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puzzles/project_euler/pe/p295.html
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puzzles/project_euler/pe/p295.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 295 - Project Euler</title>
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<h2>Lenticular holes</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 295</h3><span style="width:300px;color:#666;">Published on Saturday, 5th June 2010, 01:00 pm; Solved by 342</span></div>
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<div class="problem_content" role="problem">
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<p>We call the convex area enclosed by two circles a <i>lenticular hole</i> if:
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<ul>
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<li>The centres of both circles are on lattice points.</li>
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<li>The two circles intersect at two distinct lattice points.</li>
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<li>The interior of the convex area enclosed by both circles does not contain any lattice points.
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</li>
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</ul>
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</p>
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<p>Consider the circles:<BR />
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C<sub>0</sub>: <var>x</var><sup>2</sup>+<var>y</var><sup>2</sup>=25<BR />
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C<sub>1</sub>: (<var>x</var>+4)<sup>2</sup>+(<var>y</var>-4)<sup>2</sup>=1<BR />
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C<sub>2</sub>: (<var>x</var>-12)<sup>2</sup>+(<var>y</var>-4)<sup>2</sup>=65
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</p>
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<p>
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The circles C<sub>0</sub>, C<sub>1</sub> and C<sub>2</sub> are drawn in the picture below.</p>
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<div align='center'><img src="project/images/p295_lenticular.gif" /></div>
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<p>
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C<sub>0</sub> and C<sub>1</sub> form a lenticular hole, as well as C<sub>0</sub> and C<sub>2</sub>.</p>
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<p>
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We call an ordered pair of positive real numbers (r<sub>1</sub>, r<sub>2</sub>) a <i>lenticular pair</i> if there exist two circles with radii r<sub>1</sub> and r<sub>2</sub> that form a lenticular hole.
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We can verify that (1, 5) and (5, <img src='images/symbol_radic.gif' width='14' height='16' alt='√' border='0' style='vertical-align:middle;' />65) are the lenticular pairs of the example above.</p>
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<p>
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Let L(N) be the number of <b>distinct</b> lenticular pairs (r<sub>1</sub>, r<sub>2</sub>) for which 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> r<sub>1</sub> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> r<sub>2</sub> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> N.<br />
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We can verify that L(10) = 30 and L(100) = 3442.</p>
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<p>
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Find L(100 000).
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</p>
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