The Only Magic Hexagon

Nineteen circles are arranged in a hexagon, with rows of three, four, five, four and three. Place the numbers $1$ to $19$ in them so that every straight line of circles, in all three directions, adds to the same total.

Solution

First, the total is forced. The five horizontal rows, of three, four, five, four and three circles, between them cover all nineteen circles exactly once, and the numbers $1$ to $19$ add up to $\tfrac{19 \times 20}{2} = 190$. Five rows sharing that sum equally gives a common total of $190 / 5 = 38$. (This already shows the figure $38$, not some smaller number, is the only one that can work.) And it can indeed be achieved:

Every row reads $38$: the top row $3 + 17 + 18$, the long middle row $16 + 2 + 5 + 6 + 9$, and likewise each of the slanting lines in the other two directions.

The remarkable part is the answer to “find a second arrangement.” There is none. Apart from turning this one by rotating or reflecting the board, no other magic hexagon exists, of this or any other size, using the natural numbers in order. The single example above is the only one there is, which makes it one of the rarest objects in recreational arithmetic.