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project_euler/pe/p163.html
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project_euler/pe/p163.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 163 - Project Euler</title>
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<h2>Cross-hatched triangles</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 163</h3><span style="width:300px;color:#666;">Published on Saturday, 13th October 2007, 02:00 am; Solved by 954</span></div>
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<div class="problem_content" role="problem">
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<p>Consider an equilateral triangle in which straight lines are drawn from each vertex to the middle of the opposite side, such as in the <i>size 1</i> triangle in the sketch below.</p>
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<div style='text-align:center;'><img src='project/images/p_163.gif' width='300' height='200' alt='' /></div>
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<p>Sixteen triangles of either different shape or size or orientation or location can now be observed in that triangle. Using <i>size 1</i> triangles as building blocks, larger triangles can be formed, such as the <i>size 2</i> triangle in the above sketch. One-hundred and four triangles of either different shape or size or orientation or location can now be observed in that <i>size 2</i> triangle.</p>
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<p>It can be observed that the <i>size 2</i> triangle contains 4 <i>size 1</i> triangle building blocks. A <i>size 3</i> triangle would contain 9 <i>size 1</i> triangle building blocks and a <i>size n</i> triangle would thus contain <i>n<sup>2</sup> size 1</i> triangle building blocks.</p>
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<p>If we denote T(<var>n</var>) as the number of triangles present in a triangle of <i>size <var>n</var></i>, then</p>
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<p style="margin-left:50px;">T(1) = 16<br />
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T(2) = 104</p>
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<p>Find T(36).</p>
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