lkarch/lkarch.org
1
1
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
lkarch.org/tools/mathisfun/www.mathsisfun.com/activity/seven-bridges-konigsberg.html
Lucas Kent e39465ad2f Changes to be committed: 
new file:   Files/flashplayer_32_sa.exe
	new file:   favicon.ico
	new file:   globe.gif
	new file:   imgs/download.png
	new file:   imgs/zuck.jpg
	new file:   index.html
	new file:   other.ico
	new file:   script.js
	new file:   site.webmanifest
	new file:   sitemap.html
	new file:   styles/backround.css
	new file:   styles/border.css
	new file:   styles/fonts/Titillium_Web/OFL.txt
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
	new file:   styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
	new file:   styles/fonts/webfontkit-20221027-163353/generator_config.txt
	new file:   styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
	new file:   styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
	new file:   styles/fonts/webfontkit-20221027-163353/stylesheet.css
	new file:   styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
	new file:   styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
	new file:   styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
	new file:   styles/fonts/webfontkit-20221027-165950/generator_config.txt
	new file:   styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
	new file:   styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
	new file:   styles/fonts/webfontkit-20221027-165950/stylesheet.css
	new file:   styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
	new file:   styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
	new file:   styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
	new file:   styles/style.css
	new file:   tools/2048/.gitignore
	new file:   tools/2048/.jshintrc
	new file:   tools/2048/CONTRIBUTING.md
	new file:   tools/2048/LICENSE.txt
	new file:   tools/2048/README.md
	new file:   tools/2048/Rakefile
	new file:   tools/2048/favicon.ico
	new file:   tools/2048/index.html
	new file:   tools/2048/js/animframe_polyfill.js
	new file:   tools/2048/js/application.js
	new file:   tools/2048/js/bind_polyfill.js
	new file:   tools/2048/js/classlist_polyfill.js
	new file:   tools/2048/js/game_manager.js
	new file:   tools/2048/js/grid.js
	new file:   tools/2048/js/html_actuator.js
	new file:   tools/2048/js/keyboard_input_manager.js
	new file:   tools/2048/js/local_storage_manager.js
	new file:   tools/2048/js/tile.js
    new file:   tools/2048/meta/apple-touch-icon.png
	new file:   tools/webretro/cores/neocd_libretro.js
	new file:   tools/webretro/cores/neocd_libretro.wasm
	new file:   tools/webretro/cores/nestopia_libretro.js
	new file:   tools/webretro/cores/nestopia_libretro.wasm
	new file:   tools/webretro/cores/o2em_libretro.js
	new file:   tools/webretro/cores/o2em_libretro.wasm
	new file:   tools/webretro/cores/opera_libretro.js
	new file:   tools/webretro/cores/opera_libretro.wasm
2022-11-02 08:40:01 -04:00

481 lines
17 KiB
HTML
Raw Blame History

<!DOCTYPE html>
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
<!-- Mirrored from www.mathsisfun.com/activity/seven-bridges-konigsberg.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:58:13 GMT -->
<head>
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<!-- #BeginEditable "doctitle" -->
<title>Activity: The Seven Bridges of K&ouml;nigsberg</title>
<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
<script language="JavaScript" type="text/javascript">Author='Gates, Les Bill';</script>
<!-- #EndEditable -->
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
<meta name="HandheldFriendly" content="true">
<meta name="referrer" content="always">
<link rel="stylesheet" type="text/css" href="../style3.css">
<script src="../main3.js"></script>
</head>
<body id="bodybg">
<div class="bg">
<div id="stt"></div>
<header>
<div id="hdr"></div>
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
<div id="gtran">
<script>document.write(getTrans());</script>
</div>
<div id="adTopOuter" class="centerfull noprint">
<div id="adTop">
<script>document.write(getAdTop());</script>
</div>
</div>
<div id="adHide">
<div id="showAds1"><a href="javascript:showAds()">Show Ads</a></div>
<div id="hideAds1"><a href="javascript:hideAds()">Hide Ads</a><br>
<a href="../about-ads.html">About Ads</a></div>
</div>
</header>
<nav>
<div id="menuWide" class="menu">
<script>document.write(getMenu(0));</script>
</div>
<div id="linkto">
<div id="linktort">
<script>document.write(getLinks());</script>
</div>
</div>
<div id="search" role="search">
<script>document.write(getSearch());</script>
</div>
<div id="menuSlim" class="menu">
<script>document.write(getMenu(1));</script>
</div>
<div id="menuTiny" class="menu">
<script>document.write(getMenu(2));</script>
</div>
</nav>
<div id="extra"></div>
</div>
<article id="content" role="main">
<!-- #BeginEditable "Body" -->
<h1 class="center">Activity: The Seven Bridges of K&ouml;nigsberg</h1>
<p class="center">The old town of K&ouml;nigsberg has seven bridges:</p>
<p class="center"><img src="images/bridges1.jpg" alt="The Seven Bridges of Konigsberg" height="302" width="398"></p>
<p class="center larger">Can you take a walk through the town, visiting each part of the
town <br>
and <b>crossing each bridge only once?</b></p>
<p>&nbsp;</p>
<p>This question was given to a famous mathematician called Leonhard Euler... but let's try to answer it ourselves!</p>
<p>And along the way we will learn a little about "Graph Theory".</p>
<h2>Simplifying It</h2>
<p>We can simplify the map above to just this:</p>
<p class="center"><img src="images/bridges1.gif" alt="seven bridges of konigsberg simplified" height="134" width="261"></p>
<p>There are four areas of the town - on the
mainland north of the river, on the mainland south of the river, on the
island and on the peninsula (the piece of land on the right)</p>
<p>Let us label them A, B, C and D:</p>
<table width="100%" border="0">
<tbody>
<tr>
<td>
<p>To "visit each part of the
town" you should visit the points <b>A, B, C and D</b>.</p>
<p>And you should cross each bridge <b>p, q, r, s, t, u
and v</b> just once.</p></td>
<td>&nbsp;</td>
<td><img src="images/bridges2.gif" alt="seven bridges of konigsberg simplified with labels" height="150" width="272"></td>
</tr>
</tbody></table>
<p>And we can further simplify it to this:</p>
<p class="center"><img src="images/bridges3.gif" alt="seven bridges of konigsberg as a graph" height="172" width="175"></p>
<p class="center larger">So instead of taking long walks through the town, <br>
you can now just draw lines with a pencil.</p>
<h2>Your Turn</h2>
<div class="center80">
<p class="larger">Can you draw each line p, q, r, s, t, u and v <b>only once</b>, without removing your pencil from the paper (you may start at any point) ?</p>
</div>
<p><b>Have a try and see if you can</b>.</p>
<p>...</p>
<p>Did you succeed?</p>
<p>&nbsp;</p>
<p>Well ... let's take a step back and try some simpler shapes.</p>
<p>Try these (remember: draw all the lines, but never go over any line more than once, and don't remove your pencil from the paper.)</p>
<p class="center"><img src="images/bridges4.gif" alt="graphs 1 to 8" height="231" width="355"></p>
<p>Put your results here:</p>
<table align="center" border="1">
<tbody>
<tr>
<td align="center" height="30" width="75">Shape</td>
<td align="center" width="75">Success?</td>
</tr>
<tr>
<td align="center" height="30">1</td>
<td align="center">Yes</td>
</tr>
<tr>
<td align="center" height="30">2</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">3</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">4</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">5</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">6</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">7</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">8</td>
<td align="center">&nbsp;</td>
</tr>
</tbody></table>
<p>&nbsp;</p>
<h2>So, How Can We Know Which Ones Work
and Which Ones Do Not?</h2>
<p>Let's investigate!</p>
<h3>But first, time to learn some special words:</h3>
<table border="0">
<tbody>
<tr>
<td>
<ul>
<li>A point is called a <b>vertex</b> (plural vertices)</li>
<li>A line is called an <b>edge</b>.</li>
<li>The whole diagram is called a <b>graph</b>.</li>
</ul></td>
<td>&nbsp;</td>
<td><img src="../algebra/images/graph-vertex-edge.svg" alt="graph vertex and edge"></td>
</tr>
</tbody></table>
<table align="center" border="0">
<tbody>
<tr>
<td align="right">
<p>Yes, it is called a "Graph"... but it is <b>NOT this kind of graph</b>:</p>
<p>They are both called "graphs". <br>
But they are different things. Just how it is.</p></td>
<td>&nbsp;</td>
<td><img src="../data/images/line-graph-example.svg" alt="line graph example" width="160"></td>
</tr>
</tbody></table>
<p>&nbsp;</p>
<table width="100%" border="0">
<tbody>
<tr>
<td><img src="../algebra/images/graph-degree.svg" alt="graph degree 3 and 2"></td>
<td>
<ul>
<li>The number of edges that lead to a vertex is called the <b>degree</b>.</li>
</ul></td>
</tr>
</tbody></table>
<table width="100%" border="0">
<tbody>
<tr>
<td>
<ul>
<li>A route around a graph that visits <b>every vertex</b> once is called a <b>simple path</b>.</li>
<li>A route around a graph that visits <b>every edge</b> once is called an <b>Euler path</b>.</li>
</ul></td>
<td>&nbsp;</td>
<td><img src="../algebra/images/graph-path.svg" alt="graph simple path and euler path"></td>
</tr>
</tbody></table>
<h3>Examples:</h3>
<table width="100%" border="0">
<tbody>
<tr>
<td align="center"><img src="images/bridges-diag7.gif" alt="graph 7" height="133" width="76"></td>
<td align="center">&nbsp;</td>
<td align="center"><img src="images/bridges-diag8.gif" alt="graph 8" height="132" width="76"></td>
</tr>
<tr>
<td>
<p>Diagram 7 has</p>
<ul>
<li>5 vertices: A, B, C, D and E</li>
<li>8 edges: AB, BC, CD, DA, AE, BE, AC and BD</li>
<li>Vertices A and B have degree 4</li>
<li>Vertices C and D have degree 3</li>
<li>Vertex E has degree 2</li>
</ul></td>
<td>&nbsp;</td>
<td>
<p>Diagram 8 has</p>
<ul>
<li>6 vertices: A, B, C, D, E and F </li>
<li>10 edges: AB, BC, CD, DA, AF, BF,CF, DF, AE and BE</li>
<li>Vertices A, B and F have degree 4</li>
<li>Vertices C and D have degree 3</li>
<li>Vertex E has degree 2</li>
</ul></td>
</tr>
</tbody></table>
<h2>Euler Path</h2>
<p>OK, <span class="center">imagine the lines are bridges. If you cross them once only you have solved the puzzle, so ...</span></p>
<p class="center larger">... what we want is an "Euler Path" ...</p>
<p>... and here is a clue to help you: we can tell which graphs have an "Euler Path" by counting how many vertices have an <b>odd degree</b>.</p>
<p>So, fill out this table:</p>
<table align="center" border="1">
<tbody>
<tr>
<td align="center" height="30" width="75">Shape</td>
<td align="center" width="75">Euler Path?</td>
<td align="center" width="75">Vertices</td>
<td align="center" width="75">how many with even degree</td>
<td align="center" width="75">how many with odd degree</td>
</tr>
<tr>
<td align="center" height="30">1</td>
<td align="center">Yes</td>
<td align="center">4</td>
<td align="center">4</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center" height="30">2</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">3</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">4</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">5</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">6</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">7</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">8</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
</tbody></table>
<p><strong>Is there a pattern?</strong></p>
<h3>&nbsp;</h3>
<p>Don't read any further until you have found some kind of pattern ... the answer is in the table.</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<div class="center80">
<p><strong>OK ... the answer is ...</strong></p>
<p>The number of vertices of odd degree must
be either zero or two.</p>
<p>If not then there is no "Euler Path"</p>
<p>And if there are
two vertices with odd degree, then they are the starting and ending vertices.</p>
</div>
<p>And the reason is not hard to understand.</p>
<p class="so">A path leads
into a vertex by one edge and out by a second edge.</p>
<p class="so">So
the edges should come in pairs (an even number).</p>
<p class="so">Only the start and end point can have an odd degree.</p>
<h2>Now Back to the K&ouml;nigsberg
Bridge Question:</h2>
<p class="center"><img src="images/bridges3.gif" alt="seven bridges of konigsberg graph" height="172" width="175"></p>
<p>Vertices <strong>A</strong>, <strong>B</strong> and <strong>D</strong> have degree 3 and vertex <strong>C</strong> has degree 5, so this
graph has four vertices of odd degree. So it does <b>not have an Euler Path</b>.<br>
<br>
<strong>We have solved the K&ouml;nigsberg
bridge question just like Euler did nearly 300 years ago!</strong></p>
<p>&nbsp;</p>
<h2>Bonus Exercise: Which of the following graphs have Euler Paths?</h2>
<p class="center"><img alt="Bridges8" src="images/bridges8.jpg"></p>
<table align="center" border="1">
<tbody>
<tr>
<td align="center" height="30" width="75">Shape</td>
<td align="center" width="75">Euler Path?</td>
<td align="center" width="75">Vertices</td>
<td align="center" width="75">How many with even degree</td>
<td align="center" width="75">How many with odd degree</td>
</tr>
<tr>
<td align="center" height="30">9</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">10</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">11</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">12</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">13</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
<tr>
<td align="center" height="30">14</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
<td align="center">&nbsp;</td>
</tr>
</tbody></table>
<p class="center">&nbsp;</p>
<p class="center">&nbsp;</p>
<div class="center80">
<h3>Footnotes</h3>
<p><b>Leonhard Euler</b> (1707 - 1783), a Swiss mathematician, was one of the
greatest and most prolific mathematicians of all time. Euler spent much
of his working life at the Berlin Academy in Germany, and it was
during that time that he was given the "The Seven Bridges of K&ouml;nigsberg" question to solve that has become
famous.</p>
<p>&nbsp;</p>
<p><b>The town of K&ouml;nigsberg</b> straddles the Pregel River. It was
formerly in Prussia, but is now known as Kaliningrad and is in Russia.
K&ouml;nigsberg was situated close to the mouth of the river and had seven
bridges joining the two sides of the river and also an island and a
peninsula.</p>
<p>&nbsp;</p>
<p><b>Answer</b> to the diagrams table:</p>
<table align="center" border="1">
<tbody>
<tr>
<td align="center" height="30" width="60">Shape</td>
<td align="center" width="60">Success?</td>
<td align="center" width="60">evens</td>
<td align="center" width="60">odds</td>
</tr>
<tr>
<td align="center" height="30">1</td>
<td align="center">Yes</td>
<td align="center">4</td>
<td align="center">0</td>
</tr>
<tr>
<td align="center" height="30">2</td>
<td align="center">Yes</td>
<td align="center">2</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center" height="30">3</td>
<td align="center">NO</td>
<td align="center">0</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center" height="30">4</td>
<td align="center">NO</td>
<td align="center">1</td>
<td align="center">4</td>
</tr>
<tr>
<td align="center" height="30">5</td>
<td align="center">Yes</td>
<td align="center">2</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center" height="30">6</td>
<td align="center">Yes</td>
<td align="center">3</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center" height="30">7</td>
<td align="center">Yes</td>
<td align="center">3</td>
<td align="center">2</td>
</tr>
<tr>
<td align="center" height="30">8</td>
<td align="center">Yes</td>
<td align="center">4</td>
<td align="center">2</td>
</tr>
</tbody></table>
</div>
<p>&nbsp;</p>
<div class="related">
<a href="index.html">Activities Index</a>
</div>
<!-- #EndEditable -->
</article>
<div id="adend" class="centerfull noprint">
<script>document.write(getAdEnd());</script>
</div>
<footer id="footer" class="centerfull noprint">
<script>document.write(getFooter());</script>
</footer>
<div id="copyrt">
Copyright &copy; 2020 MathsIsFun.com
</div>
<script>document.write(getBodyEnd());</script>
</body><!-- #EndTemplate -->
<!-- Mirrored from www.mathsisfun.com/activity/seven-bridges-konigsberg.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:58:15 GMT -->
</html>
Reference in New Issue View Git Blame Copy Permalink