http://www.neverendingbooks.org/index.php/the-15-puzzle-groupoid-2.html lieven le bruyn's blog Wed, 28 Jul 2010 16:42:15 +0200 hourly 1 http://wordpress.org/?v=3.0 http://www.neverendingbooks.org/index.php/the-15-puzzle-groupoid-2.html/comment-page-1#comment-147 Michel Van den Bergh Wed, 20 Jun 2007 07:44:57 +0000 http://www.neverendingbooks.org/?p=16#comment-147 <p>It think there is an even cleaner proof which does not require GAP. By laying out the "snake like path" in a cycle we see that we can right multiply the permutation describing a position by (123) and (1234...15). It is trivial to see that these two permutations generate A15. Indeed by conjugation we get (123), (234), (345),.... which can be used to put any permutation [a1...a15] in the form [123...15] or [213....15] (using the three-cycles we can first move 15 to the end, then 14 etc...).</p> It think there is an even cleaner proof which does not require GAP. By laying out the “snake like path” in a cycle we see that we can right multiply the permutation describing a position by (123) and (1234…15). It is trivial to see that these two permutations generate A15. Indeed by conjugation we get (123), (234), (345),…. which can be used to put any permutation [a1...a15] in the form [123...15] or [213....15] (using the three-cycles we can first move 15 to the end, then 14 etc…).

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