106 lines
6.3 KiB
HTML
106 lines
6.3 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 426 - 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=5715c"><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>Box-ball system</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 426</h3><span style="width:300px;color:#666;">Published on Saturday, 4th May 2013, 07:00 pm; Solved by 118</span></div>
|
|
<div class="problem_content" role="problem">
|
|
<P>
|
|
Consider an infinite row of boxes. Some of the boxes contain a ball. For example, an initial configuration of 2 consecutive occupied boxes followed by 2 empty boxes, 2 occupied boxes, 1 empty box, and 2 occupied boxes can be denoted by the sequence (2, 2, 2, 1, 2), in which the number of consecutive occupied and empty boxes appear alternately.
|
|
</P>
|
|
<P>
|
|
A turn consists of moving each ball exactly once according to the following rule: Transfer the leftmost ball which has not been moved to the nearest empty box to its right.
|
|
</P>
|
|
<P>
|
|
After one turn the sequence (2, 2, 2, 1, 2) becomes (2, 2, 1, 2, 3) as can be seen below; note that we begin the new sequence starting at the first occupied box.
|
|
</P>
|
|
|
|
<div align='center'>
|
|
<img src="project/images/p_426_baxball1.gif" />
|
|
</div>
|
|
|
|
<P>
|
|
A system like this is called a <B>Box-Ball System</B> or <B>BBS</B> for short.
|
|
</P>
|
|
<P>
|
|
It can be shown that after a sufficient number of turns, the system evolves to a state where the consecutive numbers of occupied boxes is invariant. In the example below, the consecutive numbers of <B>occupied boxes</B> evolves to [1, 2, 3]; we shall call this the final state.
|
|
</P>
|
|
|
|
<div align='center'>
|
|
<img src="project/images/p_426_baxball2.gif" />
|
|
</div>
|
|
|
|
<P>
|
|
We define the sequence {<var>t</var><sub><var>i</var></sub>}:<BR />
|
|
<ul>
|
|
<li><var>s</var><sub>0</sub> = 290797
|
|
<li><var>s</var><sub><var>k</var>+1</sub> = <var>s</var><sub><var>k</var></sub><sup>2</sup> mod 50515093
|
|
<li><var>t</var><sub><var>k</var></sub> = (<var>s</var><sub><var>k</var></sub> mod 64) + 1
|
|
</ul>
|
|
</P>
|
|
<P>
|
|
Starting from the initial configuration (<var>t</var><sub>0</sub>, <var>t</var><sub>1</sub>, …, <var>t</var><sub>10</sub>), the final state becomes [1, 3, 10, 24, 51, 75].<BR />
|
|
Starting from the initial configuration (<var>t</var><sub>0</sub>, <var>t</var><sub>1</sub>, …, <var>t</var><sub>10 000 000</sub>), find the final state.<BR />
|
|
Give as your answer the sum of the squares of the elements of the final state. For example, if the final state is [1, 2, 3] then 14 ( = 1<sup>2</sup> + 2<sup>2</sup> + 3<sup>2</sup>) is your answer.
|
|
</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> |