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project_euler/pe/p400.html
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project_euler/pe/p400.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 400 - Project Euler</title>
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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>Fibonacci tree game</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 400</h3><span style="width:300px;color:#666;">Published on Saturday, 27th October 2012, 02:00 pm; Solved by 243</span></div>
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<div class="problem_content" role="problem">
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<p>
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A <b>Fibonacci tree</b> is a binary tree recursively defined as:<ul>
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<li>T(0) is the empty tree.
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<li>T(1) is the binary tree with only one node.
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<li>T(<var>k</var>) consists of a root node that has T(<var>k</var>-1) and T(<var>k</var>-2) as children.
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</ul>
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</p>
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<p>
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On such a tree two players play a take-away game. On each turn a player selects a node and removes that node along with the subtree rooted at that node.<br />
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The player who is forced to take the root node of the entire tree loses.<p>
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<p>
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Here are the winning moves of the first player on the first turn for T(<var>k</var>) from <var>k</var>=1 to <var>k</var>=6.
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<p align=center><img src=project/images/p400_winning.png></p>
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</p>
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</p>
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Let <var>f</var>(<var>k</var>) be the number of winning moves of the first player (i.e. the moves for which the second player has no winning strategy) on the first turn of the game when this game is played on T(<var>k</var>).
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</p>
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<p>
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For example, <var>f</var>(5) = 1 and <var>f</var>(10) = 17.
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</p>
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<p>
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Find <var>f</var>(10000). Give the last 18 digits of your answer.
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</p>
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