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puzzles/project_euler/pe/p316.html
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puzzles/project_euler/pe/p316.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 316 - Project Euler</title>
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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>Numbers in decimal expansions</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 316</h3><span style="width:300px;color:#666;">Published on Saturday, 25th December 2010, 01:00 pm; Solved by 389</span></div>
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<div class="problem_content" role="problem">
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<p>Let <var>p</var> = <var>p<sub><font size="-2">1</font></sub> p<sub><font size="-2">2</font></sub> p<sub><font size="-2">3</font></sub></var> ... be an infinite sequence of random digits, selected from {0,1,2,3,4,5,6,7,8,9} with equal probability.<br />
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It can be seen that <var>p</var> corresponds to the real number 0.<var>p<sub><font size="-2">1</font></sub> p<sub><font size="-2">2</font></sub> p<sub><font size="-2">3</font></sub></var> .... <br />
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It can also be seen that choosing a random real number from the interval [0,1) is equivalent to choosing an infinite sequence of random digits selected from {0,1,2,3,4,5,6,7,8,9} with equal probability.</p>
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<p>For any positive integer <var>n</var> with <var>d</var> decimal digits, let <var>k</var> be the smallest index such that <br /> <var>p<sub><small>k</small>, </sub></var><var>p<sub><small>k+1</small></sub></var>, ...<var>p<sub><small>k+d-1</small></sub></var> are the decimal digits of <var>n</var>, in the same order.<br />
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Also, let <var>g</var>(<var>n</var>) be the expected value of <var>k</var>; it can be proven that <var>g</var>(<var>n</var>) is always finite and, interestingly, always an integer number.</p>
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<p>For example, if <var>n</var> = 535, then<br />
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for <var>p</var> = 31415926<b>535</b>897...., we get <var>k</var> = 9<br />
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for <var>p</var> = 35528714365004956000049084876408468<b>535</b>4..., we get <var>k</var> = 36<br />
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etc and we find that <var>g</var>(535) = 1008.</p>
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<p>Given that <img style="vertical-align:middle" src="project/images/p_316_decexp1.gif" />, find <img style="vertical-align:middle" src="project/images/p_316_decexp2.gif" /></p>
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<u><i>Note</i></u>: <img style="vertical-align:middle" src="project/images/p_316_decexp3.gif" /> represents the floor function.
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