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Magic hexagon
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<table><tbody><tr><td></td><td></td></tr><tr><td>Order <i>n</i> = 1<i>M</i> = 1</td><td>Order <i>n</i> = 3<i>M</i> = 38</td></tr></tbody></table> <p>A magic hexagon of order <i>n</i> is an arrangement of numbers in a <a href="/facts/Centered_hexagonal_number/M5eUvBGG">centered hexagonal pattern</a> with <i>n</i> cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same <a href="/facts/Magic_constant/UgdHE47U">magic constant</a> <i>M</i>. A normal magic hexagon contains the consecutive <a href="/facts/Integer/PUwwrawx">integers</a> from 1 to 3<i>n</i>2 − 3<i>n</i> + 1. Normal magic hexagons exist only for <i>n</i> = 1 (which is trivial, as it is composed of only 1 cell) and <i>n</i> = 3. Moreover, the solution of order 3 is essentially unique. Meng gives a less intricate constructive <a href="/facts/Mathematical_proof/7ECDrU80">proof</a>. </p><p>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). </p>