A magic hexagon of order n is a 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. A normal magic hexagon contains the consecutive integers from 1 to 3n² − 3n + 1. It turns out that magic hexagons exist only for n = 1 (which is trivial) and n = 3. Moreover, the solution of order 3 is essentially unique.

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

What follows is a proof that no magic hexagons exist except those of order 1 and 3.

The magic constant M of a normal magic hexagon can be determined as follows. The numbers in the hexagon are consecutive, so their sum is a triangular number, namely

The rows run in three directions, so each number is counted three times. The sum of all rows is therefore 3s. But there are r = 3(2n − 1) rows in the hexagon, so the sum in each row must be

.

Rewriting this as

shows that 5/(2n − 1) must be an integer. The only n ≥ 1 that meet this condition are n = 1 and n = 3.