81 lines
5.3 KiB
HTML
81 lines
5.3 KiB
HTML
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<h2>Hexagonal tile differences</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 128</h3><span style="width:300px;color:#666;">Published on Friday, 29th September 2006, 06:00 pm; Solved by 2257</span></div>
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<p>A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "12 o'clock" and numbering the tiles 2 to 7 in an anti-clockwise direction.</p>
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<p>New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, 38 to 61, and so on. The diagram below shows the first three rings.</p>
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<p>By finding the difference between tile <i>n</i> and each its six neighbours we shall define PD(<i>n</i>) to be the number of those differences which are prime.</p>
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<p>For example, working clockwise around tile 8 the differences are 12, 29, 11, 6, 1, and 13. So PD(8) = 3.</p>
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<p>In the same way, the differences around tile 17 are 1, 17, 16, 1, 11, and 10, hence PD(17) = 2.</p>
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<p>It can be shown that the maximum value of PD(<i>n</i>) is 3.</p>
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<p>If all of the tiles for which PD(<i>n</i>) = 3 are listed in ascending order to form a sequence, the 10th tile would be 271.</p>
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<p>Find the 2000th tile in this sequence.</p>
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