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project_euler/pe/p335.html
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project_euler/pe/p335.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="utf-8" />
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<meta name="author" content="Colin Hughes" />
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<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
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<meta name="keywords" content="programming,mathematics,problems,puzzles" />
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<title>Problem 335 - Project Euler</title>
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<link rel="shortcut icon" href="http://projecteuler.net/favicon.ico" />
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<link rel="stylesheet" type="text/css" href="style_main.css" />
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<li><a href="about" title="About" accesskey="h">About</a></li>
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<div style="text-align:center;" class="print"><img src="images/pe_banner.png" alt="projecteuler.net" style="border:none;" /></div>
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<h2>Gathering the beans</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 335</h3><span style="width:300px;color:#666;">Published on Saturday, 23rd April 2011, 04:00 pm; Solved by 309</span></div>
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<div class="problem_content" role="problem">
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<p>Whenever Peter feels bored, he places some bowls, containing one bean each, in a circle. After this, he takes all the beans out of a certain bowl and drops them one by one in the bowls going clockwise. He repeats this, starting from the bowl he dropped the last bean in, until the initial situation appears again. For example with 5 bowls he acts as follows:</p>
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<div align='center'><img src="project/images/p_335_mancala.gif" /></div>
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<p>So with 5 bowls it takes Peter 15 moves to return to the initial situation.</p>
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<p>Let <var>M</var>(<var>x</var>) represent the number of moves required to return to the initial situation, starting with <var>x</var> bowls. Thus, <var>M</var>(5) = 15. It can also be verified that <var>M</var>(100) = 10920.</p>
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<p>Find <img style="vertical-align:middle" src="project/images/p_335_sum.gif" /><var>M</var>(2<sup><var>k</var></sup>+1). Give your answer modulo 7<sup>9</sup>.</p>
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