<p>Perhaps birthday wishes should not be wishes asked of a genie, but rather, resolutions, like the New Year ones, which are more like goals that we work hard to achieve.</p>
<p>In hoping for the world peace, perhaps we should start from improving our own personal relationships, by at least reducing our hostility toward others, and ourselves.</p>
<p>Today I make a wish to honor that lost life by starting a new journey to reduce my negativity.<br />
</p>

    ]]></content>
  </entry><entry>
    <title>Anything Goes</title>
    <link href="https://blog.tklee.org/en/posts/2031-anything-goes/"/>
    <updated>2020-10-11T20:31:56Z</updated>
    <id>https://blog.tklee.org/en/posts/2031-anything-goes/</id>
    <content type="html"><![CDATA[
      

<div class="post-teaser-box" data-lang="en">
<p>Someone shared a black-and-white stones puzzle for elementary students on Facebook, inviting everyone to solve it. There was a lively discussion among my friends. I spent some time thinking about it and came to an interesting conclusion.<br />
</p></div><p></p>
<slide-deck mode="slide">
<slide-page>
<h1>The Black and White Stones Puzzle</h1>
<p>The puzzle consists of sixteen small squares, each with a different patterns of black and white stones. The question asks which of the four options on the right is the answer for the last square.<br />
<img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle.png" alt="" /></p>
<p>If you're interested, don't read ahead just yet—try to solve it yourself first.<br />
</p></slide-page><br />
<slide-page><p></p>
<p>Before solving this puzzle, let's look at another one.</p>
<h1>The Kids and Apples Distribution Puzzle</h1>
<p>This is a puzzle that has been circulating online recently: &quot;There are seven children and three apples. Please make four cuts so that the children won't fight over the apples.&quot;</p>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/7kids-3apples.jpg" alt="" /></p>
<p>The standard answer is on the next page. If you're interested, you can try solving it first.</p>
</slide-page>
<slide-page>
<h1>The &quot;Standard Answer&quot; to the Apples Puzzle</h1>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/7kids-3apples-standard-solution.png" alt="" /></p>
<p>Three apples divided among seven children means each child should get <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{3}{7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> of an apple.</p>
<p>The first cut slices each apple in half horizontally, and then each half is cut at an angle of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mn>7</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p>
<p>The <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>3</mn><mn>7</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{3}{7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> portions are given to the first six children, and the remaining six small pieces add up to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>3</mn><mn>7</mn></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>14</mn></mfrac><mo>∗</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">\frac{3}{7} = \frac{1}{14} * 6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">6</span></span></span></span>.<br />
</p></slide-page><br />
<slide-page><p></p>
<h1>Katana (The Samurai Sword)</h1>
<p>Wait a minute, the puzzle didn't say we could slice multiple apples at once, nor were we told that we have access to high-precision tools. Most of us don't have that kind of cutting skills, like a Katana expert.<br />
<img src="https://upload.wikimedia.org/wikipedia/commons/f/f5/Sensei_iaido-rework.jpg" alt="Katana" /><br />
Well, with access to an Katana expert, we can have much a simpler solution.<br />
</p></slide-page><p></p>
<slide-page>
<h1>The &quot;Katana Solution&quot; to the Apples Puzzle</h1>
<p>Now that we have a Katana, we can cut beyond just apples.<br />
<img src="https://blog.tklee.org/assets/2020-10-11-2031/7kids-3apples-alt-solution.jpg" alt="" /><br />
Actually, if you &quot;cut&quot; the first child, I believe the remaining children won't dare to complain regardless of how you distribute the apples. Goal achieved, puzzle solved. :)</p>
<p>These types of &quot;math&quot; puzzles often suffer from loosely defined rules and conditions.<br />
</p></slide-page><br />
<slide-page><p></p>
<h1>Back to the Stones Puzzle</h1>
<p>For the sake of discussion, I've added red annotations.<br />
<img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle-marked.png" alt="" /></p>
</slide-page>
<slide-page>
<h1>Stones Puzzle Solution 1: The &quot;Standard&quot; Solution (Answer is x)</h1>
<p>The standard answer is x. The logic is: after adding x, if we swap the colors of the black and white stones in each square, we can find a matching square elsewhere. For example, reading <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">b2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mord">2</span></span></span></span> clockwise from the top-left gives &quot;empty, empty, white, black&quot;. If we swap the colors, it becomes &quot;empty, empty, black, white&quot;, which matches <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">d1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span><span class="mord">1</span></span></span></span>.</p>
<p>Pairing them up:</p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mn>1</mn><mo separator="true">,</mo><mi>a</mi><mn>3</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>a</mi><mn>2</mn><mo separator="true">,</mo><mi>d</mi><mn>3</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>a</mi><mn>4</mn><mo separator="true">,</mo><mi>c</mi><mn>4</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>b</mi><mn>2</mn><mo separator="true">,</mo><mi>d</mi><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>b</mi><mn>3</mn><mo separator="true">,</mo><mi>c</mi><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>b</mi><mn>4</mn><mo separator="true">,</mo><mi>c</mi><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>c</mi><mn>3</mn><mo separator="true">,</mo><mi>d</mi><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a1, a3), (a2, d3), (a4, c4), (b2, d1), (b3, c2), (b4, c1), (c3, d2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">a</span><span class="mord">3</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mord">3</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mord">4</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathnormal">c</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span><span class="mord">2</span><span class="mclose">)</span></span></span></span></p>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle-standard-solution.png" alt="" /></p>
<p>The only one left is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi><mn>1</mn></mrow><annotation encoding="application/x-tex">b1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mord">1</span></span></span></span>, which is &quot;white, empty, white, black&quot;. After swapping, it becomes &quot;black, empty, black, white&quot;, which is exactly x.</p>
<p>But this is just one possible logic.<br />
</p></slide-page><br />
<slide-page><p></p>
<h1>Solution 2: Equal Number of Go Stones (Answer can be x or y)</h1>
<p>Let's change the logic: make the total number of black and white stones on the board equal. Currently, there are 23 black stones and 24 white stones on the board.</p>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle-marked.png" alt="" /></p>
<p>So we can add y, bringing both black and white stones to 24 each.</p>
<p>Or we can add x, bringing both black and white stones to 25 each.</p>
</slide-page>
<slide-page>
<h1>Solution 3: Equal Weight (Answers w, x, y, z are all valid)</h1>
<p>We can change the logic again to make any answer correct.</p>
<p>For example, let's set the rule: draw a diagonal line from top-right to bottom-left, such that the total weight of the black and white stones on both sides is equal.</p>
<p>The stones directly on the line are divided equally between both sides, so they can be ignored.</p>
<p>For the rest, the top-left triangle has 9 black stones and 12 white stones; the bottom-right triangle has 11 black stones and 8 white stones.</p>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle-diagonal.png" alt="" /></p>
<h2>Answer is w</h2>
<p>If we want the answer to be w, placing w makes the top-left triangle have 9 black stones and 12 white stones; the bottom-right triangle will have 13 black stones and 8 white stones.</p>
<p>Let each black stone weigh 4 grams and each white stone weigh 5 grams, and it balances: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><mo>∗</mo><mn>4</mn><mo>+</mo><mn>12</mn><mo>∗</mo><mn>5</mn><mo>=</mo><mn>13</mn><mo>∗</mo><mn>4</mn><mo>+</mo><mn>8</mn><mo>∗</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">9*4+12*5 = 13*4+8*5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span></span></span></span></p>
<p>So we can state: when every black stone on this board weighs 4 grams and every white stone weighs 5 grams, the last square must be w for the total weight of the stones on the top-left and bottom-right of the blue line to balance.</p>
<h2>Answer is x</h2>
<p>Similarly, if we want x, the top-left triangle has 9 black stones and 12 white stones; the bottom-right triangle will have 13 black stones and 9 white stones. If black stones weigh 3 grams and white stones weigh 4 grams, it balances: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><mo>∗</mo><mn>3</mn><mo>+</mo><mn>12</mn><mo>∗</mo><mn>4</mn><mo>=</mo><mn>13</mn><mo>∗</mo><mn>3</mn><mo>+</mo><mn>9</mn><mo>∗</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">9*3+12*4 = 13*3+9*4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span></span></span></span></p>
<h2>Answer is y</h2>
<p>For y, the top-left triangle has 9 black stones and 12 white stones; the bottom-right triangle will have 12 black stones and 8 white stones. If black stones weigh 4 grams and white stones weigh 3 grams, it balances: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><mo>∗</mo><mn>4</mn><mo>+</mo><mn>12</mn><mo>∗</mo><mn>3</mn><mo>=</mo><mn>12</mn><mo>∗</mo><mn>4</mn><mo>+</mo><mn>8</mn><mo>∗</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">9*4+12*3 = 12*4+8*3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span></p>
<h2>Answer is z</h2>
<p>For z, the top-left triangle has 9 black stones and 12 white stones; the bottom-right triangle will have 12 black stones and 10 white stones. If black stones weigh 2 grams and white stones weigh 3 grams, it balances: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn><mo>∗</mo><mn>2</mn><mo>+</mo><mn>12</mn><mo>∗</mo><mn>3</mn><mo>=</mo><mn>12</mn><mo>∗</mo><mn>2</mn><mo>+</mo><mn>10</mn><mo>∗</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">9*2+12*3 = 12*2+10*3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">3</span></span></span></span></p>
<hr />
<p>Therefore, whatever answer you want, you just come up with a different weight ratio.<br />
</p></slide-page><br />
<slide-page><p></p>
<h1>Solution 4: Leverage</h1>
<p>If we want to factor in the leverage length based on the position of each stone, we can even balance it around the center point.</p>
<p>This solution is similar to the previous one, except each stone has a weighting coefficient (its perpendicular distance from the center line; the original squares can be subdivided into a 3x3 grid, calculate the center coordinates using trigonometry, and then use the Pythagorean theorem for the distance). In short, you just solve a two-variable equation to find the required weight ratio for the black and white stones. I will skip the details here.</p>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle-perpendicular-lines.png" alt="" /><br />
</p></slide-page><br />
<slide-page><p></p>
<h1>Solution 5: Polynomial Solution (I must have had too much time on my hand)</h1>
<p>This method uses a high-degree polynomial. Again, any answer can be achieved. However, for the brevity, we won't consider y, which is the only option with a dot in the center (if you wanted to consider it, you could treat the exact center of all other squares as empty, similar to the pip arrangement on a standard six-sided die).</p>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle-marked.png" alt="" /></p>
<p>Let's just pick z as the answer and see how the rules change.</p>
<h2>Encoding</h2>
<p>To use polynomials, we first need to convert each square into a numerical value. To ensure uniqueness, we encode all possibilities. Take b1 as an example; starting from the top left and reading clockwise, this square is &quot;white, empty, white, black&quot;. We replace these with numbers: empty=0, white=1, black=2, giving us 1012. This is a base-3 number, which converts to 32 in decimal.</p>
<table>
<thead>
<tr>
<th>Index</th>
<th>Square</th>
<th>Content</th>
<th>Base-3</th>
<th>Decimal</th>
</tr>
</thead>
<tbody>
<tr>
<td>1</td>
<td>a1</td>
<td>black, white, black, black</td>
<td>2122</td>
<td>71</td>
</tr>
<tr>
<td>2</td>
<td>a2</td>
<td>empty, empty, white, white</td>
<td>0011</td>
<td>4</td>
</tr>
<tr>
<td>3</td>
<td>a3</td>
<td>white, black, white, white</td>
<td>1211</td>
<td>49</td>
</tr>
<tr>
<td>4</td>
<td>a4</td>
<td>white, black, empty, empty</td>
<td>1200</td>
<td>45</td>
</tr>
<tr>
<td>5</td>
<td>b1</td>
<td>white, empty, white, black</td>
<td>1012</td>
<td>32</td>
</tr>
<tr>
<td>6</td>
<td>b2</td>
<td>empty, empty, white, black</td>
<td>0012</td>
<td>5</td>
</tr>
<tr>
<td>7</td>
<td>b3</td>
<td>black, white, black, white</td>
<td>2121</td>
<td>70</td>
</tr>
<tr>
<td>8</td>
<td>b4</td>
<td>white, white, black, black</td>
<td>1122</td>
<td>44</td>
</tr>
<tr>
<td>9</td>
<td>c1</td>
<td>black, black, white, white</td>
<td>2211</td>
<td>76</td>
</tr>
<tr>
<td>10</td>
<td>c2</td>
<td>white, black, white, black</td>
<td>1212</td>
<td>50</td>
</tr>
<tr>
<td>11</td>
<td>c3</td>
<td>white, black, black, white</td>
<td>1221</td>
<td>52</td>
</tr>
<tr>
<td>12</td>
<td>c4</td>
<td>black, white, empty, empty</td>
<td>2100</td>
<td>63</td>
</tr>
<tr>
<td>13</td>
<td>d1</td>
<td>empty, empty, black, white</td>
<td>0021</td>
<td>7</td>
</tr>
<tr>
<td>14</td>
<td>d2</td>
<td>black, white, white, black</td>
<td>2112</td>
<td>68</td>
</tr>
<tr>
<td>15</td>
<td>d3</td>
<td>empty, empty, black, black</td>
<td>0022</td>
<td>8</td>
</tr>
<tr>
<td>16</td>
<td>d4</td>
<td>????</td>
<td>????</td>
<td>?</td>
</tr>
<tr>
<td></td>
<td>w</td>
<td>black, black, empty, empty</td>
<td>2200</td>
<td>72</td>
</tr>
<tr>
<td></td>
<td>x</td>
<td>black, empty, black, white</td>
<td>2021</td>
<td>61</td>
</tr>
<tr>
<td></td>
<td>z</td>
<td>empty, white, black, white</td>
<td>0121</td>
<td>16</td>
</tr>
</tbody>
</table>
<p>Actually, any encoding method is fine, as long as there is a one-to-one mapping between the number and the square. You could list all possibilities and arbitrarily assign unique numbers.</p>
<p>Here, we're just taking advantage of the convenience of the <a href="https://en.wikipedia.org/wiki/Ternary_numeral_system">Ternary numeral system</a>.</p>
<p>Additionally, once encoded, you could actually use any machine learning (ML) method to solve it. However, while many ML methods can find a solution, the resulting models are not intuitive and their underlying principles are hard to explain. Therefore, we avoid probability-based ML models here (like neural networks).</p>
<h2>The Crazy Polynomial</h2>
<p>Okay, the solution is simple: just plug it into this polynomial:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>96419</mn><msup><mi>n</mi><mn>15</mn></msup></mrow><mn>326918592000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>367963</mn><msup><mi>n</mi><mn>14</mn></msup></mrow><mn>9686476800</mn></mfrac><mo>−</mo><mfrac><mrow><mn>83433193</mn><msup><mi>n</mi><mn>13</mn></msup></mrow><mn>37362124800</mn></mfrac><mo>+</mo><mfrac><mrow><mn>76018199</mn><msup><mi>n</mi><mn>12</mn></msup></mrow><mn>958003200</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>3905893151</mn><msup><mi>n</mi><mn>11</mn></msup></mrow><mn>2052864000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2834174039</mn><msup><mi>n</mi><mn>10</mn></msup></mrow><mn>87091200</mn></mfrac><mo>−</mo><mfrac><mrow><mn>748402863991</mn><msup><mi>n</mi><mn>9</mn></msup></mrow><mn>1828915200</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mfrac><mrow><mn>780710080883</mn><msup><mi>n</mi><mn>8</mn></msup></mrow><mn>203212800</mn></mfrac><mo>−</mo><mfrac><mrow><mn>35360802693773</mn><msup><mi>n</mi><mn>7</mn></msup></mrow><mn>1306368000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>6209534662753</mn><msup><mi>n</mi><mn>6</mn></msup></mrow><mn>43545600</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>56937672552137</mn><msup><mi>n</mi><mn>5</mn></msup></mrow><mn>102643200</mn></mfrac><mo>+</mo><mfrac><mrow><mn>186397079118703</mn><msup><mi>n</mi><mn>4</mn></msup></mrow><mn>119750400</mn></mfrac><mo>−</mo><mfrac><mrow><mn>4585643519077501</mn><msup><mi>n</mi><mn>3</mn></msup></mrow><mn>1513512000</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mfrac><mrow><mn>289798498820219</mn><msup><mi>n</mi><mn>2</mn></msup></mrow><mn>75675600</mn></mfrac><mo>−</mo><mfrac><mrow><mn>125147426197</mn><msup><mi>n</mi><mn>1</mn></msup></mrow><mn>45045</mn></mfrac><mo>+</mo><mn>857840</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
&amp; - \frac{96419 n^{15}}{326918592000} + \frac{367963 n^{14}}{9686476800} - \frac{83433193 n^{13}}{37362124800} + \frac{76018199 n^{12}}{958003200} \\
&amp; - \frac{3905893151 n^{11}}{2052864000} + \frac{2834174039 n^{10}}{87091200} - \frac{748402863991 n^{9}}{1828915200} \\
&amp; + \frac{780710080883 n^{8}}{203212800} - \frac{35360802693773 n^{7}}{1306368000} + \frac{6209534662753 n^{6}}{43545600} \\
&amp; - \frac{56937672552137 n^{5}}{102643200} + \frac{186397079118703 n^{4}}{119750400} - \frac{4585643519077501 n^{3}}{1513512000} \\
&amp; + \frac{289798498820219 n^{2}}{75675600} - \frac{125147426197 n^{1}}{45045} + 857840
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.385540000000002em;vertical-align:-5.94277em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.442770000000001em;"><span style="top:-8.442770000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-5.965662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-3.4885540000000006em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-1.0114460000000012em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:1.465662em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.94277em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.442770000000001em;"><span style="top:-8.442770000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">2</span><span class="mord">6</span><span class="mord">9</span><span class="mord">1</span><span class="mord">8</span><span class="mord">5</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">6</span><span class="mord">4</span><span class="mord">1</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">6</span><span class="mord">8</span><span class="mord">6</span><span class="mord">4</span><span class="mord">7</span><span class="mord">6</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">6</span><span class="mord">7</span><span class="mord">9</span><span class="mord">6</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">7</span><span class="mord">3</span><span class="mord">6</span><span class="mord">2</span><span class="mord">1</span><span class="mord">2</span><span class="mord">4</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span><span class="mord">3</span><span class="mord">4</span><span class="mord">3</span><span class="mord">3</span><span class="mord">1</span><span class="mord">9</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">5</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span><span class="mord">3</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">6</span><span class="mord">0</span><span class="mord">1</span><span class="mord">8</span><span class="mord">1</span><span class="mord">9</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-5.965662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">5</span><span class="mord">2</span><span class="mord">8</span><span class="mord">6</span><span class="mord">4</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">9</span><span class="mord">0</span><span class="mord">5</span><span class="mord">8</span><span class="mord">9</span><span class="mord">3</span><span class="mord">1</span><span class="mord">5</span><span class="mord">1</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span><span class="mord">7</span><span class="mord">0</span><span class="mord">9</span><span class="mord">1</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">8</span><span class="mord">3</span><span class="mord">4</span><span class="mord">1</span><span class="mord">7</span><span class="mord">4</span><span class="mord">0</span><span class="mord">3</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">8</span><span class="mord">2</span><span class="mord">8</span><span class="mord">9</span><span class="mord">1</span><span class="mord">5</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">4</span><span class="mord">8</span><span class="mord">4</span><span class="mord">0</span><span class="mord">2</span><span class="mord">8</span><span class="mord">6</span><span class="mord">3</span><span class="mord">9</span><span class="mord">9</span><span class="mord">1</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.4885540000000006em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">3</span><span class="mord">2</span><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">8</span><span class="mord">0</span><span class="mord">7</span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mord">8</span><span class="mord">8</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">3</span><span class="mord">0</span><span class="mord">6</span><span class="mord">3</span><span class="mord">6</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">5</span><span class="mord">3</span><span class="mord">6</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mord">2</span><span class="mord">6</span><span class="mord">9</span><span class="mord">3</span><span class="mord">7</span><span class="mord">7</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">3</span><span class="mord">5</span><span class="mord">4</span><span class="mord">5</span><span class="mord">6</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord">2</span><span class="mord">0</span><span class="mord">9</span><span class="mord">5</span><span class="mord">3</span><span class="mord">4</span><span class="mord">6</span><span class="mord">6</span><span class="mord">2</span><span class="mord">7</span><span class="mord">5</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.0114460000000012em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">6</span><span class="mord">4</span><span class="mord">3</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span><span class="mord">6</span><span class="mord">9</span><span class="mord">3</span><span class="mord">7</span><span class="mord">6</span><span class="mord">7</span><span class="mord">2</span><span class="mord">5</span><span class="mord">5</span><span class="mord">2</span><span class="mord">1</span><span class="mord">3</span><span class="mord">7</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">1</span><span class="mord">9</span><span class="mord">7</span><span class="mord">5</span><span class="mord">0</span><span class="mord">4</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">8</span><span class="mord">6</span><span class="mord">3</span><span class="mord">9</span><span class="mord">7</span><span class="mord">0</span><span class="mord">7</span><span class="mord">9</span><span class="mord">1</span><span class="mord">1</span><span class="mord">8</span><span class="mord">7</span><span class="mord">0</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">5</span><span class="mord">1</span><span class="mord">3</span><span class="mord">5</span><span class="mord">1</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">5</span><span class="mord">8</span><span class="mord">5</span><span class="mord">6</span><span class="mord">4</span><span class="mord">3</span><span class="mord">5</span><span class="mord">1</span><span class="mord">9</span><span class="mord">0</span><span class="mord">7</span><span class="mord">7</span><span class="mord">5</span><span class="mord">0</span><span class="mord">1</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:1.465662em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">5</span><span class="mord">6</span><span class="mord">7</span><span class="mord">5</span><span class="mord">6</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">8</span><span class="mord">9</span><span class="mord">7</span><span class="mord">9</span><span class="mord">8</span><span class="mord">4</span><span class="mord">9</span><span class="mord">8</span><span class="mord">8</span><span class="mord">2</span><span class="mord">0</span><span class="mord">2</span><span class="mord">1</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">5</span><span class="mord">0</span><span class="mord">4</span><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">5</span><span class="mord">1</span><span class="mord">4</span><span class="mord">7</span><span class="mord">4</span><span class="mord">2</span><span class="mord">6</span><span class="mord">1</span><span class="mord">9</span><span class="mord">7</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">8</span><span class="mord">5</span><span class="mord">7</span><span class="mord">8</span><span class="mord">4</span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.94277em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Here, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span></span></span></span> represents the position index, as shown in the image:</p>
<p><img src="https://blog.tklee.org/assets/2020-10-11-2031/stone-puzzle-numbered.png" alt="" /></p>
<h2>&quot;Are You Kidding Me?&quot;</h2>
<p>Seriously, the polynomial above can accurately calculate the content of every square.</p>
<p>For example, take the 10th square, which is the original c2 (white, black, white, black, which is 1212 in base-3), and the resulting decimal value should be 50 (see the encoding table). Substitute <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">n=10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">0</span></span></span></span> into the polynomial above, and the calculated result is exactly 50.</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>96419</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>15</mn></msup></mrow><mn>326918592000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>367963</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>14</mn></msup></mrow><mn>9686476800</mn></mfrac><mo>−</mo><mfrac><mrow><mn>83433193</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>13</mn></msup></mrow><mn>37362124800</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mfrac><mrow><mn>76018199</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>12</mn></msup></mrow><mn>958003200</mn></mfrac><mo>−</mo><mfrac><mrow><mn>3905893151</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>11</mn></msup></mrow><mn>2052864000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2834174039</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>10</mn></msup></mrow><mn>87091200</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>748402863991</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup></mrow><mn>1828915200</mn></mfrac><mo>+</mo><mfrac><mrow><mn>780710080883</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>8</mn></msup></mrow><mn>203212800</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>35360802693773</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>7</mn></msup></mrow><mn>1306368000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>6209534662753</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup></mrow><mn>43545600</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>56937672552137</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><mn>102643200</mn></mfrac><mo>+</mo><mfrac><mrow><mn>186397079118703</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><mn>119750400</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>4585643519077501</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>3</mn></msup></mrow><mn>1513512000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>289798498820219</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>2</mn></msup></mrow><mn>75675600</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>125147426197</mn><mo>∗</mo><mn>1</mn><msup><mn>0</mn><mn>1</mn></msup></mrow><mn>45045</mn></mfrac><mo>+</mo><mn>857840</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
&amp; - \frac{96419 * 10^{15}}{326918592000} + \frac{367963 * 10^{14}}{9686476800} - \frac{83433193 * 10^{13}}{37362124800} \\
&amp; + \frac{76018199 * 10^{12}}{958003200} - \frac{3905893151 * 10^{11}}{2052864000} + \frac{2834174039 * 10^{10}}{87091200} \\
&amp; - \frac{748402863991 * 10^{9}}{1828915200} + \frac{780710080883 * 10^{8}}{203212800} \\
&amp; - \frac{35360802693773 * 10^{7}}{1306368000} + \frac{6209534662753 * 10^{6}}{43545600} \\
&amp; - \frac{56937672552137 * 10^{5}}{102643200} + \frac{186397079118703 * 10^{4}}{119750400} \\
&amp; - \frac{4585643519077501 * 10^{3}}{1513512000} + \frac{289798498820219 * 10^{2}}{75675600} \\
&amp; - \frac{125147426197 * 10^{1}}{45045} + 857840
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:17.339756em;vertical-align:-8.419878em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:8.919878000000002em;"><span style="top:-10.919878000000002em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-8.442770000000003em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-5.965662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-3.4885540000000015em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-1.0114460000000003em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:1.465662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:3.94277em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:8.419878em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:8.919878000000002em;"><span style="top:-10.919878000000002em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">2</span><span class="mord">6</span><span class="mord">9</span><span class="mord">1</span><span class="mord">8</span><span class="mord">5</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">6</span><span class="mord">4</span><span class="mord">1</span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">6</span><span class="mord">8</span><span class="mord">6</span><span class="mord">4</span><span class="mord">7</span><span class="mord">6</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">6</span><span class="mord">7</span><span class="mord">9</span><span class="mord">6</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">7</span><span class="mord">3</span><span class="mord">6</span><span class="mord">2</span><span class="mord">1</span><span class="mord">2</span><span class="mord">4</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span><span class="mord">3</span><span class="mord">4</span><span class="mord">3</span><span class="mord">3</span><span class="mord">1</span><span class="mord">9</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-8.442770000000003em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">5</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span><span class="mord">3</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">6</span><span class="mord">0</span><span class="mord">1</span><span class="mord">8</span><span class="mord">1</span><span class="mord">9</span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">5</span><span class="mord">2</span><span class="mord">8</span><span class="mord">6</span><span class="mord">4</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">9</span><span class="mord">0</span><span class="mord">5</span><span class="mord">8</span><span class="mord">9</span><span class="mord">3</span><span class="mord">1</span><span class="mord">5</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span><span class="mord">7</span><span class="mord">0</span><span class="mord">9</span><span class="mord">1</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">8</span><span class="mord">3</span><span class="mord">4</span><span class="mord">1</span><span class="mord">7</span><span class="mord">4</span><span class="mord">0</span><span class="mord">3</span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-5.965662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">8</span><span class="mord">2</span><span class="mord">8</span><span class="mord">9</span><span class="mord">1</span><span class="mord">5</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">4</span><span class="mord">8</span><span class="mord">4</span><span class="mord">0</span><span class="mord">2</span><span class="mord">8</span><span class="mord">6</span><span class="mord">3</span><span class="mord">9</span><span class="mord">9</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">3</span><span class="mord">2</span><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">8</span><span class="mord">0</span><span class="mord">7</span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mord">8</span><span class="mord">8</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.4885540000000015em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">3</span><span class="mord">0</span><span class="mord">6</span><span class="mord">3</span><span class="mord">6</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">5</span><span class="mord">3</span><span class="mord">6</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mord">2</span><span class="mord">6</span><span class="mord">9</span><span class="mord">3</span><span class="mord">7</span><span class="mord">7</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">3</span><span class="mord">5</span><span class="mord">4</span><span class="mord">5</span><span class="mord">6</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord">2</span><span class="mord">0</span><span class="mord">9</span><span class="mord">5</span><span class="mord">3</span><span class="mord">4</span><span class="mord">6</span><span class="mord">6</span><span class="mord">2</span><span class="mord">7</span><span class="mord">5</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.0114460000000003em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">6</span><span class="mord">4</span><span class="mord">3</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span><span class="mord">6</span><span class="mord">9</span><span class="mord">3</span><span class="mord">7</span><span class="mord">6</span><span class="mord">7</span><span class="mord">2</span><span class="mord">5</span><span class="mord">5</span><span class="mord">2</span><span class="mord">1</span><span class="mord">3</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">1</span><span class="mord">9</span><span class="mord">7</span><span class="mord">5</span><span class="mord">0</span><span class="mord">4</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">8</span><span class="mord">6</span><span class="mord">3</span><span class="mord">9</span><span class="mord">7</span><span class="mord">0</span><span class="mord">7</span><span class="mord">9</span><span class="mord">1</span><span class="mord">1</span><span class="mord">8</span><span class="mord">7</span><span class="mord">0</span><span class="mord">3</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:1.465662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">5</span><span class="mord">1</span><span class="mord">3</span><span class="mord">5</span><span class="mord">1</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">5</span><span class="mord">8</span><span class="mord">5</span><span class="mord">6</span><span class="mord">4</span><span class="mord">3</span><span class="mord">5</span><span class="mord">1</span><span class="mord">9</span><span class="mord">0</span><span class="mord">7</span><span class="mord">7</span><span class="mord">5</span><span class="mord">0</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">5</span><span class="mord">6</span><span class="mord">7</span><span class="mord">5</span><span class="mord">6</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">8</span><span class="mord">9</span><span class="mord">7</span><span class="mord">9</span><span class="mord">8</span><span class="mord">4</span><span class="mord">9</span><span class="mord">8</span><span class="mord">8</span><span class="mord">2</span><span class="mord">0</span><span class="mord">2</span><span class="mord">1</span><span class="mord">9</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:3.94277em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">5</span><span class="mord">0</span><span class="mord">4</span><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">5</span><span class="mord">1</span><span class="mord">4</span><span class="mord">7</span><span class="mord">4</span><span class="mord">2</span><span class="mord">6</span><span class="mord">1</span><span class="mord">9</span><span class="mord">7</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">8</span><span class="mord">5</span><span class="mord">7</span><span class="mord">8</span><span class="mord">4</span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:8.419878em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Note: If you don't want to verify it by hand, you can paste the expression below into an online <a href="https://www.mathsisfun.com/calculator-precision.html">Full Precision Calculator</a>.</p>
<pre class="wrap language-text"><code class="language-text"><span class="highlight-line">-96419*10^15/326918592000+367963*10^14/9686476800-83433193*10^13/37362124800+76018199*10^12/958003200-3905893151*10^11/2052864000+2834174039*10^10/87091200-748402863991*10^9/1828915200+780710080883*10^8/203212800-35360802693773*10^7/1306368000+6209534662753*10^6/43545600-56937672552137*10^5/102643200+186397079118703*10^4/119750400-4585643519077501*10^3/1513512000+289798498820219*10^2/75675600-125147426197*10^1/45045+857840</span></code></pre>
<p>Let's look at another example: square 2 (a2), substitute <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span>:</p>
<pre class="wrap language-text"><code class="language-text"><span class="highlight-line">-96419*2^15/326918592000+367963*2^14/9686476800-83433193*2^13/37362124800+76018199*2^12/958003200-3905893151*2^11/2052864000+2834174039*2^10/87091200-748402863991*2^9/1828915200+780710080883*2^8/203212800-35360802693773*2^7/1306368000+6209534662753*2^6/43545600-56937672552137*2^5/102643200+186397079118703*2^4/119750400-4585643519077501*2^3/1513512000+289798498820219*2^2/75675600-125147426197*2^1/45045+857840</span></code></pre>
<p>The calculated total is 4 in decimal. Converting this to base-3 yields 11, and padding it with leading zeros gives 0011, which translates to &quot;empty, empty, white, white&quot;. This is indeed the content of a2.</p>
<h2>Now, z</h2>
<p>Lastly, let's calculate the 16th square. Substitute <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">n=16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">6</span></span></span></span>, and the result is the decimal value 16. Converting this to base-3 gives 121, padding it with a zero yields 0121, or &quot;empty, white, black, white&quot;. This is exactly the content of z.</p>
<pre class="wrap language-text"><code class="language-text"><span class="highlight-line">-96419*16^15/326918592000+367963*16^14/9686476800-83433193*16^13/37362124800+76018199*16^12/958003200-3905893151*16^11/2052864000+2834174039*16^10/87091200-748402863991*16^9/1828915200+780710080883*16^8/203212800-35360802693773*16^7/1306368000+6209534662753*16^6/43545600-56937672552137*16^5/102643200+186397079118703*16^4/119750400-4585643519077501*16^3/1513512000+289798498820219*16^2/75675600-125147426197*16^1/45045+857840</span></code></pre>
<h2>Where Did the Polynomial Come From?</h2>
<p>Although the dimension and coefficients of the polynomial look intimidating, it's actually not that big of a deal. The entire solution is just solving the following system of equations:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo>∗</mo><msup><mi>n</mi><mn>15</mn></msup><mo>+</mo><mi>B</mi><mo>∗</mo><msup><mi>n</mi><mn>14</mn></msup><mo>+</mo><mi>C</mi><mo>∗</mo><msup><mi>n</mi><mn>13</mn></msup><mo>+</mo><mi>D</mi><mo>∗</mo><msup><mi>n</mi><mn>12</mn></msup><mo>+</mo><mi>E</mi><mo>∗</mo><msup><mi>n</mi><mn>11</mn></msup><mo>+</mo><mi>F</mi><mo>∗</mo><msup><mi>n</mi><mn>10</mn></msup><mo>+</mo><mi>G</mi><mo>∗</mo><msup><mi>n</mi><mn>9</mn></msup><mo>+</mo><mi>H</mi><mo>∗</mo><msup><mi>n</mi><mn>8</mn></msup><mo>+</mo><mi>I</mi><mo>∗</mo><msup><mi>n</mi><mn>7</mn></msup><mo>+</mo><mi>J</mi><mo>∗</mo><msup><mi>n</mi><mn>6</mn></msup><mo>+</mo><mi>K</mi><mo>∗</mo><msup><mi>n</mi><mn>5</mn></msup><mo>+</mo><mi>L</mi><mo>∗</mo><msup><mi>n</mi><mn>4</mn></msup><mo>+</mo><mi>M</mi><mo>∗</mo><msup><mi>n</mi><mn>3</mn></msup><mo>+</mo><mi>N</mi><mo>∗</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>O</mi><mo>∗</mo><msup><mi>n</mi><mn>1</mn></msup><mo>+</mo><mi>P</mi><mo>∗</mo><msup><mi>n</mi><mn>0</mn></msup></mrow><annotation encoding="application/x-tex">f(n)=A*n^{15}+ B*n^{14}+ C*n^{13}+ D*n^{12}+ E*n^{11}+ F*n^{10}+ G*n^{9}+ H*n^{8}+ I*n^{7}+ J*n^{6}+ K*n^{5}+ L*n^{4}+ M*n^{3}+ N*n^{2}+ O*n^{1}+ P*n^{0}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">G</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9474379999999999em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Substitute n from 1 to 16 respectively, and according to the decimal values in the original encoding table, we get the following sixteen equations:</p>
<p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>71</mn><mo separator="true">;</mo><mi>f</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mo separator="true">;</mo><mi>f</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>49</mn><mo separator="true">;</mo><mo>…</mo><mtext> </mtext><mo separator="true">;</mo><mi>f</mi><mo stretchy="false">(</mo><mn>15</mn><mo stretchy="false">)</mo><mo>=</mo><mn>8</mn><mo separator="true">;</mo><mi>f</mi><mo stretchy="false">(</mo><mn>16</mn><mo stretchy="false">)</mo><mo>=</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">f(1) = 71; f(2)=4; f(3)=49; \dots; f(15)=8; f(16)=16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">7</span><span class="mord">1</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">3</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">4</span><span class="mord">9</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">1</span><span class="mord">5</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">8</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord">1</span><span class="mord">6</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">6</span></span></span></span> (decimal value for z)</p>
<p>By solving this system of sixteen equations, we can calculate the sixteen coefficients <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo separator="true">,</mo><mi>C</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">A, B, C, \dots, P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</p>
<h2>The Matrix Method</h2>
<p>We can rewrite the above system of equations into the following matrix format. Substitute <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> for the desired answer (if the answer is z, plug in <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mn>16</mn></mrow><annotation encoding="application/x-tex">X=16</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord">6</span></span></span></span> calculated from the table), and we can solve for the coefficients <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo separator="true">,</mo><mi>C</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">A, B, C, \dots, P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span>.</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>1</mn><mn>15</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>1</mn><mn>14</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>1</mn><mn>13</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>1</mn><mn>12</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>1</mn><mn>11</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>1</mn><mn>10</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mn>1</mn><mn>9</mn></msup></mstyle></mtd><mtd><mstyle scriptlevel="0" 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displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>13</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>12</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>11</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>10</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>9</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>8</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>7</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>6</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>5</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>4</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>3</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>2</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>1</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>3</mn><mn>0</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>15</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>14</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>13</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>12</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>11</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>10</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>9</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>8</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>7</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>6</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>5</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>4</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>3</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>2</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>1</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>4</mn><mn>0</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>15</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>14</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>13</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>12</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>11</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>10</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>9</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>8</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>7</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>6</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>5</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>4</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>3</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>2</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>1</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>5</mn><mn>0</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>15</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>14</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>13</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>12</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>11</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>10</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>9</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>8</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>7</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>6</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>5</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>4</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>3</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>2</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>1</mn></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><msup><mn>6</mn><mn>0</mn></msup></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>A</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>B</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>C</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>D</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>E</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>F</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>G</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>H</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>I</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>J</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>K</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>L</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>M</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>N</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>O</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>P</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>71</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>49</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>45</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>32</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>70</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>44</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>76</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>50</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>52</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>63</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>68</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>X</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix}
1^{15} &amp; 1^{14} &amp; 1^{13} &amp; 1^{12} &amp; 1^{11} &amp; 1^{10} &amp; 1^{9} &amp; 1^{8} &amp; 1^{7} &amp; 1^{6} &amp; 1^{5} &amp; 1^{4} &amp; 1^{3} &amp; 1^{2} &amp; 1^{1} &amp; 1^{0} \\
2^{15} &amp; 2^{14} &amp; 2^{13} &amp; 2^{12} &amp; 2^{11} &amp; 2^{10} &amp; 2^{9} &amp; 2^{8} &amp; 2^{7} &amp; 2^{6} &amp; 2^{5} &amp; 2^{4} &amp; 2^{3} &amp; 2^{2} &amp; 2^{1} &amp; 2^{0} \\
3^{15} &amp; 3^{14} &amp; 3^{13} &amp; 3^{12} &amp; 3^{11} &amp; 3^{10} &amp; 3^{9} &amp; 3^{8} &amp; 3^{7} &amp; 3^{6} &amp; 3^{5} &amp; 3^{4} &amp; 3^{3} &amp; 3^{2} &amp; 3^{1} &amp; 3^{0} \\
4^{15} &amp; 4^{14} &amp; 4^{13} &amp; 4^{12} &amp; 4^{11} &amp; 4^{10} &amp; 4^{9} &amp; 4^{8} &amp; 4^{7} &amp; 4^{6} &amp; 4^{5} &amp; 4^{4} &amp; 4^{3} &amp; 4^{2} &amp; 4^{1} &amp; 4^{0} \\
5^{15} &amp; 5^{14} &amp; 5^{13} &amp; 5^{12} &amp; 5^{11} &amp; 5^{10} &amp; 5^{9} &amp; 5^{8} &amp; 5^{7} &amp; 5^{6} &amp; 5^{5} &amp; 5^{4} &amp; 5^{3} &amp; 5^{2} &amp; 5^{1} &amp; 5^{0} \\
6^{15} &amp; 6^{14} &amp; 6^{13} &amp; 6^{12} &amp; 6^{11} &amp; 6^{10} &amp; 6^{9} &amp; 6^{8} &amp; 6^{7} &amp; 6^{6} &amp; 6^{5} &amp; 6^{4} &amp; 6^{3} &amp; 6^{2} &amp; 6^{1} &amp; 6^{0} \\
7^{15} &amp; 7^{14} &amp; 7^{13} &amp; 7^{12} &amp; 7^{11} &amp; 7^{10} &amp; 7^{9} &amp; 7^{8} &amp; 7^{7} &amp; 7^{6} &amp; 7^{5} &amp; 7^{4} &amp; 7^{3} &amp; 7^{2} &amp; 7^{1} &amp; 7^{0} \\
8^{15} &amp; 8^{14} &amp; 8^{13} &amp; 8^{12} &amp; 8^{11} &amp; 8^{10} &amp; 8^{9} &amp; 8^{8} &amp; 8^{7} &amp; 8^{6} &amp; 8^{5} &amp; 8^{4} &amp; 8^{3} &amp; 8^{2} &amp; 8^{1} &amp; 8^{0} \\
9^{15} &amp; 9^{14} &amp; 9^{13} &amp; 9^{12} &amp; 9^{11} &amp; 9^{10} &amp; 9^{9} &amp; 9^{8} &amp; 9^{7} &amp; 9^{6} &amp; 9^{5} &amp; 9^{4} &amp; 9^{3} &amp; 9^{2} &amp; 9^{1} &amp; 9^{0} \\
10^{15} &amp; 10^{14} &amp; 10^{13} &amp; 10^{12} &amp; 10^{11} &amp; 10^{10} &amp; 10^{9} &amp; 10^{8} &amp; 10^{7} &amp; 10^{6} &amp; 10^{5} &amp; 10^{4} &amp; 10^{3} &amp; 10^{2} &amp; 10^{1} &amp; 10^{0} \\
11^{15} &amp; 11^{14} &amp; 11^{13} &amp; 11^{12} &amp; 11^{11} &amp; 11^{10} &amp; 11^{9} &amp; 11^{8} &amp; 11^{7} &amp; 11^{6} &amp; 11^{5} &amp; 11^{4} &amp; 11^{3} &amp; 11^{2} &amp; 11^{1} &amp; 11^{0} \\
12^{15} &amp; 12^{14} &amp; 12^{13} &amp; 12^{12} &amp; 12^{11} &amp; 12^{10} &amp; 12^{9} &amp; 12^{8} &amp; 12^{7} &amp; 12^{6} &amp; 12^{5} &amp; 12^{4} &amp; 12^{3} &amp; 12^{2} &amp; 12^{1} &amp; 12^{0} \\
13^{15} &amp; 13^{14} &amp; 13^{13} &amp; 13^{12} &amp; 13^{11} &amp; 13^{10} &amp; 13^{9} &amp; 13^{8} &amp; 13^{7} &amp; 13^{6} &amp; 13^{5} &amp; 13^{4} &amp; 13^{3} &amp; 13^{2} &amp; 13^{1} &amp; 13^{0} \\
14^{15} &amp; 14^{14} &amp; 14^{13} &amp; 14^{12} &amp; 14^{11} &amp; 14^{10} &amp; 14^{9} &amp; 14^{8} &amp; 14^{7} &amp; 14^{6} &amp; 14^{5} &amp; 14^{4} &amp; 14^{3} &amp; 14^{2} &amp; 14^{1} &amp; 14^{0} \\
15^{15} &amp; 15^{14} &amp; 15^{13} &amp; 15^{12} &amp; 15^{11} &amp; 15^{10} &amp; 15^{9} &amp; 15^{8} &amp; 15^{7} &amp; 15^{6} &amp; 15^{5} &amp; 15^{4} &amp; 15^{3} &amp; 15^{2} &amp; 15^{1} &amp; 15^{0} \\
16^{15} &amp; 16^{14} &amp; 16^{13} &amp; 16^{12} &amp; 16^{11} &amp; 16^{10} &amp; 16^{9} &amp; 16^{8} &amp; 16^{7} &amp; 16^{6} &amp; 16^{5} &amp; 16^{4} &amp; 16^{3} &amp; 16^{2} &amp; 16^{1} &amp; 16^{0}
\end{bmatrix}
\begin{bmatrix}
A \\
B \\
C \\
D \\
E \\
F \\
G \\
H \\
I \\
J \\
K \\
L \\
M \\
N \\
O \\
P
\end{bmatrix} = \begin{bmatrix}
71 \\
4 \\
49 \\
45 \\
32 \\
5 \\
70 \\
44 \\
76 \\
50 \\
52 \\
63 \\
7 \\
68 \\
8 \\
X
\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:19.22703em;vertical-align:-9.35015em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.876880000000002em;"><span style="top:5.550129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:4.400129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:3.804129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:3.208129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:2.6121299999999987em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:2.0161299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:1.4201299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.8241299999999985em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.2281299999999984em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.3678700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.9638700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5598700000000019em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.155870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.751870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3478700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9438700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.539870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.135870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.7318700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.3278700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.923870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.519870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.115870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.711870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-9.307870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-9.903870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-10.499870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-10.635860000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-11.876880000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.35015em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">7</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">8</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord">9</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">1</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">3</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">4</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">5</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.876880000000002em;"><span style="top:5.550129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:4.400129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:3.804129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:3.208129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:2.6121299999999987em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:2.0161299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:1.4201299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.8241299999999985em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.2281299999999984em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.3678700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.9638700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5598700000000019em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.155870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.751870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3478700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9438700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.539870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.135870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.7318700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.3278700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.923870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.519870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.115870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.711870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-9.307870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-9.903870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-10.499870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-10.635860000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-11.876880000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.35015em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.876880000000002em;"><span style="top:5.550129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:4.400129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:3.804129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:3.208129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:2.6121299999999987em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:2.0161299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:1.4201299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.8241299999999985em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.2281299999999984em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.3678700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.9638700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5598700000000019em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.155870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.751870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3478700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9438700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.539870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.135870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.7318700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.3278700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.923870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.519870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.115870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.711870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-9.307870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-9.903870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-10.499870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-10.635860000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-11.876880000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.35015em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">G</span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">I</span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.09618em;">J</span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">L</span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.876880000000002em;"><span style="top:5.550129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:4.400129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:3.804129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:3.208129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:2.6121299999999987em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:2.0161299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:1.4201299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.8241299999999985em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.2281299999999984em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.3678700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.9638700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5598700000000019em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.155870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.751870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3478700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9438700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.539870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.135870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.7318700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.3278700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.923870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.519870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.115870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.711870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-9.307870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-9.903870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-10.499870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-10.635860000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-11.876880000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.35015em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:19.22703em;vertical-align:-9.35015em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.876880000000002em;"><span style="top:5.550129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:4.400129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:3.804129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:3.208129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:2.6121299999999987em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:2.0161299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:1.4201299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.8241299999999985em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:0.2281299999999984em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.3678700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-0.9638700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.5598700000000019em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.155870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.751870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.3478700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.9438700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.539870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.135870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.7318700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.3278700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.923870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-7.519870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.115870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-8.711870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-9.307870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-9.903870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-10.499870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-10.635860000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-11.876880000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.35015em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.849999999999998em;"><span style="top:-12.009999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">1</span></span></span><span style="top:-10.809999999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-9.609999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">9</span></span></span><span style="top:-8.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">5</span></span></span><span style="top:-7.209999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">2</span></span></span><span style="top:-6.009999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-4.809999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">0</span></span></span><span style="top:-3.6099999999999963em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">4</span></span></span><span style="top:-2.409999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">6</span></span></span><span style="top:-1.209999999999998em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span><span class="mord">0</span></span></span><span style="top:-0.009999999999998621em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span><span class="mord">2</span></span></span><span style="top:1.1900000000000006em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord">3</span></span></span><span style="top:2.39em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span><span style="top:3.590000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord">8</span></span></span><span style="top:4.79em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:5.990000000000001em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.350000000000001em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:9.876880000000002em;"><span style="top:5.550129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:4.400129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:3.804129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:3.208129999999999em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:2.6121299999999987em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:2.0161299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:1.4201299999999986em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.8241299999999985em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:0.2281299999999984em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.3678700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-0.9638700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.5598700000000019em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.155870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.751870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.3478700000000017em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.9438700000000018em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.539870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.135870000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.7318700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.3278700000000025em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.923870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-7.519870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.115870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-8.711870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-9.307870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-9.903870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-10.499870000000003em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-10.635860000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-11.876880000000002em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:9.35015em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Using the matrix to solve the system, the resulting values for <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi><mo separator="true">,</mo><mi>C</mi><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">A, B, C, \dots, P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> are the polynomial coefficients we listed earlier. A is the first term's coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mo>−</mo><mn>96419</mn></mrow><mn>326918592000</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{-96419}{326918592000}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mord mtight">2</span><span class="mord mtight">6</span><span class="mord mtight">9</span><span class="mord mtight">1</span><span class="mord mtight">8</span><span class="mord mtight">5</span><span class="mord mtight">9</span><span class="mord mtight">2</span><span class="mord mtight">0</span><span class="mord mtight">0</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">9</span><span class="mord mtight">6</span><span class="mord mtight">4</span><span class="mord mtight">1</span><span class="mord mtight">9</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>, P is the last term's coefficient <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>857840</mn></mrow><annotation encoding="application/x-tex">857840</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">8</span><span class="mord">5</span><span class="mord">7</span><span class="mord">8</span><span class="mord">4</span><span class="mord">0</span></span></span></span>, and so on.</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>96419</mn><msup><mi>n</mi><mn>15</mn></msup></mrow><mn>326918592000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>367963</mn><msup><mi>n</mi><mn>14</mn></msup></mrow><mn>9686476800</mn></mfrac><mo>−</mo><mfrac><mrow><mn>83433193</mn><msup><mi>n</mi><mn>13</mn></msup></mrow><mn>37362124800</mn></mfrac><mo>+</mo><mfrac><mrow><mn>76018199</mn><msup><mi>n</mi><mn>12</mn></msup></mrow><mn>958003200</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>3905893151</mn><msup><mi>n</mi><mn>11</mn></msup></mrow><mn>2052864000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2834174039</mn><msup><mi>n</mi><mn>10</mn></msup></mrow><mn>87091200</mn></mfrac><mo>−</mo><mfrac><mrow><mn>748402863991</mn><msup><mi>n</mi><mn>9</mn></msup></mrow><mn>1828915200</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mfrac><mrow><mn>780710080883</mn><msup><mi>n</mi><mn>8</mn></msup></mrow><mn>203212800</mn></mfrac><mo>−</mo><mfrac><mrow><mn>35360802693773</mn><msup><mi>n</mi><mn>7</mn></msup></mrow><mn>1306368000</mn></mfrac><mo>+</mo><mfrac><mrow><mn>6209534662753</mn><msup><mi>n</mi><mn>6</mn></msup></mrow><mn>43545600</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>−</mo><mfrac><mrow><mn>56937672552137</mn><msup><mi>n</mi><mn>5</mn></msup></mrow><mn>102643200</mn></mfrac><mo>+</mo><mfrac><mrow><mn>186397079118703</mn><msup><mi>n</mi><mn>4</mn></msup></mrow><mn>119750400</mn></mfrac><mo>−</mo><mfrac><mrow><mn>4585643519077501</mn><msup><mi>n</mi><mn>3</mn></msup></mrow><mn>1513512000</mn></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>+</mo><mfrac><mrow><mn>289798498820219</mn><msup><mi>n</mi><mn>2</mn></msup></mrow><mn>75675600</mn></mfrac><mo>−</mo><mfrac><mrow><mn>125147426197</mn><msup><mi>n</mi><mn>1</mn></msup></mrow><mn>45045</mn></mfrac><mo>+</mo><mn>857840</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
&amp; - \frac{96419 n^{15}}{326918592000} + \frac{367963 n^{14}}{9686476800} - \frac{83433193 n^{13}}{37362124800} + \frac{76018199 n^{12}}{958003200} \\
&amp; - \frac{3905893151 n^{11}}{2052864000} + \frac{2834174039 n^{10}}{87091200} - \frac{748402863991 n^{9}}{1828915200} \\
&amp; + \frac{780710080883 n^{8}}{203212800} - \frac{35360802693773 n^{7}}{1306368000} + \frac{6209534662753 n^{6}}{43545600} \\
&amp; - \frac{56937672552137 n^{5}}{102643200} + \frac{186397079118703 n^{4}}{119750400} - \frac{4585643519077501 n^{3}}{1513512000} \\
&amp; + \frac{289798498820219 n^{2}}{75675600} - \frac{125147426197 n^{1}}{45045} + 857840
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.385540000000002em;vertical-align:-5.94277em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.442770000000001em;"><span style="top:-8.442770000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-5.965662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-3.4885540000000006em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:-1.0114460000000012em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span><span style="top:1.465662em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.94277em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.442770000000001em;"><span style="top:-8.442770000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">2</span><span class="mord">6</span><span class="mord">9</span><span class="mord">1</span><span class="mord">8</span><span class="mord">5</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">6</span><span class="mord">4</span><span class="mord">1</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">6</span><span class="mord">8</span><span class="mord">6</span><span class="mord">4</span><span class="mord">7</span><span class="mord">6</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">6</span><span class="mord">7</span><span class="mord">9</span><span class="mord">6</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">7</span><span class="mord">3</span><span class="mord">6</span><span class="mord">2</span><span class="mord">1</span><span class="mord">2</span><span class="mord">4</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span><span class="mord">3</span><span class="mord">4</span><span class="mord">3</span><span class="mord">3</span><span class="mord">1</span><span class="mord">9</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span><span class="mord">5</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span><span class="mord">3</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">6</span><span class="mord">0</span><span class="mord">1</span><span class="mord">8</span><span class="mord">1</span><span class="mord">9</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-5.965662000000001em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">5</span><span class="mord">2</span><span class="mord">8</span><span class="mord">6</span><span class="mord">4</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">9</span><span class="mord">0</span><span class="mord">5</span><span class="mord">8</span><span class="mord">9</span><span class="mord">3</span><span class="mord">1</span><span class="mord">5</span><span class="mord">1</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span><span class="mord">7</span><span class="mord">0</span><span class="mord">9</span><span class="mord">1</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">8</span><span class="mord">3</span><span class="mord">4</span><span class="mord">1</span><span class="mord">7</span><span class="mord">4</span><span class="mord">0</span><span class="mord">3</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">8</span><span class="mord">2</span><span class="mord">8</span><span class="mord">9</span><span class="mord">1</span><span class="mord">5</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">4</span><span class="mord">8</span><span class="mord">4</span><span class="mord">0</span><span class="mord">2</span><span class="mord">8</span><span class="mord">6</span><span class="mord">3</span><span class="mord">9</span><span class="mord">9</span><span class="mord">1</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.4885540000000006em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">3</span><span class="mord">2</span><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">8</span><span class="mord">0</span><span class="mord">7</span><span class="mord">1</span><span class="mord">0</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mord">8</span><span class="mord">8</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">8</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">3</span><span class="mord">0</span><span class="mord">6</span><span class="mord">3</span><span class="mord">6</span><span class="mord">8</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span><span class="mord">5</span><span class="mord">3</span><span class="mord">6</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mord">2</span><span class="mord">6</span><span class="mord">9</span><span class="mord">3</span><span class="mord">7</span><span class="mord">7</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">3</span><span class="mord">5</span><span class="mord">4</span><span class="mord">5</span><span class="mord">6</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord">2</span><span class="mord">0</span><span class="mord">9</span><span class="mord">5</span><span class="mord">3</span><span class="mord">4</span><span class="mord">6</span><span class="mord">6</span><span class="mord">2</span><span class="mord">7</span><span class="mord">5</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.0114460000000012em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">6</span><span class="mord">4</span><span class="mord">3</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span><span class="mord">6</span><span class="mord">9</span><span class="mord">3</span><span class="mord">7</span><span class="mord">6</span><span class="mord">7</span><span class="mord">2</span><span class="mord">5</span><span class="mord">5</span><span class="mord">2</span><span class="mord">1</span><span class="mord">3</span><span class="mord">7</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">1</span><span class="mord">9</span><span class="mord">7</span><span class="mord">5</span><span class="mord">0</span><span class="mord">4</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">8</span><span class="mord">6</span><span class="mord">3</span><span class="mord">9</span><span class="mord">7</span><span class="mord">0</span><span class="mord">7</span><span class="mord">9</span><span class="mord">1</span><span class="mord">1</span><span class="mord">8</span><span class="mord">7</span><span class="mord">0</span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">5</span><span class="mord">1</span><span class="mord">3</span><span class="mord">5</span><span class="mord">1</span><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">5</span><span class="mord">8</span><span class="mord">5</span><span class="mord">6</span><span class="mord">4</span><span class="mord">3</span><span class="mord">5</span><span class="mord">1</span><span class="mord">9</span><span class="mord">0</span><span class="mord">7</span><span class="mord">7</span><span class="mord">5</span><span class="mord">0</span><span class="mord">1</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:1.465662em;"><span class="pstrut" style="height:3.491108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span><span class="mord">5</span><span class="mord">6</span><span class="mord">7</span><span class="mord">5</span><span class="mord">6</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">8</span><span class="mord">9</span><span class="mord">7</span><span class="mord">9</span><span class="mord">8</span><span class="mord">4</span><span class="mord">9</span><span class="mord">8</span><span class="mord">8</span><span class="mord">2</span><span class="mord">0</span><span class="mord">2</span><span class="mord">1</span><span class="mord">9</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.491108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord">5</span><span class="mord">0</span><span class="mord">4</span><span class="mord">5</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">5</span><span class="mord">1</span><span class="mord">4</span><span class="mord">7</span><span class="mord">4</span><span class="mord">2</span><span class="mord">6</span><span class="mord">1</span><span class="mord">9</span><span class="mord">7</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">8</span><span class="mord">5</span><span class="mord">7</span><span class="mord">8</span><span class="mord">4</span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.94277em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>In summary, whatever answer you want for the final square, just encode it into a decimal representation, plug it into the matrix as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>, calculate the coefficients, and then you can use the resulting polynomial to calculate the content of any of the sixteen squares.</p>
</slide-page>
<slide-page>
<h1>Epilogue</h1>
<p>If a puzzle's conditions and rules are incompletely defined, the answer can be anything at all, even totally arbitrary without breaking any rules. Most puzzles found online, and even many exam questions in schools, suffer from this flaw. Asking people to guess which of the many possible solutions the author had in mind is quite meaningless. However, asking the solver to explain their thought process can be a good exercise for developing analytical skills, imagination, and rigorous execution.</p>
<p>Using a high-degree polynomial to capture the puzzle and its desired solution is actually a bit like cheating. Frankly, it's just encoding a massive amount of information into these coefficients. It offer a description for a phenomenon, not an explanation for it, and is likely against the <a href="https://en.wikipedia.org/wiki/Occam%27s_razor">Occam's razor</a> principle.</p>
<p>By the way, in machine learning, this is called <em>over-fitting</em>; meaning the model perfectly captures observed phenomena, but will fail miserably when predicting unobserved ones. For example, using this polynomial, we can even calculate the base-3 value for the 100th square:</p>
<pre class="wrap language-text"><code class="language-text"><span class="highlight-line">-221102221101221221122102022021222022122010021211</span></code></pre>
<p>But it's meaningless, because our roots were only defined up to the 16th square, and we didn't specifically model relationships like sequence or adjacency between the squares. Therefore, any result after the 17th square has nothing to do with our puzzle.</p>
<h1>Anything Goes</h1>
<p>I wrote the original post in Chinese, and its title was simply &quot;Any solution works&quot;. However, I use the famous musical number's title here because its melody does remind me of an intricate puzzle.</p>
<div class="youtube-widget" style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden; border-radius: 12px; margin: 1.5rem 0; box-shadow: 0 4px 24px rgba(0,0,0,0.2);">
<iframe src="https://www.youtube.com/embed/tN2KHeEWqZM?start=39" style="position: absolute; top: 0; left: 0; width: 100%; height: 100%; border: 0;" allowfullscreen=""></iframe>
</div>
<div class="sheet-music-container engine-abcjs mode-client" data-audio="false">
<pre class="sheet-music-raw-source" style="display:none;">X: 1
T: Anything Goes (Chorus)
C: Cole Porter
M: 4/4
L: 1/4
Q: "medium swing"
K: C
G/G/- | G A E3/2 G/ | A G/A/- A/G/ E | G/A E/ G A/c/- | c/A/ c/d/- d/ e3/2 |
w: In old--en days a glimpse of stock--ing Was looked on as some-thing shock--ing, But now, _ God
d4- | d z/ c/- c/c/ c | c4- | c]
w: knows, _ An--y-thing goes! _
</pre>
</div>
<p>Its simple pentatonic notes (and the simple chords in this old recording) are paired with a highly syncopated rhythm, just as a simple puzzle can be approached by very complex solutions.</p>
</slide-page>
</slide-deck>


    ]]></content>
  </entry><entry>
    <title>Streaming Custom Videos in Google Hangouts</title>
    <link href="https://blog.tklee.org/en/posts/0635-streaming-custom-videos-in-google-hangouts/"/>
    <updated>2019-04-30T06:35:00.002Z</updated>
    <id>https://blog.tklee.org/en/posts/0635-streaming-custom-videos-in-google-hangouts/</id>
    <content type="html"><![CDATA[
      <p>With this method, I was able to send Google Hangouts any video contents, such as me with a news broadcast background, while the small TV in that background is showing my desktop playing a live Youtube video.</p>
<p><a href="https://obsproject.com/">OBS</a> is a fine tool that enables us to overlay various visual/audio sources, on the fly, for streaming and/or recording. Unfortunately, it doesn't support a virtual cam under Linux (there <em>is</em> <a href="https://github.com/CatxFish/obs-virtual-cam">a plugin</a> that only works under Windows), i.e. I can't send a complex composition of a live desktop screen, webcam, video, etc. to Google Hangouts.</p>
<p><a href="https://webcamoid.github.io/">Webcamoid</a> is another fine tool that supports output to a virtual cam under Linux, which Google Hangouts can directly use.</p>
<p><a href="https://github.com/CatxFish/obs-virtual-cam/issues/17">Several attempts</a> were made, but I wasn't able to get a result with low enough latency for video conferencing. The video latency caused an unacceptable out-of-sync problem with the audio.</p>
<p>I reached an easy workaround, which is resource heavy, but works quite well.</p>
<ol>
<li>Use OBS to construct the target scene, say, a chroma keyed webcam, with a video background, and a logo foreground, as well as a live chat overlay.</li>
<li>In OBS, right click on the preview, and project it under a separate screen.</li>
<li>In Webcamoid, choose that screen as the source.</li>
<li>In Google Hangouts, choose Webcamoid's virtual cam instead of the real webcam.</li>
<li>Voila! Now I have full control of what others see in my video conferencing.</li>
</ol>
<p>This solution needs one whole screen just to link OBS and webcamoid. Any of the following improvement of either project will make this workflow even better:</p>
<ul>
<li>When Webcamoid supports <a href="https://github.com/webcamoid/webcamoid/issues/180">a window as a source</a> (right now only whole screens can be sources), we can hide the window so it doesn't take up our screen real estate.</li>
<li>When OBS can <a href="https://github.com/CatxFish/obs-virtual-cam/issues/17#issuecomment-486121295">correctly piping a video stream into a linux virtual cam</a>, or even better, correctly implements its own virtual cam.</li>
</ul>
<p>Hopefully either will come soon.</p>
<p>Update: now there's an easier way: use OBS's v4l2sink plugin + v4l2loopback kernel module. See this <a href="https://srcco.de/posts/using-obs-studio-with-v4l2-for-google-hangouts-meet.html">article</a>.</p>

    ]]></content>
  </entry><entry>
    <title>uwsgi wrong python interpreter</title>
    <link href="https://blog.tklee.org/en/posts/2102-uwsgi-wrong-python-interpreter/"/>
    <updated>2017-10-13T21:02:00Z</updated>
    <id>https://blog.tklee.org/en/posts/2102-uwsgi-wrong-python-interpreter/</id>
    <content type="html"><![CDATA[
      <p>In some projects, I use uwsgi behind nginx to drive Django. I also use virtualenv to manage my environments (by the way, Panda's introducing .ix and then deprecating it wasn't fun). In a recent update, I was pulling my hair out trying to figure out why I'm getting the following error message, and a wrong Python interpreter.</p>
<pre class="language-text"><code class="language-text"><span class="highlight-line">Could not find platform independent libraries </span>
<span class="highlight-line">  </span>
<span class="highlight-line">Could not find platform dependent libraries </span>
<span class="highlight-line">  </span>
<span class="highlight-line">Consider setting $PYTHONHOME to [:]</span>
<span class="highlight-line">  </span>
<span class="highlight-line">Python version: 3.5.4 |Continuum Analytics, Inc.| (default, Aug 14 2017, 13:26:58)  [GCC 4.4.7 20120313 (Red Hat 4.4.7-1)]</span>
<span class="highlight-line">  </span>
<span class="highlight-line">Set PythonHome to /opt/anaconda</span>
<span class="highlight-line">  </span>
<span class="highlight-line">Fatal Python error: Py_Initialize: Unable to get the locale encoding</span>
<span class="highlight-line">  </span>
<span class="highlight-line">ImportError: No module named 'encodings'</span></code></pre>
<p>It turned out that the problem was in the binary code of uwsgi in my environment. It was compiled for another environment, with the environment path hard-coded in. Yes, in the binary. I had done that in another environment and the compiled package stayed in the pip cache, and the new environment just picked it up from the cache, causing inconsistency!</p>
<p>The solution:</p>
<pre class="language-bash"><code class="language-bash"><span class="highlight-line">pip <span class="token function">install</span> --no-cache uwsgi</span></code></pre>

    ]]></content>
  </entry><entry>
    <title>Ergodox EZ Configurator Layer Manipulation</title>
    <link href="https://blog.tklee.org/en/posts/0300-ergodox-ez-configurator-layer-manipulation/"/>
    <updated>2017-07-12T03:00:00Z</updated>
    <id>https://blog.tklee.org/en/posts/0300-ergodox-ez-configurator-layer-manipulation/</id>
    <content type="html"><![CDATA[
      <p>Ergodox EZ provides a great <a href="http://configure.ergodox-ez.com/keyboard_layouts/qgbavy/edit">configurator</a>; however, it doesn't provide all editing functions I need. Unfortunately, the source code <a href="https://github.com/ErgoDox-EZ/fusion">disappeared</a> from github (multiple unofficial documents have links to that repo so I assume the source was once available; this is rather disappointing, since the company made it <a href="https://ergodox-ez.com/pages/our-firmware">look like</a> that they are using all open source resources.)</p>
<p>One of the functions that I need the most is swapping the layers.</p>
<p>Here's what I did to achieve it:</p>
<p>Upon examining the minimized javascript, it was written against the <a href="https://facebook.github.io/react/">React</a> framework. Therefore, I installed the <a href="https://chrome.google.com/webstore/detail/react-developer-tools/fmkadmapgofadopljbjfkapdkoienihi?hl=en">React debugger extension for chrome</a>, in order to find and manipulate the variable that stores the layer information.</p>
<p>Once the extension is installed, reload the page, and open up the Developer Tools in Chrome.</p>
<p>Click on React to open the debugging panel.</p>
<p>Make sure the correct scope &quot;App&quot; is selected.</p>
<p>Now the variable $r will contain the React object, including the state, for editing.</p>
<p>The layers (including all key assignment) information is in the array: <code>$r.state.layout.layers</code></p>
<p>Here's an example of swapping layer 3 and 4.</p>
<pre class="language-javascript"><code class="language-javascript"><span class="highlight-line">$r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers <span class="token operator">=</span> <span class="token punctuation">[</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">0</span><span class="token punctuation">]</span><span class="token punctuation">,</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">,</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">2</span><span class="token punctuation">]</span><span class="token punctuation">,</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">4</span><span class="token punctuation">]</span><span class="token punctuation">,</span>  <span class="token comment">// notice the swapping</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">3</span><span class="token punctuation">]</span><span class="token punctuation">,</span>  <span class="token comment">// notice the swapping</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">5</span><span class="token punctuation">]</span><span class="token punctuation">,</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">6</span><span class="token punctuation">]</span><span class="token punctuation">,</span></span>
<span class="highlight-line">  $r<span class="token punctuation">.</span>state<span class="token punctuation">.</span>layout<span class="token punctuation">.</span>layers<span class="token punctuation">[</span><span class="token number">7</span><span class="token punctuation">]</span><span class="token punctuation">,</span></span>
<span class="highlight-line"><span class="token punctuation">]</span></span></code></pre>
<p>Note that if there are layer switching keys, you'll also need to manually modify those from the GUI.</p>
<p>I also did some batch editing offline, and then re-assign the whole <code>$r.state.layout.layers</code> by first converting the variable to JSON and write it to a local file, with <a href="http://bgrins.github.io/devtools-snippets/#console-save">this snippet</a>.</p>
<p>Also when you copy/create a layer, make sure to use the GUI to first allocate the variable <code>$r.state.layout.layers[x].keyboard_layout_id</code> on their server and then carefully copy those values.</p>
<p>Here's my <a href="http://configure.ergodox-ez.com/keyboard_layouts/qbbndp/edit">current setup</a>, if you are interested.</p>
<p><a href="https://blog.tklee.org/assets/2017-07-12-0300/ergodox_layers_manipulation.png"><img src="https://blog.tklee.org/assets/2017-07-12-0300/ergodox_layers_manipulation.png" alt="" /></a></p>

    ]]></content>
  </entry><entry>
    <title>[QA] Socio-economic class</title>
    <link href="https://blog.tklee.org/en/posts/1657-socio-economic-class-社經地位/"/>
    <updated>2016-08-02T16:57:00Z</updated>
    <id>https://blog.tklee.org/en/posts/1657-socio-economic-class-社經地位/</id>
    <content type="html"><![CDATA[
      

<div class="post-teaser-box" data-lang="en">
A Taiwanese netizen asks about professions with lower socio-economic class (in Taiwan) but with stable income even higher than average people.
</div>
<div class="conversation-other">
Quote from omare:  
Do you know if there are any professions with socio-economic class in the middle or lower, but with stable income even higher than average people? Any gossips?
</div>
<div class="conversation-author">
Almost all the professions that our culture looks down upon can we find wealthy ones. Some examples around me are the funeral industry, resource recycling, fruit farmers, and dairy farmers.
<p>Back then, all the people I knew started from the bottom, but each of them had an entrepreneurial talent. I believe that no matter which industry they were thrown into, they would be able to manage it well.</p>
<p>&quot;Low socio-economic class&quot; is an inaccurate term to use here, since it already implies a low income. However, the term itself ties social and economic status together, actually pointing out the blind spots of our culture, the narrowness of our values, and the contradictions in the ratio of effort and reward between managers and laborers under capitalism.</p>
<p>It is a pity that our culture rarely notices those who live truly peaceful and happy lives. Even if they occasionally make the news, the focus is on the contrast between their low social status, presenting a superficial attitude of &quot;it is rare for someone with low social status to be content and happy.&quot; Unexpectedly, the quality of life of these people is actually beyond the reach of ordinary people.</p>
</div>


    ]]></content>
  </entry><entry>
    <title>Por una Cabeza (arranged for guitar, viola, piano) (2016)</title>
    <link href="https://blog.tklee.org/en/posts/2253-por-una-cabeza-arranged-for-guitar-viola-piano-2016/"/>
    <updated>2016-06-17T22:53:13.000Z</updated>
    <id>https://blog.tklee.org/en/posts/2253-por-una-cabeza-arranged-for-guitar-viola-piano-2016/</id>
    <content type="html"><![CDATA[
      
<p>My brother and I live on two different continents, so we don't see each other very often. For three decades, we had not been able to spend quality time together until the summer of 2016, when he finally managed to visit me for more than just a few days.</p>
<p>During his visit, we dug out a lot of childhood video games to play together and had a lot of fun and laughs.</p>
<p>Besides playing games, we decided to play music together. He picked <em>Por una Cabeza</em>.</p>
<p>We both took a few years of piano lessons when we were little. In adulthood, our musical paths diverged: he plays the guitar, and I mostly put my energy into composing and arranging. My wife is a professional violinist and violist. I quickly arranged the tune in the style of chamber music for guitar, viola, and piano. It was simple enough that we were able to record it during his visit.</p>
<p>Making music with family is such a special experience. We had so much fun that I decided to memorialize our time together with this video.</p>
<div class="youtube-widget" style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden; border-radius: 12px; margin: 1.5rem 0; box-shadow: 0 4px 24px rgba(0,0,0,0.2);">
<iframe src="https://www.youtube.com/embed/cowUQRpDs24" style="position: absolute; top: 0; left: 0; width: 100%; height: 100%; border: 0;" allowfullscreen=""></iframe>
</div>


    ]]></content>
  </entry><entry>
    <title>Fixing Lenovo U31-70 Wi Fi Problem</title>
    <link href="https://blog.tklee.org/en/posts/0624-fixing-lenovo-u31-70-wifi-problem/"/>
    <updated>2016-02-06T06:24:00Z</updated>
    <id>https://blog.tklee.org/en/posts/0624-fixing-lenovo-u31-70-wifi-problem/</id>
    <content type="html"><![CDATA[
      
<div class="post-teaser-box" data-lang="en">
<p>Lenovo U31-70 is a lightweight laptop. Its Wi Fi performance has been a headache to many people. Most people report unstable signal dropping. For those who use Windows and are experiencing unstable Wi Fi problem, you just need to solve the hardware problem. I use Linux, so I had to go a different route to solve my driver problem.<br />
</p></div><p></p>
<h2>The Hardware Problem</h2>
<p>I opened up the laptop, and found that the two antenna cables were not properly attached to the card. There was a tape (not shown in the picture) trying to keep the two cables attached, but failed due to the tension from the bending; also, the screw to attach the card onto the main board is too close to one of the antenna contacts so it automatically grounds it, making that antenna useless. This is a design problem.</p>
<p><a href="https://blog.tklee.org/assets/2016-02-06-0624/original_wifi_card.jpg"><img src="https://blog.tklee.org/assets/2016-02-06-0624/original_wifi_card.jpg" alt="" /></a></p>
<p>I first re-attached the two cables back to their contacts and re-tape them, and put some non-conductive material between the screw and the contact to prevent the grounding. It helped.</p>
<h2>The Atheros 10K Driver Problem under Linux</h2>
<p>I had used several Thinkpads since its IBM days, so I was quite surprised to have found that this Lenovo's Atheros 10K chip was not supported under Linux. People were reporting stable driver performance under Windows, and some efforts were made to bring the firmware from Windows driver to Linux. I waited a few months for Linux kernel to come to version 4.3 and spent a lot of time trying various hacks to get the driver working. I was never able to get stable performance.</p>
<p>I decided to replace the card with Intel Dual Band Wireless AC 3160, which is well supported by native Linux.</p>
<p><a href="https://blog.tklee.org/assets/2016-02-06-0624/replaced_wifi.jpg"><img src="https://blog.tklee.org/assets/2016-02-06-0624/replaced_wifi.jpg" alt="" /></a></p>
<p>(I have no idea which antenna cable is intended for Wi Fi and which is for bluetooth, but this works for me; besides, tension-wise, this is the only way I could keep the contacts isolated.)</p>
<p>It has been working flawlessly.<br />
</p>
<div class="post-comments" style="margin-top: 4rem; padding-top: 2rem; border-top: 2px dashed var(--glass-border);">
<h3 class="comments-title" style="font-size: 1.5rem; margin-bottom: 2rem; color: var(--primary-color); text-shadow: 0 0 10px rgba(56,189,248,0.2);">Comments</h3>

<p></p><div class="conversation-other"><br />
<div class="comment-meta" style="font-size: 0.9em; color: var(--text-muted); margin-bottom: 0.8rem; display: flex; gap: 0.5rem; align-items: center;"><br />
<strong style="color: var(--text-main); font-weight: 600;"><a href="https://www.blogger.com/profile/11849094639209265284" target="_blank" rel="nofollow">Rui Ramos</a></strong> <span class="comment-date" style="opacity: 0.7;">on Sunday, July 21, 2019 at 10:25:00 AM EDT</span><br />
</div><br />
<div class="comment-body" style="line-height: 1.6; color: var(--text-main); opacity: 0.9;"><br />
<p>Hi there,</p><p></p>
<p>I am doing the same thing.<br />
But I have found 2 models of 3160:</p>
<ul>
<li>NGW</li>
<li>NGFF<br />
How can I know which is right for my u31-70?</li>
</ul>
<p>Thank you.</p>
<p></p></div><br />
</div><p></p>
<p></p><div class="conversation-author"><br />
<div class="comment-meta" style="font-size: 0.9em; color: var(--text-muted); margin-bottom: 0.8rem; display: flex; gap: 0.5rem; align-items: center;"><br />
<strong style="color: var(--text-main); font-weight: 600;"><a href="https://www.blogger.com/profile/11411703278692679285" target="_blank" rel="nofollow">TK</a></strong> <span class="comment-date" style="opacity: 0.7;">on Tuesday, July 23, 2019 at 3:32:00 PM EDT</span><br />
</div><br />
<div class="comment-body" style="line-height: 1.6; color: var(--text-main); opacity: 0.9;"><br />
<p>If you click on the photo that shows the new card, it's the NGW version. I have no idea what the differences are between the two models. I'm using the NGW and it works. Good luck.</p><p></p>
<p></p></div><br />
</div><p></p>
<p></p><div class="conversation-other"><br />
<div class="comment-meta" style="font-size: 0.9em; color: var(--text-muted); margin-bottom: 0.8rem; display: flex; gap: 0.5rem; align-items: center;"><br />
<strong style="color: var(--text-main); font-weight: 600;"><a href="https://www.blogger.com/profile/11849094639209265284" target="_blank" rel="nofollow">Rui Ramos</a></strong> <span class="comment-date" style="opacity: 0.7;">on Tuesday, July 23, 2019 at 5:04:00 PM EDT</span><br />
</div><br />
<div class="comment-body" style="line-height: 1.6; color: var(--text-main); opacity: 0.9;"><br />
<p>Yup, I see it now.<br /><br />
I will test mine when it gets home.<br /><br />
Thank you!</p><p></p>
<p></p></div><br />
</div><p></p>
</div>

    ]]></content>
  </entry><entry>
    <title>Coordinate Transformation Matrix for Touchscreen in multi-screen setup in X11</title>
    <link href="https://blog.tklee.org/en/posts/0831-coordinate-transformation-matrix-for-touchscreen-in-multi-screen-setup-in-x11/"/>
    <updated>2016-01-30T08:31:00.002Z</updated>
    <id>https://blog.tklee.org/en/posts/0831-coordinate-transformation-matrix-for-touchscreen-in-multi-screen-setup-in-x11/</id>
    <content type="html"><![CDATA[
      <p>Currently I have a twin screen setup in Arch Linux, like this:</p>
<p><a href="https://blog.tklee.org/assets/2016-01-30-0831/arrangement_of_screens.png"><img src="https://blog.tklee.org/assets/2016-01-30-0831/arrangement_of_screens.png" alt="" /></a></p>
<p>On the left, DELL 1901FP is turned 90 degree counter-clockwise in its portrait mode. On the right, DELL P2314T, a touch screen, is in its regular landscape orientation, with its bottom line aligning with the bottom of the rotated screen on the left.</p>
<p>Here's my xorg.conf for that. (Although one can also setup screen arrangement in KDE/Plasma/Gnome/etc., for KDM/SDDM/GDM/etc. to recognize it in the login screen, it's important to put this in xorg.conf)</p>
<pre class="language-apacheconf"><code class="language-apacheconf"><span class="highlight-line"><span class="token comment"># /etc/X11/xorg.conf</span></span>
<span class="highlight-line"><span class="token comment"># relevant section of screen rotation</span></span>
<span class="highlight-line">Section <span class="token string">"Screen"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Screen0"</span></span>
<span class="highlight-line">Device         <span class="token string">"Device0"</span></span>
<span class="highlight-line">Monitor        <span class="token string">"Monitor0"</span></span>
<span class="highlight-line">DefaultDepth    24</span>
<span class="highlight-line">Option         <span class="token string">"Stereo" "0"</span></span>
<span class="highlight-line">Option         <span class="token string">"nvidiaXineramaInfoOrder" "DFP-2"</span></span>
<span class="highlight-line">Option         <span class="token string">"metamodes" "DVI-I-1: 1280x1024_75 +0+0 {rotation=right}, HDMI-0: nvidia-auto-select +1024+200"</span></span>
<span class="highlight-line">Option         <span class="token string">"SLI" "Off"</span></span>
<span class="highlight-line">Option         <span class="token string">"MultiGPU" "Off"</span></span>
<span class="highlight-line">Option         <span class="token string">"BaseMosaic" "off"</span></span>
<span class="highlight-line">SubSection     <span class="token string">"Display"</span></span>
<span class="highlight-line">Depth       24</span>
<span class="highlight-line">EndSubSection</span>
<span class="highlight-line">EndSection</span></code></pre>
<p>X11 sees a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1024</mn><mo>+</mo><mn>1920</mn><mo stretchy="false">)</mo><mtext> wide</mtext><mo>×</mo><mn>1280</mn><mtext> high</mtext></mrow><annotation encoding="application/x-tex">(1024+1920) \text{ wide} \times 1280 \text{ high}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mclose">)</span><span class="mord text"><span class="mord"> wide</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mord text"><span class="mord"> high</span></span></span></span></span> DISPLAY, with an upper-right corner being invisible.</p>
<p>Since the touchscreen is just a regular pointer device (through its driver), when I put my finger on the left border of the touchscreen on the right, the cursor goes all the way to the very left of the DISPLAY. Touchscreen is only useful if the cursor follows my finger. Therefore, we need to do some Input Coordinate Transformation.</p>
<p><a href="https://wiki.ubuntu.com/X/InputCoordinateTransformation">Coordinate Transformation Matrix</a> is precisely the needed configuration.</p>
<p>From the documentation above, our goal is to map the following coordinates <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> on the left (X11's default behavior) to the ones on the right:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.15999999999999992em" columnalign="right left center right left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⇒</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>1024</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1280</mn><mo>−</mo><mn>1080</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>1024</mn><mo>+</mo><mn>1920</mn><mo>−</mo><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1280</mn><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⇒</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>1024</mn><mo>+</mo><mn>1920</mn><mo>−</mo><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1280</mn><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>1024</mn><mo>+</mo><mn>1920</mn><mo>−</mo><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⇒</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>1024</mn><mo>+</mo><mn>1920</mn><mo>−</mo><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1280</mn><mo>−</mo><mn>1080</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1280</mn><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⇒</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mn>1024</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1280</mn><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{r l c r l}
(0, &amp; 0) &amp; \Rightarrow &amp; (1024, &amp; 1280-1080) \\
(1024+1920-1, &amp; 1280-1) &amp; \Rightarrow &amp; (1024+1920-1, &amp; 1280-1) \\
(1024+1920-1, &amp; 0) &amp; \Rightarrow &amp; (1024+1920-1, &amp; 1280-1080) \\
(0, &amp; 1280-1) &amp; \Rightarrow &amp; (1024, &amp; 1280-1)
\end{array}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.800000000000001em;vertical-align:-2.1500000000000004em;"></span><span class="mord"><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">⇒</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">⇒</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">⇒</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mrel">⇒</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mpunct">,</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6500000000000004em;"><span style="top:-4.8100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord">0</span><span class="mord">8</span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-1.2099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span></span></span></span></span></p>
<p>So that means we are solving the following system:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>b</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>c</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>b</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>c</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>b</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>c</mi><mn>3</mn></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2943</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2943</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1279</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1279</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1024</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2943</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2943</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1024</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>200</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1279</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>200</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1279</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix}
a_1 &amp; b_1 &amp; c_1 \\
a_2 &amp; b_2 &amp; c_2 \\
a_3 &amp; b_3 &amp; c_3
\end{bmatrix}
\begin{bmatrix}
0 &amp; 2943 &amp; 2943 &amp; 0 \\
0 &amp; 1279 &amp; 0    &amp; 1279 \\
1 &amp; 1    &amp; 1    &amp; 1
\end{bmatrix}
=
\begin{bmatrix}
1024 &amp; 2943 &amp; 2943 &amp; 1024 \\
200  &amp; 1279 &amp;  200 &amp; 1279 \\
1 &amp; 1    &amp; 1    &amp; 1
\end{bmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6010299999999997em;vertical-align:-1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">9</span><span class="mord">4</span><span class="mord">3</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">7</span><span class="mord">9</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">9</span><span class="mord">4</span><span class="mord">3</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">7</span><span class="mord">9</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.6010299999999997em;vertical-align:-1.55002em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">9</span><span class="mord">4</span><span class="mord">3</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">7</span><span class="mord">9</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">9</span><span class="mord">4</span><span class="mord">3</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">7</span><span class="mord">9</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Here's the solution and the approximate float values:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>b</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>c</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>b</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>c</mi><mn>2</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>b</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>c</mi><mn>3</mn></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>1919</mn><mn>2943</mn></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1024</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>1079</mn><mn>1279</mn></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>200</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>≈</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0.652055725450221</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1024</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0.843627834245504</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>200</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\begin{bmatrix}
a_1 &amp; b_1 &amp; c_1 \\
a_2 &amp; b_2 &amp; c_2 \\
a_3 &amp; b_3 &amp; c_3
\end{bmatrix}
&amp;=
\begin{bmatrix}
\frac{1919}{2943} &amp; 0                 &amp; 1024 \\
0                 &amp; \frac{1079}{1279} &amp; 200 \\
0                 &amp; 0                 &amp; 1
\end{bmatrix}
\\
&amp;\approx
\begin{bmatrix}
0.652055725450221 &amp; 0                 &amp; 1024 \\
0                 &amp; 0.843627834245504 &amp; 200  \\
0                 &amp; 0                 &amp; 1
\end{bmatrix}
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.811246em;vertical-align:-3.6556229999999994em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.155623em;"><span style="top:-6.155623em;"><span class="pstrut" style="height:4.055108em;"></span><span class="mord"><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.249505em;"><span class="pstrut" style="height:4.055108em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6556229999999994em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.155623em;"><span style="top:-6.155623em;"><span class="pstrut" style="height:4.055108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0551079999999997em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">9</span><span class="mord mtight">4</span><span class="mord mtight">3</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">9</span><span class="mord mtight">1</span><span class="mord mtight">9</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0048919999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8048919999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5551080000000006em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0551079999999997em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0048919999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span><span class="mord mtight">7</span><span class="mord mtight">9</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">0</span><span class="mord mtight">7</span><span class="mord mtight">9</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.8048919999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5551080000000006em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0551079999999997em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span></span></span><span style="top:-3.0048919999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-1.8048919999999995em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5551080000000006em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-2.249505em;"><span class="pstrut" style="height:4.055108em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mord">.</span><span class="mord">6</span><span class="mord">5</span><span class="mord">2</span><span class="mord">0</span><span class="mord">5</span><span class="mord">5</span><span class="mord">7</span><span class="mord">2</span><span class="mord">5</span><span class="mord">4</span><span class="mord">5</span><span class="mord">0</span><span class="mord">2</span><span class="mord">2</span><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mord">.</span><span class="mord">8</span><span class="mord">4</span><span class="mord">3</span><span class="mord">6</span><span class="mord">2</span><span class="mord">7</span><span class="mord">8</span><span class="mord">3</span><span class="mord">4</span><span class="mord">2</span><span class="mord">4</span><span class="mord">5</span><span class="mord">5</span><span class="mord">0</span><span class="mord">4</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span></span></span><span style="top:-3.0099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span><span style="top:-1.8099999999999994em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0510099999999998em;"><span style="top:-2.2500000000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.8099900000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.05101em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6556229999999994em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>Now we can put these numbers into the configuration, as the document instructed.</p>
<pre class="language-bash"><code class="language-bash"><span class="highlight-line"><span class="token comment"># This does NOT work</span></span>
<span class="highlight-line">xinput set-prop <span class="token string">"Advanced Silicon S.A CoolTouch(TM) System"</span> <span class="token parameter variable">--type</span><span class="token operator">=</span>float <span class="token string">"Coordinate Transformation Matrix"</span> .652056 <span class="token number">0</span> <span class="token number">1024</span> <span class="token number">0</span> .8436278 <span class="token number">200</span> <span class="token number">0</span> <span class="token number">0</span> <span class="token number">1</span></span></code></pre>
<p>However, it doesn't work. The cursor stays on the lower right. After some debugging, I found the offsets 1024 and 200 were the culprit. They need to be a percentage, i.e. relative to the full width and height:</p>
<p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1024</mn><mrow><mn>1024</mn><mo>+</mo><mn>1920</mn></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0.3478261</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>200</mn><mn>1280</mn></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0.15625</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\dfrac{1024}{1024+1920} &amp;= 0.3478261 \\[1em]
\dfrac{200}{1280} &amp;= 0.15625
\end{aligned}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.288880000000001em;vertical-align:-2.394440000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.89444em;"><span style="top:-4.8944399999999995em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span><span class="mord">9</span><span class="mord">2</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span><span class="mord">2</span><span class="mord">4</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.9129999999999994em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span><span class="mord">8</span><span class="mord">0</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord">0</span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.394440000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.89444em;"><span style="top:-4.8944399999999995em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span><span class="mord">.</span><span class="mord">3</span><span class="mord">4</span><span class="mord">7</span><span class="mord">8</span><span class="mord">2</span><span class="mord">6</span><span class="mord">1</span></span></span><span style="top:-1.9129999999999994em;"><span class="pstrut" style="height:3.32144em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span><span class="mord">.</span><span class="mord">1</span><span class="mord">5</span><span class="mord">6</span><span class="mord">2</span><span class="mord">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.394440000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<pre class="language-bash"><code class="language-bash"><span class="highlight-line"><span class="token comment"># This works</span></span>
<span class="highlight-line">xinput set-prop <span class="token string">"Advanced Silicon S.A CoolTouch(TM) System"</span> <span class="token parameter variable">--type</span><span class="token operator">=</span>float <span class="token string">"Coordinate Transformation Matrix"</span> .652056 <span class="token number">0</span> .3478261 <span class="token number">0</span> .8436278 .15625 <span class="token number">0</span> <span class="token number">0</span> <span class="token number">1</span></span></code></pre>
<p>I put it in <code>/usr/share/sddm/scripts/Xsetup</code> (or <code>/usr/share/config/kdm/Xsetup</code> if you haven't upgraded to plasma) as well as <code>~/.xprofile</code> so it automatically runs every time.</p>
<p>Edit 2016-02-22: I ended up having 3 screens with two in the portrait orientation. SDDM correctly uses metamodes setting in xorg.conf, regardless of the screen power. However, KDE is more sensitive to that so sometimes the layout is messed up. Here's a script to quickly correct that.</p>
<pre class="language-bash"><code class="language-bash"><span class="highlight-line"><span class="token shebang important">#!/bin/sh</span></span>
<span class="highlight-line"><span class="token comment"># my xorg.conf has this line:</span></span>
<span class="highlight-line"><span class="token comment"># Option "metamodes" "DVI-I-1: 1280x1024_75 +0+0 {rotation=right}, HDMI-0: nvidia-auto-select +2944+0 {rotation=right}, DP-1: nvidia-auto-select +1024+200"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">gksu <span class="token string">"xrandr --output DP-1 --primary --auto --pos 1024x200 --output DVI-I-1 --auto --pos 0x0 --rotation right --output HDMI-0 --auto --pos 2944x0 --rotation right"</span></span></code></pre>

    ]]></content>
  </entry><entry>
    <title>Install Sibelius 8.0.1 on a non-C drive in Windows</title>
    <link href="https://blog.tklee.org/en/posts/2243-install-sibelius-8-0-1-on-a-non-c-drive-in-windows/"/>
    <updated>2015-10-19T22:43:00.002Z</updated>
    <id>https://blog.tklee.org/en/posts/2243-install-sibelius-8-0-1-on-a-non-c-drive-in-windows/</id>
    <content type="html"><![CDATA[
      <p>Since Sibelius 8.0.0, Avid changed the installer, and it has a serious bug -- it assumes you have a <code>C:</code> drive before letting you choose the installation destination. I figured out <a href="https://blog.tklee.org/posts/2015-08-05-2359-sibelius-8-installation-on-non-standard-windows-system-drive/">a hack to install it</a>. However, it doesn't work for Sibelius 8.0.1, so here's another way to approach it, before Avid's Sibelius team fixes the bug.</p>
<p>I don't have a <code>C:</code> drive, but Sibelius insists that I must do, so what we are trying to do here is to temporarily create a <code>C:</code> drive, and then create a link pointing crucial directories back to my system drive, K:. After Sibelius is successfully installed, we can then safely remove the temporary <code>C:</code> drive.</p>
<p>First, we need to create a Virtual Hard Drive (VHD) as our <code>C:\</code> drive. I learned how to do this via <a href="https://stackoverflow.com/a/18112515/701284">this answer</a>:</p>
<ul>
<li>Start → run → diskmgmt.msc (accept all defaults... I'm not doing anything special below)</li>
<li>From the menu bar select Action → Create VHD</li>
<li>Choose the location and name the file (which will be the vhd) and specify the size and click OK. My file is <code>K:\virtual_drive_c.vhd</code> and the size is 1G, but I made it dynamic so it's actual size is around 16MB.</li>
<li>Right click on the Disk # (underneath will be Unknown and the size and &quot;Not Initialized&quot;). Select &quot;Initialize Disk&quot; and click OK</li>
<li>Right click on the black bar of the unallocated disk space and select &quot;new simple volume&quot;. A wizard opens up an on the second page it lets you assign the drive letter. We want <code>C:</code> here. Complete the wizard and you're done!</li>
</ul>
<p>Now that we have a <code>C:</code> drive, we can link the important folders. I learned how to do that from <a href="http://www.sevenforums.com/tutorials/278262-mklink-create-use-links-windows.html">this post</a>:</p>
<ul>
<li>Start → run → cmd, then right click and choose &quot;Run as Administrator&quot;.</li>
<li>In the prompt, type the following:</li>
</ul>
<pre class="language-cmd"><code class="language-cmd"><span class="highlight-line">K:\>mklink /j "C:\Program Files" "K:\Program Files"</span>
<span class="highlight-line">Junction created for C:\Program Files <<===>> K:\Program Files</span>
<span class="highlight-line"></span>
<span class="highlight-line">K:\>mklink /j "C:\Program Files (x86)" "K:\Program Files (x86)"</span>
<span class="highlight-line">Junction created for C:\Program Files (x86) <<===>> K:\Program Files (x86)</span></code></pre>
<p>After that, Sibelius installation should run normally. In the installer, I still choose to install the program to K:, to avoid future problems running Sibelius, since the bug is on the installer, not Sibelius itself.</p>
<p>Afterwards, you can reboot, and safely delete the vhd file.</p>

    ]]></content>
  </entry><entry>
    <title></title>
    <link href="https://blog.tklee.org/en/posts/0519-大易二十七年-行列二十七小時/"/>
    <updated>2015-09-24T05:19:00Z</updated>
    <id>https://blog.tklee.org/en/posts/0519-大易二十七年-行列二十七小時/</id>
    <content type="html"><![CDATA[
      

    ]]></content>
  </entry><entry>
    <title>Cry Link (for Orchestra) (2015)</title>
    <link href="https://blog.tklee.org/en/posts/0849-cry-link-soundtrack-for-xonotic/"/>
    <updated>2015-08-19T08:49:09.000Z</updated>
    <id>https://blog.tklee.org/en/posts/0849-cry-link-soundtrack-for-xonotic/</id>
    <content type="html"><![CDATA[
      
<div class="post-teaser-box" data-lang="en">
<p>This is an orchestral soundtrack I composed for one of my favorite video games, <a href="https://xonotic.org/">Xonotic</a> (which I also contributed to its codebase).<br />
</p></div><p></p>
<h1>The soundtrack</h1>
<p>I composed this orchestral piece for <a href="https://xonotic.org/">Xonotic</a>, in the summer of 2015, when the community held a soundtrack competition. The soundtracks of this game had been predominately electronic music, and I wanted to challenge myself to write one with all traditional orchestral instruments, and for this particular game play.</p>
<p>Music for video game <em>play</em> is a unique genre. In concert music, which I mostly write, music is the focal point. In film scores or video game cut scene soundtrack, all the highlight events (hit points) are spotted in advanced so music can easily match the visual and development of the story. However, music for a live game play has to be generic enough to cover unexpected events, yet interesting enough not to cause aural fatigue. This is indeed challenging for me.</p>
<p>The title <em>Cry Link</em> is a word play of the game's unique weapon &quot;Crylink&quot;, in which &quot;cry&quot; is short for &quot;crystal&quot;, a lasing medium in the game. Compare to other weapons, Crylink does not deal the most damage; instead, its reflective projectile and the special force can pull back the opponents, making it very difficult for them to escape from a slow death. Such sense of doom of a courageous warrior inspired me to write &quot;Cry Link.&quot;</p>
<p>Instrumentation:</p>
<ul>
<li>2 flutes</li>
<li>2 oboes</li>
<li>2 Bb clarinets</li>
<li>2 bassoons</li>
<li>3-4 horns</li>
<li>2 trumpets</li>
<li>2 trombones</li>
<li>timpani</li>
<li>triangle</li>
<li>drum set</li>
<li>piano</li>
<li>full string sections (double-bass with C extension).</li>
</ul>
<h1>About the video</h1>
<ul>
<li>Game: <a href="https://xonotic.org/">Xonotic</a></li>
<li>Game Map: Erbium (by <a href="https://www.conradjustin.art/">Conrad 'Justin' Slawinski</a>)</li>
<li>Game Font: Xolonium (by <a href="https://github.com/sevmeyer">Severin 'Sev' Meyer</a>)</li>
<li>Players: SPLAT vs. smoke.antibody</li>
</ul>
<p>Although the main purpose of this video is to showcase the music, Xonotic players will enjoy watching these two virtuosi dueling in the video. This map just won a recent mapping contest so it's new to both players. Observe not only their incredible aiming and movement skills, but also how they quickly familiarize themselves with the map, identify the opponent's pattern, and develop strategies against each other. Also notice how one small wrong decision would lead to point loss much later on. Between two masters, there's little space for mistakes!</p>
<p>I chose Justin's map Erbium to showcase this piece because both our works blend opposite elements: nature growing in an industrial environment; traditional acoustic instruments for the futuristic action.</p>
<div class="youtube-widget" style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden; border-radius: 12px; margin: 1.5rem 0; box-shadow: 0 4px 24px rgba(0,0,0,0.2);">
<iframe src="https://www.youtube.com/embed/IAXUvnUaS-4" style="position: absolute; top: 0; left: 0; width: 100%; height: 100%; border: 0;" allowfullscreen=""></iframe>
</div>


    ]]></content>
  </entry><entry>
    <title>Sibelius 8 Installation on Non-standard Windows System Drive</title>
    <link href="https://blog.tklee.org/en/posts/2359-sibelius-8-installation-on-non-standard-windows-system-drive/"/>
    <updated>2015-08-05T23:59:00.001Z</updated>
    <id>https://blog.tklee.org/en/posts/2359-sibelius-8-installation-on-non-standard-windows-system-drive/</id>
    <content type="html"><![CDATA[
      <p>UPDATE: This doesn't work for Sibelius 8.0.1. See <a href="https://blog.tklee.org/posts/2015-10-19-2243-install-sibelius-8-0-1-on-a-non-c-drive-in-windows/">this post</a> instead for that.</p>
<p>Over the years I have many generations of audio software and plugins installed on Windows, which itself has gone through many version upgrades. I can't remember why but one upgrade called for changing the system drive from C: to K: (probably for a backup reason), and ever since then K: has been my system drive.</p>
<p>Between projects, I decided to give Sibelius 8 a try. However, the installation file (<code>Install_Sibelius.exe</code>, unpacked from <code>Sibelius_8.0.0_Win.zip</code>) downloaded from Avid gave me this error:</p>
<p><code>Error 1327 invalid drive C:\.</code></p>
<p>By inspecting the log, I concluded that was hard-coded in the installer (FAIL! To tell the truth, it has been a familiar feeling ever since Avid acquired Sibelius).</p>
<p>I really don't want to change the system drive again, since it would involve the painful process of re-configuring a million things. It's also after their support hours so I did some <a href="http://avid.force.com/pkb/articles/en_US/How_To/Sibelius-Silent-Install-Procedure">research</a>, and after some trial and error, this is what worked for me:</p>
<ol>
<li>run cmd.exe as Administrator.</li>
<li>Change into the directory where Install_Sibelius.exe resides</li>
<li>Run the following command</li>
</ol>
<p><code>J:\Users\composer\Desktop&gt;J:\Install_Sibelius.exe /s /v&quot;/qn /l*v %temp%\AvidSibeliusx64.log&quot;</code></p>
<ol start="4">
<li>Follow the standard procedure to activate the software</li>
</ol>
<p>That's it.</p>
<p>PS: I very much look forward to <a href="http://blog.steinberg.net/2015/06/development-diary-part-11/">Daniel Spreadbury's new score writer</a>.</p>
<div class="post-comments" style="margin-top: 4rem; padding-top: 2rem; border-top: 2px dashed var(--glass-border);">
<h3 class="comments-title" style="font-size: 1.5rem; margin-bottom: 2rem; color: var(--primary-color); text-shadow: 0 0 10px rgba(56,189,248,0.2);">Comments</h3>

<p></p><div class="conversation-other"><br />
<div class="comment-meta" style="font-size: 0.9em; color: var(--text-muted); margin-bottom: 0.8rem; display: flex; gap: 0.5rem; align-items: center;"><br />
<strong style="color: var(--text-main); font-weight: 600;"><a href="https://www.blogger.com/profile/14976089532177772917" target="_blank" rel="nofollow">Ralph</a></strong> <span class="comment-date" style="opacity: 0.7;">on Monday, August 19, 2019 at 6:16:00 PM EDT</span><br />
</div><br />
<div class="comment-body" style="line-height: 1.6; color: var(--text-main); opacity: 0.9;"><br />
<p>I'm still getting this error every time I need to update Sibelius.  It doesn't like ANY of the windows locations (e.g. DOcuments, Photos, Music) being mapped to a NAS drive.  It is massively frustrating and doesn't apply to any other piece of software I use!</p><p></p>
<p></p></div><br />
</div><p></p>
<p></p><div class="conversation-author"><br />
<div class="comment-meta" style="font-size: 0.9em; color: var(--text-muted); margin-bottom: 0.8rem; display: flex; gap: 0.5rem; align-items: center;"><br />
<strong style="color: var(--text-main); font-weight: 600;"><a href="https://www.blogger.com/profile/11411703278692679285" target="_blank" rel="nofollow">TK</a></strong> <span class="comment-date" style="opacity: 0.7;">on Tuesday, August 27, 2019 at 12:12:00 AM EDT</span><br />
</div><br />
<div class="comment-body" style="line-height: 1.6; color: var(--text-main); opacity: 0.9;"><br />
<p>This seems to be such a common requirement and yet they still haven't fixed the installation process.</p><p></p>
<p>However, I do have my score files on a NAS over the years but haven't got problems accessing them. What error messages are you getting?</p>
<p></p></div><br />
</div><p></p>
</div>

    ]]></content>
  </entry><entry>
    <title>X11 Multi-seat running KDE and GNOME</title>
    <link href="https://blog.tklee.org/en/posts/0243-x11-multi-seat-running-kde-and-gnome/"/>
    <updated>2015-05-07T02:43:00Z</updated>
    <id>https://blog.tklee.org/en/posts/0243-x11-multi-seat-running-kde-and-gnome/</id>
    <content type="html"><![CDATA[
      <p>I was archiving data on my old machine and decided to share my old multi-seat configurations (I don't need multi-seat anymore), in case somebody out there is trying to do the same thing.</p>
<p>It was a 7-year-old Dell PowerEdge server SC-1430 running dual Xeon (8 virtual cores), excellent for compiling, number crunching (machine learning), and video processing (Cinelerra), and on top of all that, it served as a file/media server running RAID 5 over RAID 0.</p>
<p>It was a power beast, in another sense, too, though -- it consumed a lot of power.</p>
<p>Since it was consuming power anyway, I decided to get rid of other computers and maximize its use, by setting up a multi-seat X11, so two users could use it at the same time. There were two sets of everything: two screens (on two NVIDIA cards), two keyboards, two mouses, and two USB sound cards. It worked just like two independent machines. In fact, one seat ran KDE and the other one ran GNOME. The login screen ran KDM, with two login boxes (default to the regular user login name) on respective screens and matching wallpapers. We were even able to play &quot;networked&quot; game against each other.</p>
<p>(Actually I had three sets of xorg.conf to support multi-seat, left-side dual screen, and right-side dual screen, and I used symlinks to dynamically switch to a different configuration. However, there's more info on dual screen setups out there so I skipped those here.)</p>
<p>For me, there were a few files to configure. These are the ones relevant to most people:</p>
<ul>
<li>/etc/X11/xorg.conf</li>
<li>/etc/kde/kdm/kdmrc</li>
<li>/etc/kde/kdm/backgroundrc.seat0</li>
<li>/etc/kde/kdm/backgroundrc.seat1</li>
<li>/etc/udev/rules.d/60-dell-keyboards.rules (omit here; see my <a href="https://blog.tklee.org/posts/2011-09-25-0400-create-distinguishable-device-ids-for-two-dell-keyboards/">other post on udev</a> for that)</li>
<li>/etc/udev/rules.d/90-creative-xmod.rules</li>
</ul>
<p>For the sound cards, I just used the GUI setup in KDM and Gnome, respectively, to choose the right pulse audio sound device. I did use the udev to make the names easier to tell them apart.</p>
<p>I ran Arch Linux 64 on that one so the paths may be different if you use a different distro. I apologize in advance for not explaining the details here, many of which were specific to my system back then. I also wish I had time to dig out the document sources for these configs.</p>
<details>
<summary>/etc/X11/xorg.conf</summary>
<pre class="language-apacheconf"><code class="language-apacheconf"><span class="highlight-line"><span class="token comment"># File: /etc/X11/xorg.conf</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># nvidia-settings: X configuration file generated by nvidia-settings</span></span>
<span class="highlight-line"><span class="token comment"># nvidia-settings:  version 280.11  (mockbuild@hephaestus.wilsonet.com)  Sun Jul 31 18:18:43 EDT 2011</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">#Section "ServerLayout"</span></span>
<span class="highlight-line"><span class="token comment">#    Identifier     "Layout0"</span></span>
<span class="highlight-line"><span class="token comment">#    Screen      0  "Screen0" 0 0</span></span>
<span class="highlight-line"><span class="token comment">#    Screen      1  "Screen1" RightOf "Screen0"</span></span>
<span class="highlight-line"><span class="token comment">#    InputDevice    "Keyboard0" "CoreKeyboard"</span></span>
<span class="highlight-line"><span class="token comment">#    InputDevice    "Mouse0" "CorePointer"</span></span>
<span class="highlight-line"><span class="token comment">#    Option         "Xinerama" "0"</span></span>
<span class="highlight-line"><span class="token comment">#EndSection</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"ServerLayout"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Seat0"</span></span>
<span class="highlight-line">Screen      0  <span class="token string">"Screen0"</span> 0 0</span>
<span class="highlight-line">InputDevice    <span class="token string">"Keyboard0" "CoreKeyboard"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"Keyboard0.1"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"webcam_Button"</span></span>
<span class="highlight-line"><span class="token comment">#InputDevice    "X10_Lola.keyboard"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"XmodControl.0"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"Mouse0.Naga" "CorePointer"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"MouseKeys0a.Naga"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"Mouse0.1"</span></span>
<span class="highlight-line"><span class="token comment">#InputDevice    "MouseKeys0b.Naga"</span></span>
<span class="highlight-line">Option         <span class="token string">"Xinerama" "0"</span></span>
<span class="highlight-line">Option         <span class="token string">"AutoEnableDevices" "false"</span></span>
<span class="highlight-line">Option         <span class="token string">"AutoAddDevices"   "false"</span></span>
<span class="highlight-line">Option         <span class="token string">"AllowEmptyInput" "true"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"ServerLayout"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Seat1"</span></span>
<span class="highlight-line">Screen      0  <span class="token string">"Screen1"</span> 0 0</span>
<span class="highlight-line">InputDevice    <span class="token string">"Keyboard1" "CoreKeyboard"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"XmodControl.1"</span></span>
<span class="highlight-line">InputDevice    <span class="token string">"Mouse1.Salmosa" "CorePointer"</span></span>
<span class="highlight-line">Option         <span class="token string">"Xinerama" "0"</span></span>
<span class="highlight-line">Option         <span class="token string">"AutoEnableDevices" "false"</span></span>
<span class="highlight-line">Option         <span class="token string">"AutoAddDevices"   "false"</span></span>
<span class="highlight-line">Option         <span class="token string">"AllowEmptyInput" "true"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"Files"</span></span>
<span class="highlight-line">FontPath        <span class="token string">"/usr/share/fonts/default/Type1"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Mouse0.Naga"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option         <span class="token string">"GrabDevice" "true"</span></span>
<span class="highlight-line">Option         <span class="token string">"Protocol" "auto"</span></span>
<span class="highlight-line">Option   <span class="token string">"Device" "/dev/input/by-id/usb-Razer_Razer_Naga-event-mouse"</span></span>
<span class="highlight-line">Option         <span class="token string">"Emulate3Buttons" "no"</span></span>
<span class="highlight-line">Option         <span class="token string">"ZAxisMapping" "4 5"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"MouseKeys0a.Naga"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/by-id/usb-Razer_Razer_Naga-event-kbd"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"MouseKeys0b.Naga"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/by-id/usb-Razer_Razer_Naga-if01-event-kbd"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Mouse0.1"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/by-id/usb-Logitech_USB_Receiver-event-mouse"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Mouse1.Salmosa"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option         <span class="token string">"GrabDevice" "true"</span></span>
<span class="highlight-line">Option         <span class="token string">"Protocol" "auto"</span></span>
<span class="highlight-line">Option   <span class="token string">"Device" "/dev/input/by-id/usb-Razer_Razer_Salmosa-event-mouse"</span></span>
<span class="highlight-line">Option         <span class="token string">"Emulate3Buttons" "no"</span></span>
<span class="highlight-line">Option         <span class="token string">"ZAxisMapping" "4 5"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">#Section "InputDevice"</span></span>
<span class="highlight-line"><span class="token comment">#    # generated from data in "/etc/sysconfig/keyboard"</span></span>
<span class="highlight-line"><span class="token comment">#    Identifier     "Keyboard0"</span></span>
<span class="highlight-line"><span class="token comment">#    Driver         "keyboard"</span></span>
<span class="highlight-line"><span class="token comment">#    Option         "XkbLayout" "us"</span></span>
<span class="highlight-line"><span class="token comment">#    Option         "XkbModel" "pc105"</span></span>
<span class="highlight-line"><span class="token comment">#EndSection</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Keyboard0"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/dell_keyboard_2003"</span></span>
<span class="highlight-line">Option         <span class="token string">"XkbLayout" "us"</span></span>
<span class="highlight-line">Option         <span class="token string">"XkbModel" "pc105"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Keyboard0.1"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/by-id/usb-Logitech_USB_Receiver-event-kbd"</span></span>
<span class="highlight-line">Option         <span class="token string">"XkbLayout" "us"</span></span>
<span class="highlight-line">Option         <span class="token string">"XkbModel" "pc105"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Keyboard1"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/dell_keyboard_2105"</span></span>
<span class="highlight-line">Option         <span class="token string">"XkbLayout" "us"</span></span>
<span class="highlight-line">Option         <span class="token string">"XkbModel" "pc105"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"webcam_Button"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/by-id/usb-046d_0809_BD4BC0AA-event-if00"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"X10_Lola.keyboard"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/x10_lola"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"XmodControl.0"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/by-path/pci-0000:00:1d.2-usb-0:1.2:1.3-event"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"InputDevice"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"XmodControl.1"</span></span>
<span class="highlight-line">Driver         <span class="token string">"evdev"</span></span>
<span class="highlight-line">Option     <span class="token string">"Device" "/dev/input/by-path/pci-0000:00:1d.0-usb-0:1:1.3-event"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">#Section "Monitor"</span></span>
<span class="highlight-line"><span class="token comment">#    # HorizSync source: edid, VertRefresh source: edid</span></span>
<span class="highlight-line"><span class="token comment">#    Identifier     "Monitor0"</span></span>
<span class="highlight-line"><span class="token comment">#    VendorName     "Unknown"</span></span>
<span class="highlight-line"><span class="token comment">#    ModelName      "DELL 1907FP"</span></span>
<span class="highlight-line"><span class="token comment">#    HorizSync       30.0 - 81.0</span></span>
<span class="highlight-line"><span class="token comment">#    VertRefresh     56.0 - 76.0</span></span>
<span class="highlight-line"><span class="token comment">#    Option         "DPMS"</span></span>
<span class="highlight-line"><span class="token comment">#EndSection</span></span>
<span class="highlight-line">Section <span class="token string">"Monitor"</span></span>
<span class="highlight-line"><span class="token comment"># HorizSync source: edid, VertRefresh source: edid</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Monitor0"</span></span>
<span class="highlight-line">VendorName     <span class="token string">"Unknown"</span></span>
<span class="highlight-line">ModelName      <span class="token string">"ViewSonic VA2431 Series"</span></span>
<span class="highlight-line">HorizSync       24.0 - 82.0</span>
<span class="highlight-line">VertRefresh     50.0 - 75.0</span>
<span class="highlight-line">Option         <span class="token string">"DPMS"</span></span>
<span class="highlight-line"><span class="token comment">#Option          "BlankTime"     "4"</span></span>
<span class="highlight-line"><span class="token comment">#Option          "StandbyTime"   "1"</span></span>
<span class="highlight-line"><span class="token comment">#Option          "SuspendTime"   "1"</span></span>
<span class="highlight-line"><span class="token comment">#Option          "OffTime"       "5"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"Monitor"</span></span>
<span class="highlight-line"><span class="token comment"># HorizSync source: edid, VertRefresh source: edid</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Monitor1"</span></span>
<span class="highlight-line">VendorName     <span class="token string">"Unknown"</span></span>
<span class="highlight-line">ModelName      <span class="token string">"DELL 1901FP"</span></span>
<span class="highlight-line">HorizSync       30.0 - 80.0</span>
<span class="highlight-line">VertRefresh     56.0 - 76.0</span>
<span class="highlight-line">Option         <span class="token string">"DPMS"</span></span>
<span class="highlight-line">Option          <span class="token string">"BlankTime"     "4"</span></span>
<span class="highlight-line">Option          <span class="token string">"StandbyTime"   "1"</span></span>
<span class="highlight-line">Option          <span class="token string">"SuspendTime"   "1"</span></span>
<span class="highlight-line">Option          <span class="token string">"OffTime"       "5"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"Device"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Device0"</span></span>
<span class="highlight-line">Driver         <span class="token string">"nvidia"</span></span>
<span class="highlight-line">VendorName     <span class="token string">"NVIDIA Corporation"</span></span>
<span class="highlight-line">BoardName      <span class="token string">"GeForce 8500 GT"</span></span>
<span class="highlight-line">BusID          <span class="token string">"PCI:4:0:0"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"Device"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Device1"</span></span>
<span class="highlight-line">Driver         <span class="token string">"nvidia"</span></span>
<span class="highlight-line">VendorName     <span class="token string">"NVIDIA Corporation"</span></span>
<span class="highlight-line">BoardName      <span class="token string">"ION"</span></span>
<span class="highlight-line">BusID          <span class="token string">"PCI:7:0:0"</span></span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"Screen"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Screen0"</span></span>
<span class="highlight-line">Device         <span class="token string">"Device0"</span></span>
<span class="highlight-line">Monitor        <span class="token string">"Monitor0"</span></span>
<span class="highlight-line">DefaultDepth    24</span>
<span class="highlight-line">Option         <span class="token string">"TwinView" "0"</span></span>
<span class="highlight-line">Option         <span class="token string">"metamodes" "nvidia-auto-select +0+0"</span></span>
<span class="highlight-line">SubSection     <span class="token string">"Display"</span></span>
<span class="highlight-line">Depth       24</span>
<span class="highlight-line">EndSubSection</span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line">Section <span class="token string">"Screen"</span></span>
<span class="highlight-line">Identifier     <span class="token string">"Screen1"</span></span>
<span class="highlight-line">Device         <span class="token string">"Device1"</span></span>
<span class="highlight-line">Monitor        <span class="token string">"Monitor1"</span></span>
<span class="highlight-line">DefaultDepth    24</span>
<span class="highlight-line">Option         <span class="token string">"TwinView" "0"</span></span>
<span class="highlight-line">Option         <span class="token string">"metamodes" "nvidia-auto-select +0+0"</span></span>
<span class="highlight-line">SubSection     <span class="token string">"Display"</span></span>
<span class="highlight-line">Depth       24</span>
<span class="highlight-line">EndSubSection</span>
<span class="highlight-line">EndSection</span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">#Section "ServerFlags"</span></span>
<span class="highlight-line"><span class="token comment"># # Even if mouse detection fails, X will start</span></span>
<span class="highlight-line"><span class="token comment"># Option "AllowMouseOpenFail" "yes"</span></span>
<span class="highlight-line"><span class="token comment">#</span></span>
<span class="highlight-line"><span class="token comment"># # VT switching is disabled</span></span>
<span class="highlight-line"><span class="token comment"># Option "DontVTSwitch" "yes"</span></span>
<span class="highlight-line"><span class="token comment">#</span></span>
<span class="highlight-line"><span class="token comment"># # X restart (Ctrl+Alt+Backspace) is disabled</span></span>
<span class="highlight-line"><span class="token comment"># # For some reason, this would cause Ctrl-C to terminate X</span></span>
<span class="highlight-line"><span class="token comment"># #Option "DontZap" "yes"</span></span>
<span class="highlight-line"><span class="token comment">#EndSection</span></span></code></pre>
</details>
<details>
<summary>/usr/share/config/kdm/kdmrc</summary>
<pre class="language-apacheconf"><code class="language-apacheconf"><span class="highlight-line"><span class="token comment"># File: /usr/share/config/kdm/kdmrc</span></span>
<span class="highlight-line"><span class="token comment">#</span></span>
<span class="highlight-line"><span class="token comment"># KDM master configuration file</span></span>
<span class="highlight-line"><span class="token comment">#</span></span>
<span class="highlight-line"><span class="token comment"># Definition: the greeter is the login dialog, i.e., the part of KDM</span></span>
<span class="highlight-line"><span class="token comment"># which the user sees.</span></span>
<span class="highlight-line"><span class="token comment">#</span></span>
<span class="highlight-line"><span class="token comment"># You can configure every X-display individually.</span></span>
<span class="highlight-line"><span class="token comment"># Every display has a display name, which consists of a host name</span></span>
<span class="highlight-line"><span class="token comment"># (which is empty for local displays specified in {Static|Reserve}Servers),</span></span>
<span class="highlight-line"><span class="token comment"># a colon, and a display number. Additionally, a display belongs to a</span></span>
<span class="highlight-line"><span class="token comment"># display class (which can be ignored in most cases; the control center</span></span>
<span class="highlight-line"><span class="token comment"># does not support this feature at all).</span></span>
<span class="highlight-line"><span class="token comment"># Sections with display-specific settings have the formal syntax</span></span>
<span class="highlight-line"><span class="token comment"># "[X-" host [":" number [ "_" class ]] "-" sub-section "]"</span></span>
<span class="highlight-line"><span class="token comment"># You can use the "*" wildcard for host, number, and class. You may omit</span></span>
<span class="highlight-line"><span class="token comment"># trailing components; they are assumed to be "*" then.</span></span>
<span class="highlight-line"><span class="token comment"># The host part may be a domain specification like ".inf.tu-dresden.de".</span></span>
<span class="highlight-line"><span class="token comment"># It may also be "+", which means non-empty, i.e. remote displays only.</span></span>
<span class="highlight-line"><span class="token comment"># From which section a setting is actually taken is determined by these</span></span>
<span class="highlight-line"><span class="token comment"># rules:</span></span>
<span class="highlight-line"><span class="token comment"># - an exact match takes precedence over a partial match (for the host part),</span></span>
<span class="highlight-line"><span class="token comment">#   which in turn takes precedence over a wildcard ("+" taking precedence</span></span>
<span class="highlight-line"><span class="token comment">#   over "*")</span></span>
<span class="highlight-line"><span class="token comment"># - precedence decreases from left to right for equally exact matches</span></span>
<span class="highlight-line"><span class="token comment"># Example: display name "myhost:0", class "dpy".</span></span>
<span class="highlight-line"><span class="token comment"># [X-myhost:0_dpy] precedes</span></span>
<span class="highlight-line"><span class="token comment"># [X-myhost:0_*] (same as [X-myhost:0]) precedes</span></span>
<span class="highlight-line"><span class="token comment"># [X-myhost:*_dpy] precedes</span></span>
<span class="highlight-line"><span class="token comment"># [X-myhost:*_*] (same as [X-myhost]) precedes</span></span>
<span class="highlight-line"><span class="token comment"># [X-+:0_dpy] precedes</span></span>
<span class="highlight-line"><span class="token comment"># [X-*:0_dpy] precedes</span></span>
<span class="highlight-line"><span class="token comment"># [X-*:0_*] (same as [X-*:0]) precedes</span></span>
<span class="highlight-line"><span class="token comment"># [X-*:*_*] (same as [X-*])</span></span>
<span class="highlight-line"><span class="token comment"># These sections do NOT match this display:</span></span>
<span class="highlight-line"><span class="token comment"># [X-hishost], [X-myhost:0_dec], [X-*:1], [X-:*]</span></span>
<span class="highlight-line"><span class="token comment"># If a setting is not found in any matching section, the default is used.</span></span>
<span class="highlight-line"><span class="token comment">#</span></span>
<span class="highlight-line"><span class="token comment"># Every comment applies to the following section or key. Note that all</span></span>
<span class="highlight-line"><span class="token comment"># comments will be lost if you change this file with the systemsettings frontend.</span></span>
<span class="highlight-line"><span class="token comment"># The defaults refer to KDM's built-in values, not anything set in this file.</span></span>
<span class="highlight-line"><span class="token comment">#</span></span>
<span class="highlight-line"><span class="token comment"># Special characters need to be backslash-escaped (leading and trailing</span></span>
<span class="highlight-line"><span class="token comment"># spaces (\s), tab (\t), linefeed (\n), carriage return (\r) and the</span></span>
<span class="highlight-line"><span class="token comment"># backslash itself (\\)).</span></span>
<span class="highlight-line"><span class="token comment"># In lists, fields are separated with commas without whitespace in between.</span></span>
<span class="highlight-line"><span class="token comment"># Some command strings are subject to simplified sh-style word splitting:</span></span>
<span class="highlight-line"><span class="token comment"># single quotes (') and double quotes (") have the usual meaning; the backslash</span></span>
<span class="highlight-line"><span class="token comment"># quotes everything (not only special characters). Note that the backslashes</span></span>
<span class="highlight-line"><span class="token comment"># need to be doubled because of the two levels of quoting.</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token directive-flags keyword">[General]</span></span>
<span class="highlight-line"><span class="token comment"># This option exists solely for the purpose of a clean automatic upgrade.</span></span>
<span class="highlight-line"><span class="token comment"># Do not even think about changing it!</span></span>
<span class="highlight-line">ConfigVersion=2.4</span>
<span class="highlight-line"><span class="token comment"># List of permanent displays. Displays with a hostname are foreign. A display</span></span>
<span class="highlight-line"><span class="token comment"># class may be specified separated by an underscore.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ":0"</span></span>
<span class="highlight-line">StaticServers=:0,:1</span>
<span class="highlight-line"><span class="token comment"># List of on-demand displays. See StaticServers for syntax.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">ReserveServers=:2,:3,:4</span>
<span class="highlight-line"><span class="token comment"># VTs to allocate to X-servers. A negative number means that the VT will be</span></span>
<span class="highlight-line"><span class="token comment"># used only if it is free. If all VTs in this list are used up, the next free</span></span>
<span class="highlight-line"><span class="token comment"># one greater than the last one in this list will be allocated.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">ServerVTs=-1</span>
<span class="highlight-line"><span class="token comment"># TTYs (without /dev/) to monitor for activity while in console mode.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">ConsoleTTYs=tty1,tty2,tty3,tty4,tty5,tty6</span>
<span class="highlight-line"><span class="token comment"># Where KDM should store its PID (do not store if empty).</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token directive-inline property">PidFile</span>=/var/run/kdm.pid</span>
<span class="highlight-line"><span class="token comment"># Whether KDM should lock the PID file to prevent having multiple KDM</span></span>
<span class="highlight-line"><span class="token comment"># instances running at once. Do not change unless you are brave.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#LockPidFile=false</span></span>
<span class="highlight-line"><span class="token comment"># Where to store authorization files.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/var/run/xauth"</span></span>
<span class="highlight-line"><span class="token comment">#AuthDir=/tmp</span></span>
<span class="highlight-line"><span class="token comment"># Whether KDM should automatically re-read configuration files, if it</span></span>
<span class="highlight-line"><span class="token comment"># finds them having changed.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#AutoRescan=false</span></span>
<span class="highlight-line"><span class="token comment"># Additional environment variables KDM should pass on to all programs it runs.</span></span>
<span class="highlight-line"><span class="token comment"># LD_LIBRARY_PATH and XCURSOR_THEME are good candidates;</span></span>
<span class="highlight-line"><span class="token comment"># otherwise, it should not be necessary very often.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#ExportList=LD_LIBRARY_PATH,ANOTHER_IMPORTANT_VAR</span></span>
<span class="highlight-line"><span class="token comment"># A character device KDM should read entropy from.</span></span>
<span class="highlight-line"><span class="token comment"># Empty means use the system's preferred entropy device.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#RandomDevice=/dev/altrandom</span></span>
<span class="highlight-line"><span class="token comment"># Where the command sockets should be created; make it empty to disable</span></span>
<span class="highlight-line"><span class="token comment"># them.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/var/run/xdmctl"</span></span>
<span class="highlight-line"><span class="token comment">#FifoDir=/tmp</span></span>
<span class="highlight-line"><span class="token comment"># The group to which the global command socket should belong;</span></span>
<span class="highlight-line"><span class="token comment"># can be either a name or a numerical ID.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 0</span></span>
<span class="highlight-line"><span class="token comment">#FifoGroup=xdmctl</span></span>
<span class="highlight-line"><span class="token comment"># The user the greeter should run as. Empty results in root.</span></span>
<span class="highlight-line"><span class="token comment"># Consider the impact on LogSource when setting it.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">GreeterUID=kdm</span>
<span class="highlight-line"><span class="token comment"># The directory in which KDM should store persistent working data.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/var/lib/kdm"</span></span>
<span class="highlight-line"><span class="token comment">#DataDir=</span></span>
<span class="highlight-line"><span class="token comment"># The directory in which KDM should store users' .dmrc files. This is only</span></span>
<span class="highlight-line"><span class="token comment"># needed if the home directories are not readable before actually logging in</span></span>
<span class="highlight-line"><span class="token comment"># (like with AFS).</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#DmrcDir=/nfs-shared/var/dmrcs</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token directive-flags keyword">[Xdmcp]</span></span>
<span class="highlight-line"><span class="token comment"># Whether KDM should listen to incoming XDMCP requests.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line">Enable=false</span>
<span class="highlight-line"><span class="token comment"># The UDP port on which KDM should listen for XDMCP requests. Do not change.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 177</span></span>
<span class="highlight-line"><span class="token comment">#Port=177</span></span>
<span class="highlight-line"><span class="token comment"># File with the private keys of X-terminals. Required for XDM authentication.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#KeyFile=/usr/share/config/kdm/kdmkeys</span></span>
<span class="highlight-line"><span class="token comment"># XDMCP access control file in the usual XDM-Xaccess format.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/share/config/kdm/Xaccess"</span></span>
<span class="highlight-line">Xaccess=/usr/share/config/kdm/Xaccess</span>
<span class="highlight-line"><span class="token comment"># Number of seconds to wait for display to respond after the user has</span></span>
<span class="highlight-line"><span class="token comment"># selected a host from the chooser.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 15</span></span>
<span class="highlight-line"><span class="token comment">#ChoiceTimeout=10</span></span>
<span class="highlight-line"><span class="token comment"># Strip domain name from remote display names if it is equal to the local</span></span>
<span class="highlight-line"><span class="token comment"># domain.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#RemoveDomainname=false</span></span>
<span class="highlight-line"><span class="token comment"># Use the numeric IP address of the incoming connection on multihomed hosts</span></span>
<span class="highlight-line"><span class="token comment"># instead of the host name.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#SourceAddress=true</span></span>
<span class="highlight-line"><span class="token comment"># The program which is invoked to dynamically generate replies to XDMCP</span></span>
<span class="highlight-line"><span class="token comment"># DirectQuery or BroadcastQuery requests.</span></span>
<span class="highlight-line"><span class="token comment"># If empty, no program is invoked and "Willing to manage" is sent.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">Willing=/usr/share/config/kdm/Xwilling</span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token directive-flags keyword">[Shutdown]</span></span>
<span class="highlight-line"><span class="token comment"># The command (subject to word splitting) to run to halt the system.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/sbin/shutdown -h -P now"</span></span>
<span class="highlight-line">HaltCmd=/sbin/poweroff</span>
<span class="highlight-line"><span class="token comment"># The command (subject to word splitting) to run to reboot the system.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/sbin/shutdown -r now"</span></span>
<span class="highlight-line">RebootCmd=/sbin/shutdown -r now</span>
<span class="highlight-line"><span class="token comment"># Whether it is allowed to shut down the system via the global command socket.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#AllowFifo=true</span></span>
<span class="highlight-line"><span class="token comment"># Whether it is allowed to abort active sessions when shutting down the</span></span>
<span class="highlight-line"><span class="token comment"># system via the global command socket.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#AllowFifoNow=false</span></span>
<span class="highlight-line"><span class="token comment"># The boot manager KDM should use for offering boot options in the</span></span>
<span class="highlight-line"><span class="token comment"># shutdown dialog.</span></span>
<span class="highlight-line"><span class="token comment"># "None" - no boot manager</span></span>
<span class="highlight-line"><span class="token comment"># "Grub" - Grub boot manager</span></span>
<span class="highlight-line"><span class="token comment"># "Grub2" - Grub2 boot manager</span></span>
<span class="highlight-line"><span class="token comment"># "Burg" - Burg boot manager</span></span>
<span class="highlight-line"><span class="token comment"># "Lilo" - Lilo boot manager (Linux on i386 &amp; x86-64 only)</span></span>
<span class="highlight-line"><span class="token comment"># Default is None</span></span>
<span class="highlight-line"><span class="token comment">#BootManager=Grub</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># Rough estimations about how many seconds KDM will spend at most on</span></span>
<span class="highlight-line"><span class="token comment"># - opening a connection to the X-server (OpenTime) if the attempt</span></span>
<span class="highlight-line"><span class="token comment">#   - times out: OpenTimeout</span></span>
<span class="highlight-line"><span class="token comment">#   - is refused: OpenRepeat * OpenDelay</span></span>
<span class="highlight-line"><span class="token comment"># - starting a local X-server (ServerTime):</span></span>
<span class="highlight-line"><span class="token comment">#   ServerAttempts * (ServerTimeout + OpenDelay)</span></span>
<span class="highlight-line"><span class="token comment"># - starting a display:</span></span>
<span class="highlight-line"><span class="token comment">#   - local display: ServerTime + OpenTime</span></span>
<span class="highlight-line"><span class="token comment">#   - foreign display: StartAttempts * OpenTime</span></span>
<span class="highlight-line"><span class="token comment">#   - XDMCP display: OpenTime (repeated indefinitely by client)</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># Core config for all displays</span></span>
<span class="highlight-line">[X-*-Core]</span>
<span class="highlight-line"><span class="token comment"># How long to wait before retrying to connect a display.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 15</span></span>
<span class="highlight-line"><span class="token comment">#OpenDelay=15</span></span>
<span class="highlight-line"><span class="token comment"># How long to wait before timing out a display connection attempt.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 120</span></span>
<span class="highlight-line"><span class="token comment">#OpenTimeout=120</span></span>
<span class="highlight-line"><span class="token comment"># How many connection attempts to make during a start attempt. Note that</span></span>
<span class="highlight-line"><span class="token comment"># a timeout aborts the entire start attempt.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 5</span></span>
<span class="highlight-line"><span class="token comment">#OpenRepeat=5</span></span>
<span class="highlight-line"><span class="token comment"># Try at most that many times to start a display. If this fails, the display</span></span>
<span class="highlight-line"><span class="token comment"># is disabled.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 4</span></span>
<span class="highlight-line"><span class="token comment">#StartAttempts=4</span></span>
<span class="highlight-line"><span class="token comment"># Ping remote display every that many minutes.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 5</span></span>
<span class="highlight-line"><span class="token comment">#PingInterval=5</span></span>
<span class="highlight-line"><span class="token comment"># Wait for a Pong that many minutes.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 5</span></span>
<span class="highlight-line"><span class="token comment">#PingTimeout=5</span></span>
<span class="highlight-line"><span class="token comment"># The name of this X-server's Xauth file.</span></span>
<span class="highlight-line"><span class="token comment"># If empty, a random name in the AuthDir directory will be used.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#AuthFile=</span></span>
<span class="highlight-line"><span class="token comment"># Specify a file with X-resources for the greeter, chooser and background.</span></span>
<span class="highlight-line"><span class="token comment"># The KDE frontend does not use this file, so you do not need it unless you</span></span>
<span class="highlight-line"><span class="token comment"># use another background generator than krootimage.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#Resources=</span></span>
<span class="highlight-line"><span class="token comment"># The xrdb program to use to read the above specified recources.</span></span>
<span class="highlight-line"><span class="token comment"># Subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/bin/xrdb"</span></span>
<span class="highlight-line"><span class="token comment">#Xrdb=</span></span>
<span class="highlight-line"><span class="token comment"># A program to run before the greeter is shown. Can be used to start an</span></span>
<span class="highlight-line"><span class="token comment"># xconsole or an alternative background generator. Subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">Setup=/usr/share/config/kdm/Xsetup</span>
<span class="highlight-line"><span class="token comment"># A program to run before a user session starts. Subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">Startup=/usr/share/config/kdm/Xstartup</span>
<span class="highlight-line"><span class="token comment"># A program to run after a user session exits. Subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">Reset=/usr/share/config/kdm/Xreset</span>
<span class="highlight-line"><span class="token comment"># The program which is run as the user which logs in. It is supposed to</span></span>
<span class="highlight-line"><span class="token comment"># interpret the session argument (see SessionsDirs) and start an appropriate</span></span>
<span class="highlight-line"><span class="token comment"># session according to it. Subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/bin/xterm -ls -T"</span></span>
<span class="highlight-line"><span class="token directive-inline property">Session</span>=/usr/share/config/kdm/Xsession</span>
<span class="highlight-line"><span class="token comment"># The program to run if Session fails.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/bin/xterm"</span></span>
<span class="highlight-line"><span class="token comment">#FailsafeClient=</span></span>
<span class="highlight-line"><span class="token comment"># The PATH for the Session program.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/local/bin:/usr/bin:/bin:/usr/games"</span></span>
<span class="highlight-line"><span class="token comment">#UserPath=</span></span>
<span class="highlight-line"><span class="token comment"># The PATH for Setup, Startup and Reset, etc.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin"</span></span>
<span class="highlight-line"><span class="token comment">#SystemPath=</span></span>
<span class="highlight-line"><span class="token comment"># The default system shell.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/bin/sh"</span></span>
<span class="highlight-line"><span class="token comment">#SystemShell=/bin/bash</span></span>
<span class="highlight-line"><span class="token comment"># Where to put the user's X-server authorization file if ~/.Xauthority</span></span>
<span class="highlight-line"><span class="token comment"># cannot be created.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/tmp"</span></span>
<span class="highlight-line"><span class="token comment">#UserAuthDir=</span></span>
<span class="highlight-line"><span class="token comment"># If true, UserAuthDir will be used unconditionally.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#ForceUserAuthDir=true</span></span>
<span class="highlight-line"><span class="token comment"># Whether to automatically restart sessions after X-server crashes.</span></span>
<span class="highlight-line"><span class="token comment"># Note that enabling this makes circumventing screen lockers other than</span></span>
<span class="highlight-line"><span class="token comment"># KDE's built-in one possible!</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#AutoReLogin=true</span></span>
<span class="highlight-line"><span class="token comment"># Allow root logins?</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line">AllowRootLogin=false</span>
<span class="highlight-line"><span class="token comment"># Allow to log in, when user has set an empty password?</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line">AllowNullPasswd=false</span>
<span class="highlight-line"><span class="token comment"># Who is allowed to shut down the system. This applies both to the</span></span>
<span class="highlight-line"><span class="token comment"># greeter and to the command sockets.</span></span>
<span class="highlight-line"><span class="token comment"># "None" - no "Shutdown..." menu entry is shown at all</span></span>
<span class="highlight-line"><span class="token comment"># "Root" - the root password must be entered to shut down</span></span>
<span class="highlight-line"><span class="token comment"># "All" - everybody can shut down the machine</span></span>
<span class="highlight-line"><span class="token comment"># Default is All</span></span>
<span class="highlight-line">AllowShutdown=Root</span>
<span class="highlight-line"><span class="token comment"># Who is allowed to abort active sessions when shutting down.</span></span>
<span class="highlight-line"><span class="token comment"># "None" - no forced shutdown is allowed at all</span></span>
<span class="highlight-line"><span class="token comment"># "Root" - the root password must be entered to shut down forcibly</span></span>
<span class="highlight-line"><span class="token comment"># "All" - everybody can shut down the machine forcibly</span></span>
<span class="highlight-line"><span class="token comment"># Default is All</span></span>
<span class="highlight-line"><span class="token comment">#AllowSdForceNow=Root</span></span>
<span class="highlight-line"><span class="token comment"># The default choice for the shutdown condition/timing.</span></span>
<span class="highlight-line"><span class="token comment"># "Schedule" - shut down after all active sessions exit (possibly at once)</span></span>
<span class="highlight-line"><span class="token comment"># "TryNow" - shut down, if no active sessions are open; otherwise, do nothing</span></span>
<span class="highlight-line"><span class="token comment"># "ForceNow" - shut down unconditionally</span></span>
<span class="highlight-line"><span class="token comment"># Default is Schedule</span></span>
<span class="highlight-line"><span class="token comment">#DefaultSdMode=ForceNow</span></span>
<span class="highlight-line"><span class="token comment"># How to offer shutdown scheduling options:</span></span>
<span class="highlight-line"><span class="token comment"># "Never" - not at all</span></span>
<span class="highlight-line"><span class="token comment"># "Optional" - as a button in the simple shutdown dialogs</span></span>
<span class="highlight-line"><span class="token comment"># "Always" - instead of the simple shutdown dialogs</span></span>
<span class="highlight-line"><span class="token comment"># Default is Never</span></span>
<span class="highlight-line"><span class="token comment">#ScheduledSd=Optional</span></span>
<span class="highlight-line"><span class="token comment"># The directories containing session type definitions in .desktop format,</span></span>
<span class="highlight-line"><span class="token comment"># ordered by falling priority.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/share/apps/kdm/sessions"</span></span>
<span class="highlight-line">SessionsDirs=/usr/share/config/kdm/sessions,/usr/share/apps/kdm/sessions</span>
<span class="highlight-line"><span class="token comment"># The file (relative to $HOME) to redirect the session output to. The</span></span>
<span class="highlight-line"><span class="token comment"># following character pairs are replaced:</span></span>
<span class="highlight-line"><span class="token comment"># - %d -> current display</span></span>
<span class="highlight-line"><span class="token comment"># - %u -> current user</span></span>
<span class="highlight-line"><span class="token comment"># - %r -> empty at first. See below.</span></span>
<span class="highlight-line"><span class="token comment"># - %% -> a single %</span></span>
<span class="highlight-line"><span class="token comment"># When the constructed filename cannot be used safely and the specification</span></span>
<span class="highlight-line"><span class="token comment"># contains %&lt;stuff>r, other names will be tried - this time expanding %&lt;stuff>r</span></span>
<span class="highlight-line"><span class="token comment"># to &lt;stuff> followed by a random number.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ".xsession-errors"</span></span>
<span class="highlight-line">ClientLogFile=.xsession-errors-<span class="token variable">%d</span></span>
<span class="highlight-line"><span class="token comment"># Fallback when ClientLogFile cannot be used. The same expansions are</span></span>
<span class="highlight-line"><span class="token comment"># supported. DO NOT use relative paths here.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/tmp/xerr-%u-%d%-r"</span></span>
<span class="highlight-line"><span class="token comment">#ClientLogFallback=/var/log/xsession-errors/%u-%d%-r</span></span>
<span class="highlight-line"><span class="token comment"># Whether KDM's built-in utmp/wtmp/lastlog registration should be used.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#UseSessReg=false</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># Greeter config for all displays</span></span>
<span class="highlight-line">[X-*-Greeter]</span>
<span class="highlight-line"><span class="token comment"># Specify the widget style for the greeter. Empty means to use the</span></span>
<span class="highlight-line"><span class="token comment"># built-in default which currently is "Oxygen-air".</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#GUIStyle=Plastique</span></span>
<span class="highlight-line"><span class="token comment"># Specify the widget color scheme for the greeter. Empty means to use the</span></span>
<span class="highlight-line"><span class="token comment"># built-in default which currently is "Oxygen-air".</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#ColorScheme=MidnightMeadow</span></span>
<span class="highlight-line"><span class="token comment"># What should be shown in the greeter's logo are:</span></span>
<span class="highlight-line"><span class="token comment"># "None" - nothing</span></span>
<span class="highlight-line"><span class="token comment"># "Logo" - the image specified by LogoPixmap</span></span>
<span class="highlight-line"><span class="token comment"># "Clock" - a neat analog clock</span></span>
<span class="highlight-line"><span class="token comment"># Default is Clock</span></span>
<span class="highlight-line">LogoArea=Logo</span>
<span class="highlight-line"><span class="token comment"># The image to show when LogoArea=Logo.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">LogoPixmap=/usr/share/apps/kdm/pics/kdelogo.png</span>
<span class="highlight-line"><span class="token comment"># The relative coordinates (X,Y in percent) of the center of the greeter.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "50,50"</span></span>
<span class="highlight-line"><span class="token comment">#GreeterPos=30,40</span></span>
<span class="highlight-line"><span class="token comment"># The screen the greeter should be displayed on in multi-headed and Xinerama</span></span>
<span class="highlight-line"><span class="token comment"># setups. The numbering starts with 0. For Xinerama, it corresponds to the</span></span>
<span class="highlight-line"><span class="token comment"># listing order in the active ServerLayout section of XF86Config; -1 means</span></span>
<span class="highlight-line"><span class="token comment"># to use the upper-left screen, -2 means to use the upper-right screen.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 0</span></span>
<span class="highlight-line"><span class="token comment">#GreeterScreen=-1</span></span>
<span class="highlight-line"><span class="token comment"># The headline in the greeter. The following character pairs are replaced:</span></span>
<span class="highlight-line"><span class="token comment"># - %d -> current display</span></span>
<span class="highlight-line"><span class="token comment"># - %h -> host name, possibly with domain name</span></span>
<span class="highlight-line"><span class="token comment"># - %n -> node name, most probably the host name without domain name</span></span>
<span class="highlight-line"><span class="token comment"># - %s -> the operating system</span></span>
<span class="highlight-line"><span class="token comment"># - %r -> the operating system's version</span></span>
<span class="highlight-line"><span class="token comment"># - %m -> the machine (hardware) type</span></span>
<span class="highlight-line"><span class="token comment"># - %% -> a single %</span></span>
<span class="highlight-line"><span class="token comment"># Default is "Welcome to %s at %n"</span></span>
<span class="highlight-line">GreetString=Bending Fern Glade</span>
<span class="highlight-line"><span class="token comment"># Whether the fonts used in the greeter should be antialiased.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line">AntiAliasing=true</span>
<span class="highlight-line"><span class="token comment"># The font for the greeter headline. The value is encoded.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "Serif 20pt bold"</span></span>
<span class="highlight-line">GreetFont=Serif,16,-1,5,50,0,0,0,0,0</span>
<span class="highlight-line"><span class="token comment"># The normal font used in the greeter. The value is encoded.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "Sans Serif 10pt"</span></span>
<span class="highlight-line">StdFont=Sans Serif,10,-1,5,50,0,0,0,0,0</span>
<span class="highlight-line"><span class="token comment"># The font used for the "Login Failed" message. The value is encoded.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "Sans Serif 10pt bold"</span></span>
<span class="highlight-line">FailFont=Sans Serif,10,-1,5,75,0,0,0,0,0</span>
<span class="highlight-line"><span class="token comment"># What to do with the Num Lock modifier for the time the greeter is running:</span></span>
<span class="highlight-line"><span class="token comment"># "Off" - turn off</span></span>
<span class="highlight-line"><span class="token comment"># "On" - turn on</span></span>
<span class="highlight-line"><span class="token comment"># "Keep" - do not change the state</span></span>
<span class="highlight-line"><span class="token comment"># Default is Keep</span></span>
<span class="highlight-line"><span class="token comment">#NumLock=Off</span></span>
<span class="highlight-line"><span class="token comment"># Language and locale to use in the greeter, encoded like $LANGUAGE.</span></span>
<span class="highlight-line"><span class="token comment"># If empty, the settings from the environment are used.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#Language=de_DE</span></span>
<span class="highlight-line"><span class="token comment"># Enable autocompletion in the username line edit.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line">UserCompletion=true</span>
<span class="highlight-line"><span class="token comment"># Enable user list (names along with images) in the greeter.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#UserList=false</span></span>
<span class="highlight-line"><span class="token comment"># User selection for UserCompletion and UserList:</span></span>
<span class="highlight-line"><span class="token comment"># "NotHidden" - all users except those listed in HiddenUsers</span></span>
<span class="highlight-line"><span class="token comment"># "Selected" - only the users listed in SelectedUsers</span></span>
<span class="highlight-line"><span class="token comment"># Default is NotHidden</span></span>
<span class="highlight-line"><span class="token comment">#ShowUsers=Selected</span></span>
<span class="highlight-line"><span class="token comment"># For ShowUsers=Selected. @&lt;group> means all users in that group.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#SelectedUsers=root,johndoe</span></span>
<span class="highlight-line"><span class="token comment"># For ShowUsers=NotHidden. @&lt;group> means all users in that group.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">HiddenUsers=root</span>
<span class="highlight-line"><span class="token comment"># Special case of HiddenUsers: users with a non-zero UID less than this number</span></span>
<span class="highlight-line"><span class="token comment"># will not be shown as well.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 0</span></span>
<span class="highlight-line">MinShowUID=1000</span>
<span class="highlight-line"><span class="token comment"># Complement to MinShowUID: users with a UID greater than this number will</span></span>
<span class="highlight-line"><span class="token comment"># not be shown as well.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 65535</span></span>
<span class="highlight-line">MaxShowUID=65000</span>
<span class="highlight-line"><span class="token comment"># If false, the users are listed in the order they appear in /etc/passwd.</span></span>
<span class="highlight-line"><span class="token comment"># If true, they are sorted alphabetically.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#SortUsers=false</span></span>
<span class="highlight-line"><span class="token comment"># Specify, where the users' pictures should be taken from.</span></span>
<span class="highlight-line"><span class="token comment"># "AdminOnly" - from &lt;facedir>/$USER.face[.icon]</span></span>
<span class="highlight-line"><span class="token comment"># "PreferAdmin" - prefer &lt;facedir>, fallback on $HOME</span></span>
<span class="highlight-line"><span class="token comment"># "PreferUser" - ... and the other way round</span></span>
<span class="highlight-line"><span class="token comment"># "UserOnly" - from the user's $HOME/.face[.icon]</span></span>
<span class="highlight-line"><span class="token comment"># Default is AdminOnly</span></span>
<span class="highlight-line">FaceSource=PreferUser</span>
<span class="highlight-line"><span class="token comment"># The directory containing the user images if FaceSource is not UserOnly.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/share/apps/kdm/faces"</span></span>
<span class="highlight-line"><span class="token comment">#FaceDir=/usr/share/faces</span></span>
<span class="highlight-line"><span class="token comment"># Specify, if/which user should be preselected for log in.</span></span>
<span class="highlight-line"><span class="token comment"># "None" - do not preselect any user</span></span>
<span class="highlight-line"><span class="token comment"># "Previous" - the user which successfully logged in last time</span></span>
<span class="highlight-line"><span class="token comment"># "Default" - the user specified in the DefaultUser option</span></span>
<span class="highlight-line"><span class="token comment"># Default is None</span></span>
<span class="highlight-line"><span class="token comment">#PreselectUser=Previous</span></span>
<span class="highlight-line"><span class="token comment"># If this is true, the password input line is focused automatically if</span></span>
<span class="highlight-line"><span class="token comment"># a user is preselected.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line">FocusPasswd=true</span>
<span class="highlight-line"><span class="token comment"># If this is true, the entered password is echoed as bullets. Otherwise,</span></span>
<span class="highlight-line"><span class="token comment"># no feedback is given at all.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#EchoPasswd=false</span></span>
<span class="highlight-line"><span class="token comment"># If true, krootimage will be automatically started by KDM; otherwise, the</span></span>
<span class="highlight-line"><span class="token comment"># Setup script should be used to setup the background.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#UseBackground=false</span></span>
<span class="highlight-line"><span class="token comment"># The configuration file to be used by krootimage.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/share/config/kdm/backgroundrc"</span></span>
<span class="highlight-line">BackgroundCfg=/usr/share/config/kdm/backgroundrc</span>
<span class="highlight-line"><span class="token comment"># Whether to grab keyboard and mouse while the greeter is visible. Grabs</span></span>
<span class="highlight-line"><span class="token comment"># may improve security, but make on-screen keyboards, etc. unusable.</span></span>
<span class="highlight-line"><span class="token comment"># "Never" - never grab</span></span>
<span class="highlight-line"><span class="token comment"># "IfNoAuth" - grab if the display requires no X authorization</span></span>
<span class="highlight-line"><span class="token comment"># "Always" - always grab</span></span>
<span class="highlight-line"><span class="token comment"># Default is IfNoAuth</span></span>
<span class="highlight-line"><span class="token comment">#GrabInput=Always</span></span>
<span class="highlight-line"><span class="token comment"># Hold the X-server grabbed the whole time the greeter is visible. This</span></span>
<span class="highlight-line"><span class="token comment"># may be more secure, but it will disable any background and other</span></span>
<span class="highlight-line"><span class="token comment"># X-clients started from the Setup script.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#GrabServer=true</span></span>
<span class="highlight-line"><span class="token comment"># How many seconds to wait for grab to succeed.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 3</span></span>
<span class="highlight-line"><span class="token comment">#GrabTimeout=3</span></span>
<span class="highlight-line"><span class="token comment"># Warn, if display has no X-authorization (local auth cannot be created,</span></span>
<span class="highlight-line"><span class="token comment"># XDMCP display wants no auth, or display is foreign from StaticServers).</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#AuthComplain=false</span></span>
<span class="highlight-line"><span class="token comment"># Random seed for forging saved session types, etc. of unknown users.</span></span>
<span class="highlight-line"><span class="token comment"># This value should be random but constant across the login domain.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 0</span></span>
<span class="highlight-line">ForgingSeed=1386325123</span>
<span class="highlight-line"><span class="token comment"># Specify conversation plugins for the login dialog. Each plugin can be</span></span>
<span class="highlight-line"><span class="token comment"># specified as a base name (which expands to $kde_modulesdir/kgreet_$base)</span></span>
<span class="highlight-line"><span class="token comment"># or as a full pathname.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "classic"</span></span>
<span class="highlight-line"><span class="token comment">#PluginsLogin=sign</span></span>
<span class="highlight-line"><span class="token comment"># Same as PluginsLogin, but for the shutdown dialog.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "classic"</span></span>
<span class="highlight-line"><span class="token comment">#PluginsShutdown=modern</span></span>
<span class="highlight-line"><span class="token comment"># A list of options of the form Key=Value. The conversation plugins can query</span></span>
<span class="highlight-line"><span class="token comment"># these settings; it is up to them what possible keys are.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#PluginOptions=SomeKey=randomvalue,Foo=bar</span></span>
<span class="highlight-line"><span class="token comment"># Show the "Console Login" action in the greeter (if ServerTTY/ConsoleTTYs</span></span>
<span class="highlight-line"><span class="token comment"># is configured).</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#AllowConsole=false</span></span>
<span class="highlight-line"><span class="token comment"># A program to run while the greeter is visible. It is supposed to preload</span></span>
<span class="highlight-line"><span class="token comment"># as much as possible of the session that is going to be started (most</span></span>
<span class="highlight-line"><span class="token comment"># probably).</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">Preloader=/usr/bin/preloadkde</span>
<span class="highlight-line"><span class="token comment"># Whether the greeter should be themed.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line">UseTheme=true</span>
<span class="highlight-line"><span class="token comment"># The theme to use for the greeter. Can point to either a directory or an XML</span></span>
<span class="highlight-line"><span class="token comment"># file.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">Theme=/usr/share/apps/kdm/themes/elarun</span>
<span class="highlight-line"><span class="token comment"># Enable the Alt-Ctrl-D shortcut to toggle greeter theme debugging.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#AllowThemeDebug=true</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># Core config for local displays</span></span>
<span class="highlight-line">[X-:*-Core]</span>
<span class="highlight-line"><span class="token comment"># How often to try to run the X-server. Running includes executing it and</span></span>
<span class="highlight-line"><span class="token comment"># waiting for it to come up.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 1</span></span>
<span class="highlight-line"><span class="token comment">#ServerAttempts=1</span></span>
<span class="highlight-line"><span class="token comment"># How long to wait for a local X-server to come up.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 30</span></span>
<span class="highlight-line"><span class="token comment">#ServerTimeout=30</span></span>
<span class="highlight-line"><span class="token comment"># The command line to start the X-server, without display number and VT spec.</span></span>
<span class="highlight-line"><span class="token comment"># This string is subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "/usr/bin/X"</span></span>
<span class="highlight-line">ServerCmd=/usr/bin/X -br -novtswitch -quiet</span>
<span class="highlight-line"><span class="token comment"># Additional arguments for the X-servers for local sessions.</span></span>
<span class="highlight-line"><span class="token comment"># This string is subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">ServerArgsLocal=-background none -nolisten tcp</span>
<span class="highlight-line"><span class="token comment"># Additional arguments for the X-servers for remote sessions.</span></span>
<span class="highlight-line"><span class="token comment"># This string is subject to word splitting.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#ServerArgsRemote=</span></span>
<span class="highlight-line"><span class="token comment"># The user the X-server should run as. Empty results in root.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#ServerUID=_x11</span></span>
<span class="highlight-line"><span class="token comment"># Restart instead of resetting the local X-server after session exit.</span></span>
<span class="highlight-line"><span class="token comment"># Use it if the server leaks memory etc.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#TerminateServer=false</span></span>
<span class="highlight-line"><span class="token comment"># Create X-authorizations for local displays.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line"><span class="token comment">#Authorize=false</span></span>
<span class="highlight-line"><span class="token comment"># Which X-authorization mechanisms should be used.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "MIT-MAGIC-COOKIE-1"</span></span>
<span class="highlight-line"><span class="token comment">#AuthNames=</span></span>
<span class="highlight-line"><span class="token comment"># Need to reset the X-server to make it read initial Xauth file.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#ResetForAuth=true</span></span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">AllowNullPasswd=false</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">AllowShutdown=All</span>
<span class="highlight-line"><span class="token comment"># Enable password-less logins on this display. USE WITH EXTREME CARE!</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#NoPassEnable=true</span></span>
<span class="highlight-line"><span class="token comment"># The users that do not need to provide a password to log in. NEVER list root!</span></span>
<span class="highlight-line"><span class="token comment"># "*" means all non-root users. @&lt;group> means all users in that group.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#NoPassUsers=fred,ethel</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># Greeter config for local displays</span></span>
<span class="highlight-line">[X-:*-Greeter]</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">PreselectUser=Previous</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">FocusPasswd=true</span>
<span class="highlight-line"><span class="token comment"># Specify whether the greeter of local displays should start up in host chooser</span></span>
<span class="highlight-line"><span class="token comment"># (remote) or login (local) mode and whether it is allowed to switch to the</span></span>
<span class="highlight-line"><span class="token comment"># other mode.</span></span>
<span class="highlight-line"><span class="token comment"># "LocalOnly" - only local login possible</span></span>
<span class="highlight-line"><span class="token comment"># "DefaultLocal" - start up in local mode, but allow switching to remote mode</span></span>
<span class="highlight-line"><span class="token comment"># "DefaultRemote" - ... and the other way round</span></span>
<span class="highlight-line"><span class="token comment"># "RemoteOnly" - only choice of remote host possible</span></span>
<span class="highlight-line"><span class="token comment"># Default is LocalOnly</span></span>
<span class="highlight-line">LoginMode=DefaultLocal</span>
<span class="highlight-line"><span class="token comment"># A list of hosts to be automatically added to the remote login menu. The</span></span>
<span class="highlight-line"><span class="token comment"># special name "*" means broadcast.</span></span>
<span class="highlight-line"><span class="token comment"># Default is "*"</span></span>
<span class="highlight-line"><span class="token comment">#ChooserHosts=*,host1,host2,host3.local,login.domain.com</span></span>
<span class="highlight-line"><span class="token comment"># Show the "Restart X Server"/"Close Connection" action in the greeter.</span></span>
<span class="highlight-line"><span class="token comment"># Default is true</span></span>
<span class="highlight-line">AllowClose=false</span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># Core config for 1st local display</span></span>
<span class="highlight-line">[X-:0-Core]</span>
<span class="highlight-line"><span class="token comment"># The VT the X-server should run on; auto-assign if zero, don't assign if -1.</span></span>
<span class="highlight-line"><span class="token comment"># Better leave it zero and use ServerVTs.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 0</span></span>
<span class="highlight-line">ServerVT=7</span>
<span class="highlight-line"><span class="token comment"># Enable automatic login. USE WITH EXTREME CARE!</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#AutoLoginEnable=true</span></span>
<span class="highlight-line"><span class="token comment"># If true, auto-login after logout. If false, auto-login is performed only</span></span>
<span class="highlight-line"><span class="token comment"># when a display session starts up.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#AutoLoginAgain=true</span></span>
<span class="highlight-line"><span class="token comment"># The delay in seconds before automatic login kicks in.</span></span>
<span class="highlight-line"><span class="token comment"># Default is 0</span></span>
<span class="highlight-line"><span class="token comment">#AutoLoginDelay=10</span></span>
<span class="highlight-line"><span class="token comment"># The user to log in automatically. NEVER specify root!</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#AutoLoginUser=fred</span></span>
<span class="highlight-line"><span class="token comment"># The password for the user to log in automatically. This is NOT required</span></span>
<span class="highlight-line"><span class="token comment"># unless the user is logged into a NIS or Kerberos domain. If you use this</span></span>
<span class="highlight-line"><span class="token comment"># option, you should "chmod 600 kdmrc" for obvious reasons.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line"><span class="token comment">#AutoLoginPass=secret!</span></span>
<span class="highlight-line"><span class="token comment"># Immediately lock the automatically started session. This works only with</span></span>
<span class="highlight-line"><span class="token comment"># KDE sessions.</span></span>
<span class="highlight-line"><span class="token comment"># Default is false</span></span>
<span class="highlight-line"><span class="token comment">#AutoLoginLocked=true</span></span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">ClientLogFile=.xsession-errors</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">ServerArgsLocal=-nolisten tcp -novtswitch -keeptty -config xorg.conf -layout Seat0</span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># Greeter config for 1st local display</span></span>
<span class="highlight-line">[X-:0-Greeter]</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">PreselectUser=Default</span>
<span class="highlight-line"><span class="token comment"># The user to preselect if PreselectUser=Default.</span></span>
<span class="highlight-line"><span class="token comment"># Default is ""</span></span>
<span class="highlight-line">DefaultUser=user1</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">BackgroundCfg=/usr/share/config/kdm/backgroundrc.seat0</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">FocusPasswd=true</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">GreeterPos=0,100</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">LoginMode=DefaultLocal</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">LogoArea=None</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">UserCompletion=false</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">UserList=false</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">Preloader=/usr/bin/preloadkde</span>
<span class="highlight-line"></span>
<span class="highlight-line">[X-:1-Greeter]</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">BackgroundCfg=/usr/share/config/kdm/backgroundrc.seat1</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">DefaultUser=user2</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">FocusPasswd=true</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">GreeterPos=100,100</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">LoginMode=DefaultLocal</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">LogoArea=None</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">PreselectUser=Default</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">UserCompletion=false</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">UserList=false</span>
<span class="highlight-line"></span>
<span class="highlight-line">[X-:1-Core]</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">AutoLoginEnable=false</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">AutoLoginLocked=false</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">AutoLoginUser=</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">ServerArgsLocal=-nolisten tcp -sharevts -keeptty -config xorg.conf -layout Seat1</span>
<span class="highlight-line"><span class="token comment"># See above</span></span>
<span class="highlight-line">ServerVT=9</span></code></pre>
</details>
<details>
<summary>/usr/share/config/kdm/backgroundrc.seat0</summary>
<pre class="language-apacheconf"><code class="language-apacheconf"><span class="highlight-line"><span class="token comment"># File: /usr/share/config/kdm/backgroundrc.seat0</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token directive-flags keyword">[Desktop0]</span></span>
<span class="highlight-line">BackgroundMode=Flat</span>
<span class="highlight-line">BlendBalance=100</span>
<span class="highlight-line">BlendMode=NoBlending</span>
<span class="highlight-line">ChangeInterval=60</span>
<span class="highlight-line">Color1=0,0,200</span>
<span class="highlight-line">Color2=192,192,0</span>
<span class="highlight-line">CurrentWallpaper=/usr/share/config/kdm/20100519-190000.jpg</span>
<span class="highlight-line">LastChange=0</span>
<span class="highlight-line">MinOptimizationDepth=1</span>
<span class="highlight-line">MultiWallpaperMode=NoMulti</span>
<span class="highlight-line">Pattern=fish</span>
<span class="highlight-line">Program=</span>
<span class="highlight-line">ReverseBlending=false</span>
<span class="highlight-line">UseSHM=false</span>
<span class="highlight-line">Wallpaper=/usr/share/config/kdm/20100519-190000.jpg</span>
<span class="highlight-line">WallpaperMode=ScaleAndCrop</span></code></pre>
</details>
<details>
<summary>/usr/share/config/kdm/backgroundrc.seat1</summary>
<pre class="language-apacheconf"><code class="language-apacheconf"><span class="highlight-line"><span class="token comment"># File: /usr/share/config/kdm/backgroundrc.seat1</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token directive-flags keyword">[Desktop0]</span></span>
<span class="highlight-line">BackgroundMode=Flat</span>
<span class="highlight-line">BlendBalance=100</span>
<span class="highlight-line">BlendMode=NoBlending</span>
<span class="highlight-line">ChangeInterval=60</span>
<span class="highlight-line">Color1=0,0,200</span>
<span class="highlight-line">Color2=192,192,192</span>
<span class="highlight-line">CurrentWallpaper=/usr/share/config/kdm/20100519-185835.jpg</span>
<span class="highlight-line">LastChange=0</span>
<span class="highlight-line">MinOptimizationDepth=1</span>
<span class="highlight-line">MultiWallpaperMode=NoMulti</span>
<span class="highlight-line">Pattern=fish</span>
<span class="highlight-line">Program=</span>
<span class="highlight-line">ReverseBlending=false</span>
<span class="highlight-line">UseSHM=false</span>
<span class="highlight-line">Wallpaper=/usr/share/config/kdm/20100519-185835.jpg</span>
<span class="highlight-line">WallpaperMode=ScaleAndCrop</span></code></pre>
</details>
<details>
<summary>/etc/udev/rules.d/90-creative-xmod.rules</summary>
<pre class="language-apacheconf"><code class="language-apacheconf"><span class="highlight-line"><span class="token comment"># File: /etc/udev/rules.d/90-creative-xmod.rules</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># udev rules to disginguish the two Creative usb sound cards that plug into the usb hub</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">#DEVPATH=="/devices/pci0000:00/0000:00:1d.7/usb1/1-6/1-6.7/1-6.7.1*", GOTO="CreativeXmodLEFT"</span></span>
<span class="highlight-line"><span class="token comment">#DEVPATH=="/devices/pci0000:00/0000:00:1d.7/usb1/1-6/1-6.4*", GOTO="CreativeXmodRIGHT"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">DEVPATH==<span class="token string">"/devices/pci0000:00/0000:00:1d.7/usb*/*-6/*-6.7/*-6.7.1*", GOTO="CreativeXmodLEFT"</span></span>
<span class="highlight-line">DEVPATH==<span class="token string">"/devices/pci0000:00/0000:00:1d.7/usb*/*-6/*-6.4*", GOTO="CreativeXmodRIGHT"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">GOTO=<span class="token string">"CreativeXmodEnd"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">##### LEFT SEAT</span></span>
<span class="highlight-line">LABEL=<span class="token string">"CreativeXmodLEFT"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">ENV{ID_MODEL}=<span class="token string">"Creative_Xmod_LeftSeat", ENV{ID_MODEL_ENC}="Creative\x20Xmod\x20LeftSeat", ENV{ID_SERIAL}="Creative_Technology_Ltd_Creative_Xmod_LeftSeat", ENV{NAME}="Creative Xmod leftSeat"</span></span>
<span class="highlight-line">SYMLINK==<span class="token string">"input/by-path/pci-0000:00:1d.7-usb-0:6.7.1:1.3-event", SYMLINK+="input/by-id/CreativeXmod-Button-LeftSeat"</span></span>
<span class="highlight-line">SYMLINK==<span class="token string">"snd/by-path/pci-0000:00:1d.7-usb-0:6.7.1:1.0", SYMLINK+="snd/by-id/usb-Creative_Technology_Ltd_Creative_Xmod-LeftSeat"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">#ENV{ID_MODEL}="Creative_Xmod", SYMLINK+="snd/by-id/usb-Creative_Technology_Ltd_Creative_Xmod-LeftSeat"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">GOTO=<span class="token string">"CreativeXmodEnd"</span></span>
<span class="highlight-line"><span class="token comment">### END LEFT ###</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment"># RIGHT SEAT</span></span>
<span class="highlight-line">LABEL=<span class="token string">"CreativeXmodRIGHT"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">ENV{ID_MODEL}=<span class="token string">"Creative_Xmod_RightSeat", ENV{ID_MODEL_ENC}="Creative\x20Xmod\x20RightSeat", ENV{ID_SERIAL}="Creative_Technology_Ltd_Creative_Xmod_RightSeat", NAME="Creative Xmod RightSeat"</span></span>
<span class="highlight-line">SYMLINK==<span class="token string">"input/by-path/pci-0000:00:1d.7-usb-0:6.4:1.3-event", SYMLINK+="input/by-id/CreativeXmod-Button-RightSeat"</span></span>
<span class="highlight-line">SYMLINK==<span class="token string">"snd/by-path/pci-0000:00:1d.7-usb-0:6.4:1.0", SYMLINK+="snd/by-id/usb-Creative_Technology_Ltd_Creative_Xmod-RightSeat"</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">### END RIGHT ###</span></span>
<span class="highlight-line"></span>
<span class="highlight-line"><span class="token comment">### common ened</span></span>
<span class="highlight-line"></span>
<span class="highlight-line">LABEL=<span class="token string">"CreativeXmodEnd"</span></span></code></pre>
</details>

    ]]></content>
  </entry><entry>
    <title>憶老方</title>
    <link href="https://blog.tklee.org/en/posts/0543-憶老方/"/>
    <updated>2015-03-06T05:43:00.001Z</updated>
    <id>https://blog.tklee.org/en/posts/0543-憶老方/</id>
    <content type="html"><![CDATA[
      <div class="post-teaser-box" data-lang="en">
老方是我師大附中三年的導師。年初他病逝，我們694班同學在臉書上私人社團裡，紛紛表達對他的懷念。我人在遠方，無法親自參加公祭，託同學把我對他的懷念和感謝帶上。
</div>
<p>憶老方</p>
<p>老方是我師大附中三年的導師。年初他病逝，我們694班同學在臉書上私人社團裡，紛紛表達對他的懷念。我人在遠方，無法親自參加公祭，託同學把我對他的懷念和感謝帶上。</p>
<p>在那個時代，給老師取綽號算是不敬。老方知道我們在背後這麼叫他，卻一笑置之。回想起來，「老方」的暱稱，和益師益友的他其實很相襯。</p>
<p>老方教數學，扎實的數學底子，讓我一生受益無窮。但我特別欣賞的，是他在課堂上不做作的態度。有時候他導公式寫了滿黑板，最後發現結果不對，他會請同學幫忙一起看，然後發現之前錯了一步時，他會拍頭一笑，回過重來。這種和學生一起探索的態度，和我與國外博士班指導教授討論想法，竟有異曲同工之妙。後來在工作上，我也用一樣的態度來帶新進的人員。我帶出來的人，特別有實驗精神，敢於表達意見，也讓我的團隊贏得最有創意的名聲。</p>
<p>老方對同學們一視同仁，但我個人有個印象深刻的小故事。</p>
<p>社團是附中特色之一。學生們在社團裡學習經營，發揮創意，結交朋友。但不少社團幹部會因此而影響學業，所以如果成績太離譜，學校會發一張「紅單子」，讓學業成績太離譜的學生不得再擔任幹部。有趣的是，在社聯會（所有社團幹部的聯誼會）裡，「拿紅單子」有時候反而被視為努力經營社團的光榮象徵。</p>
<p>高二時我擔任社團社長，把大部分心力都用在社團經營上，成績一度跌落谷底。有一天，老方把我叫到他辦公室；我當然知道他要說什麼，但我覺得反正他不可能了解社團人的執著，準備好硬著頭皮聽訓，聽完就算了。</p>
<p>進到他辦公室，他看著我笑了笑，問我知不知道他為什麼叫我來。我點了點頭。沒想到他說，「那就好。沒其他事了。」</p>
<p>霎時間，所有我事前的心戰策略全無用武之地。他輕輕一點，讓簡單的真理乍現：你成績怎樣是你的事，你知道自己在幹什麼就好。</p>
<p>雖然這好像是廢話，但我之前在一個逼讀文化（不管是當面逼，還是用制度逼）下成長學習，這是第一次看到師長直接移步側站，讓我自己去面對問題、抉擇及後果。</p>
<p>老方那句話，是我這輩子最後一次讓別人來擔心我的學業。</p>
<p>傳道，授業，解惑，老方也。</p>
<p>(My high school homeroom teacher passed away, and this is my tribute and eulogy to him. There's no English translation from me, yet.)</p>

    ]]></content>
  </entry>
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