Magic hexagon

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. 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. 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).

The numbers in the hexagon are consecutive, and run from 1 to ${\displaystyle (3n^{2}-3n+1)}$. Hence their sum is a triangular number, namely

There are r = (2n − 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 ${\displaystyle {\frac {5}{2n-1}}}$ must be an integer, hence 2n-1 must be a factor of 5, namely 2n-1 = 1 or 2n-1 = 5. The only ${\displaystyle n\geq 1}$ that meet this condition are ${\displaystyle n=1}$ and ${\displaystyle n=3}$, proving that there are no normal magic hexagons except those of order 1 and 3.

...
Wikipedia