Magic hexagon

From Freepedia

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.

Image:MagicHexagon-Order1.png	Image:MagicHexagon-Order3.png
Order 1 M = 1	Order 3 M = 38

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 normal 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

<math>s={1\over2}(9n^4-18n^3+18n^2-9n+2).</math>

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

<math>M={3s\over r}={9n^4-18n^3+18n^2-9n+2\over2(2n-1)}</math>.

Rewriting this as

<math>32M=72n^3-108n^2+90n-27+{5\over2n-1}</math>

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

Another Type of Magic Hexagon

Hexagons can also be constructed with triangles, as the following diagrams show.

Image:Thex2.gif	Image:Thex3.gif
Order 2	Order 2 with numbers 1–24

This type of configuration can be called a T-hexagon and it has many more properties than the hexagon of hexagons.

As with the above, the rows of triangles run in three directions and there are 24 triangles in a T-hexagon of order 2. In general, a T-hexagon of order n has <math>6n^2</math> triangles. The sum of all these numbers is given by:

<math>{S}={6n^2(6n^2 + 1)\over 2}</math>

If we try to construct a magic T-hexagon of side n, we have to choose n to be even, because there are r = 2n rows so the sum in each row must be

<math>M={S\over R}={3n^2(6n^2+1)\over 2n}</math>

For this to be an integer, n has to be even. To date, magic T-hexagons of order 2, 4, 6 and 8 have been discovered. The first was a magic T-hexagon of order 2, discovered by John Baker on 13th September, 2003. Since that time, John has been collaborating with David King, who discovered that there are 59,674,527 non-congruent magic T-hexagons of order 2.

Magic T-hexagons have a number of properties in common with magic squares, but they also have their own special features. The most surprising of these is that the sum of the numbers in the triangles that point upwards is the same as the sum of those in triangles that point downwards (no matter how large the T-hexagon). In the above example,

<math>{17 + 20 + 22 + 21 + 2 + 6 + 10 + 14 + 3 + 16 + 12 + 7}</math>

<math> {= 5 + 11 + 19 + 9 + 8 + 13 + 4 + 1 + 24 + 15 + 23 + 18}</math>

<math> {= 150}</math>

To find out more about magic T-hexagons, visit Hexagonia [1] or the Hall of Hexagons [[2].

References

Baker. J. E. and King, D. R. (2004) "The use of visual schema to find properties of a hexagon" Visual Mathematics, Volume 5, Number 3

Baker, J. E. and Baker, A. J. (2004) "The hexagon, nature's choice" Archimedes, Volume 4