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project_euler/pe/p247.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 247 - 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>Squares under a hyperbola</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 247</h3><span style="width:300px;color:#666;">Published on Friday, 29th May 2009, 09:00 pm; Solved by 871</span></div>
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<div class="problem_content" role="problem">
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<p>Consider the region constrained by 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>x</var> and 0 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>y</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <sup>1</sup>/<sub><var>x</var></sub>.
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</p><p>
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Let S<sub>1</sub> be the largest square that can fit under the curve.<br />
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Let S<sub>2</sub> be the largest square that fits in the remaining area, and so on. <br />
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Let the <i>index</i> of S<sub><var>n</var></sub> be the pair (left, below) indicating the number of squares to the left of S<sub><var>n</var></sub> and the number of squares below S<sub><var>n</var></sub>.
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</p>
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<div style="text-align:center;">
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<img src="project/images/p_247_hypersquares.gif" alt="" /></div>
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<p>
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The diagram shows some such squares labelled by number. <br />
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S<sub>2</sub> has one square to its left and none below, so the index of S<sub>2</sub> is (1,0).<br />
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It can be seen that the index of S<sub>32</sub> is (1,1) as is the index of S<sub>50</sub>. <br />
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50 is the largest <var>n</var> for which the index of S<sub><var>n</var></sub> is (1,1).
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</p>
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<p>
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What is the largest <var>n</var> for which the index of S<sub><var>n</var></sub> is (3,3)?
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</p>
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