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 3n2 − 3n + 1. Normal magic hexagons exist only for n = 1 (which is trivial, as it is composed of only 1 cell) and n = 3. Moreover, the solution of order 3 is unique up to reflections and rotations. Meng gives a less intricate constructive proof. The order-3 magic hexagon with numbers 1 through 19 and magic sum 38 has been published many times as a 'new' discovery. An early reference, and possibly the first discoverer, is Ernst von Haselberg (1887).
Proof of normal magic hexagons The numbers in the hexagon are consecutive, and run from 1 to 3 n 2 − 3 n + 1 {\displaystyle 3n^{2}-3n+1} . Hence their sum is a triangular number, namely
s = 1 2 ( 3 n 2 − 3 n + 1 ) ( 3 n 2 − 3 n + 2 ) = 9 n 4 − 18 n 3 + 18 n 2 − 9 n + 2 2 {\displaystyle s={1 \over {2}}(3n^{2}-3n+1)(3n^{2}-3n+2)={9n^{4}-18n^{3}+18n^{2}-9n+2 \over {2}}}
There are r = 2n − 1 rows running along any given direction (E-W, NE-SW, or NW-SE). Each of these rows sums up to the same number M. Therefore:
M = s r = 9 n 4 − 18 n 3 + 18 n 2 − 9 n + 2 2 ( 2 n − 1 ) {\displaystyle M={s \over {r}}={9n^{4}-18n^{3}+18n^{2}-9n+2 \over {2(2n-1)}}}
This can be rewritten as
M = ( 9 n 3 4 − 27 n 2 8 + 45 n 16 − 27 32 ) + 5 32 ( 2 n − 1 ) {\displaystyle M=\left({\frac {9n^{3}}{4}}-{\frac {27n^{2}}{8}}+{\frac {45n}{16}}-{\frac {27}{32}}\right)+{\frac {5}{32\left(2n-1\right)}}}
Multiplying throughout by 32 gives
32 M = 72 n 3 − 108 n 2 + 90 n − 27 + 5 2 n − 1 {\displaystyle 32M=72n^{3}-108n^{2}+90n-27+{5 \over 2n-1}}
which shows that 5 2 n − 1 {\displaystyle {\frac {5}{2n-1}}} must be an integer, hence 2n − 1 must be a factor of 5, namely 2n − 1 = ±1 or 2n − 1 = ±5. The only n ≥ 1 {\displaystyle n\geq 1} that meet this condition are n = 1 {\displaystyle n=1} and n = 3 {\displaystyle n=3} , proving that there are no normal magic hexagons except those of order 1 and 3.
Abnormal magic hexagons Although there are no normal magical hexagons with order greater than 3, certain abnormal ones do exist. In this case, abnormal means starting the sequence of numbers other than with 1. Here is an order 3 hexagon starting from -9 and summing to 0:
Arsen Zahray discovered these order 4 and 5 hexagons:
The order 4 hexagon starts with 3 and ends with 39, its rows summing to 111. The order 5 hexagon starts with 6 and ends with 66 and sums to 244. An order 5 hexagon starting with 15, ending with 75 and summing to 305 is this:
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