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puzzles/project_euler/pe/p301.html
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puzzles/project_euler/pe/p301.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta name="author" content="Colin Hughes" />
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<title>Problem 301 - Project Euler</title>
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<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>
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<div class="problem_content" role="problem">
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<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>
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<p>We'll consider the three-heap normal-play version of Nim, which works as follows:<br />
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- At the start of the game there are three heaps of stones.<br />
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- On his turn the player removes any positive number of stones from any single heap.<br />
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- The first player unable to move (because no stones remain) loses.</p>
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<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:
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<ul><li>zero if, with perfect strategy, the player about to move will eventually lose; or</li>
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<li>non-zero if, with perfect strategy, the player about to move will eventually win.</li></ul>
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</p>
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<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 />
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- current player moves to (1,2,1)<br />
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- opponent moves to (1,0,1)<br />
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- current player moves to (0,0,1)<br />
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- opponent moves to (0,0,0), and so wins.</p>
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<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 ?
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</p>
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