89 lines
6.1 KiB
HTML
89 lines
6.1 KiB
HTML
<!DOCTYPE html>
|
|
<html lang="en">
|
|
<head>
|
|
<meta charset="utf-8" />
|
|
<meta name="author" content="Colin Hughes" />
|
|
<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
|
|
<meta name="keywords" content="programming,mathematics,problems,puzzles" />
|
|
|
|
<title>Problem 301 - Project Euler</title>
|
|
<link rel="shortcut icon" href="http://projecteuler.net/favicon.ico" />
|
|
<link rel="stylesheet" type="text/css" href="style_main.css" />
|
|
<link rel="stylesheet" type="text/css" href="style_light.css" />
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
jax: ["input/TeX", "output/HTML-CSS"],
|
|
tex2jax: {
|
|
inlineMath: [ ["$","$"], ["\\(","\\)"] ],
|
|
displayMath: [ ["$$","$$"], ["\\[","\\]"] ],
|
|
processEscapes: true
|
|
},
|
|
"HTML-CSS": { availableFonts: ["TeX"] }
|
|
});
|
|
</script>
|
|
|
|
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML">
|
|
</script>
|
|
</head>
|
|
|
|
<body>
|
|
<div id="container">
|
|
|
|
<div id="nav" class="noprint">
|
|
<ul>
|
|
<li><a href="about" title="About" accesskey="h">About</a></li>
|
|
<li><a href="register" title="Register" accesskey="1">Register</a></li>
|
|
<li id="current"><a href="problems" title="Problems" accesskey="2">Problems</a></li>
|
|
<li><a href="login" title="Login" accesskey="3">Login</a></li>
|
|
</ul>
|
|
</div>
|
|
<div id="info_panel"><a href="rss2_euler.xml"><img src="images/icon_rss.png" alt="RSS Feed" title="RSS Feed" /></a><a href="secure=afb8f"><img src="images/icon_lock.png" alt="Use secure connection" title="Use secure connection" /></a></div>
|
|
|
|
<div id="logo" class="noprint">
|
|
<img src="images/pe_banner_light.png" alt="Project Euler .net" />
|
|
</div>
|
|
|
|
<div id="content">
|
|
<div style="text-align:center;" class="print"><img src="images/pe_banner.png" alt="projecteuler.net" style="border:none;" /></div>
|
|
<h2>Nim</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 301</h3><span style="width:300px;color:#666;">Published on Saturday, 11th September 2010, 04:00 pm; Solved by 2515</span></div>
|
|
<div class="problem_content" role="problem">
|
|
<p><i>Nim</i> is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.</p>
|
|
|
|
<p>We'll consider the three-heap normal-play version of Nim, which works as follows:<br />
|
|
- At the start of the game there are three heaps of stones.<br />
|
|
- On his turn the player removes any positive number of stones from any single heap.<br />
|
|
- The first player unable to move (because no stones remain) loses.</p>
|
|
|
|
<p> If (<var>n</var><sub>1</sub>,<var>n</var><sub>2</sub>,<var>n</var><sub>3</sub>) indicates a Nim position consisting of heaps of size <var>n</var><sub>1</sub>, <var>n</var><sub>2</sub> and <var>n</var><sub>3</sub> then there is a simple function <var>X</var>(<var>n</var><sub>1</sub>,<var>n</var><sub>2</sub>,<var>n</var><sub>3</sub>) — that you may look up or attempt to deduce for yourself — that returns:
|
|
<ul><li>zero if, with perfect strategy, the player about to move will eventually lose; or</li>
|
|
<li>non-zero if, with perfect strategy, the player about to move will eventually win.</li></ul>
|
|
</p>
|
|
|
|
<p>For example <var>X</var>(1,2,3) = 0 because, no matter what the current player does, his opponent can respond with a move that leaves two heaps of equal size, at which point every move by the current player can be mirrored by his opponent until no stones remain; so the current player loses. To illustrate:<br />
|
|
- current player moves to (1,2,1)<br />
|
|
- opponent moves to (1,0,1)<br />
|
|
- current player moves to (0,0,1)<br />
|
|
- opponent moves to (0,0,0), and so wins.</p>
|
|
|
|
<p>For how many positive integers <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 2<sup>30</sup> does <var>X</var>(<var>n</var>,2<var>n</var>,3<var>n</var>) = 0 ?
|
|
</p>
|
|
|
|
</div><br />
|
|
<br /></div>
|
|
|
|
|
|
|
|
|
|
|
|
<div id="footer" class="noprint">
|
|
<a href="copyright">Project Euler Copyright Information</a>
|
|
<!--/Creative Commons License--><!-- <rdf:RDF xmlns="http://web.resource.org/cc/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#">
|
|
<Work rdf:about="">
|
|
<license rdf:resource="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/" />
|
|
<dc:type rdf:resource="http://purl.org/dc/dcmitype/Text" />
|
|
</Work>
|
|
<License rdf:about="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/"><permits rdf:resource="http://web.resource.org/cc/Reproduction"/><permits rdf:resource="http://web.resource.org/cc/Distribution"/><requires rdf:resource="http://web.resource.org/cc/Notice"/><requires rdf:resource="http://web.resource.org/cc/Attribution"/><prohibits rdf:resource="http://web.resource.org/cc/CommercialUse"/><permits rdf:resource="http://web.resource.org/cc/DerivativeWorks"/><requires rdf:resource="http://web.resource.org/cc/ShareAlike"/></License></rdf:RDF> -->
|
|
</div>
|
|
</div>
|
|
<div style="height:1px;"> </div></body>
|
|
</html> |