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puzzles/project_euler/pe/p156.html
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puzzles/project_euler/pe/p156.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="keywords" content="programming,mathematics,problems,puzzles" />
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<title>Problem 156 - 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>Counting Digits</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 156</h3><span style="width:300px;color:#666;">Published on Friday, 25th May 2007, 10:00 pm; Solved by 1163</span></div>
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<div class="problem_content" role="problem">
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<p>Starting from zero the natural numbers are written down in base 10 like this:
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<br />
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0 1 2 3 4 5 6 7 8 9 10 11 12....
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</p>
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<p>Consider the digit <var>d</var>=1. After we write down each number <var>n</var>, we will update the number of ones that have occurred and call this number <var>f</var>(<var>n</var>,1). The first values for <var>f</var>(<var>n</var>,1), then, are as follows:</p>
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<div style='text-align:center;'>
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<table style='text-align:center;' align='center'>
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<tr>
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<td><var>n</var></td><td><var>f</var>(<var>n</var>,1)</td>
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</tr><tr>
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<td>0</td><td>0</td>
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</tr><tr>
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<td>1</td><td>1</td>
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</tr><tr>
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<td>2</td><td>1</td>
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</tr><tr>
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<td>3</td><td>1</td>
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</tr><tr>
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<td>4</td><td>1</td>
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</tr><tr>
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<td>5</td><td>1</td>
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</tr><tr>
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<td>6</td><td>1</td>
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</tr><tr>
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<td>7</td><td>1</td>
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</tr><tr>
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<td>8</td><td>1</td>
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</tr><tr>
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<td>9</td><td>1</td>
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</tr><tr>
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<td>10</td><td>2</td>
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</tr><tr>
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<td>11</td><td>4</td>
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</tr><tr>
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<td>12</td><td>5</td>
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</tr></table></div>
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<p>Note that <var>f</var>(<var>n</var>,1) never equals 3.
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<br />
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So the first two solutions of the equation <var>f</var>(<var>n</var>,1)=<var>n</var> are <var>n</var>=0 and <var>n</var>=1. The next solution is <var>n</var>=199981.</p>
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<p>In the same manner the function <var>f</var>(<var>n,d</var>) gives the total number of digits <var>d</var> that have been written down after the number <var>n</var> has been written.
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<br />
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In fact, for every digit <var>d</var> <img src='images/symbol_ne.gif' width='11' height='10' alt='≠' border='0' style='vertical-align:middle;' /> 0, 0 is the first solution of the equation <var>f</var>(<var>n,d</var>)=<var>n</var>.</p>
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<p>Let <var>s</var>(<var>d</var>) be the sum of all the solutions for which <var>f</var>(<var>n,d</var>)=<var>n</var>.
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<br />
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You are given that <var>s</var>(1)=22786974071.</p>
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<p>Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /> <var>s</var>(<var>d</var>) for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> d <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 9.</p>
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<p>Note: if, for some <var>n</var>, <var>f</var>(<var>n,d</var>)=<var>n</var>
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for more than one value of <var>d</var> this value of <var>n</var> is counted again for every value of <var>d</var> for which <var>f</var>(<var>n,d</var>)=<var>n</var>.</p>
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</div><br />
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