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A magic hexagon of order ''n'' is an arrangement of numbers in a centered hexagonal pattern with ''n'' cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant ''M''. A normal magic hexagon contains the consecutive integers from 1 to 3''n''2 − 3''n'' + 1. It turns out that normal magic hexagons exist only for ''n'' = 1 (which is trivial) and ''n'' = 3. Moreover, the solution of order 3 is essentially unique.〔Trigg, C. W. ("A Unique Magic Hexagon" ), ''Recreational Mathematics Magazine'', January–February 1964. Retrieved on 2009-12-16.〕 Meng also gave a less intricate constructive proof.〔 The order-3 magic hexagon has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is Ernst von Haselberg (1887). ==Proof that there are no normal magic hexagons except those of order 1 and 3== The numbers in the hexagon are consecutive, and run from 1 to . Hence their sum is a triangular number, namely : There are ''r'' = (2''n'' − 1) rows running along any given direction (E-W, NE-SW, or NW-SE). Each of these rows sum up to the same number ''M''. Therefore: : This can be rewritten as : Multiplying throughout by 32 gives : which shows that must be an integer, hence 2n-1 must be a factor of 5, namely 2n-1 = 1 or 2n-1 = 5. The only that meet this condition are and . QED. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Magic hexagon」の詳細全文を読む スポンサード リンク
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