Magic hexagon

From Wikipedia, the free encyclopedia

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. 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.


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). Even if there is no normal magical hexagon with order greater than 3, we can always seek for hexagons that are a bit “innormal”, which means that we start not with 1, but with an other number. Here are order 4 and 5 hexagons, discovered by Zahray Arsen:

The order 4 hexagon starts with 3 and ends with 38. The sum number is 111. Order5 hexagon starts with 6 and ends with 66. It’s sum number is 244. The order 6 hexagon can be found under http://www.geocities.com/notlkh/worlds_largest.html. Currently world largest magic hexagon is beed discovered by Zahray Arsen on 22 March 2006:

It starts with 2 and ends with 128. Its sum is 635.

[edit] Proof

Here is a proof sketch 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

$s={1\over{2}}(9n^4-18n^3+18n^2-9n+2).$

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

$M={3s\over{r}}={9n^4-18n^3+18n^2-9n+2\over{2(2n-1)}}.$

Rewriting this as

$32M=72n^3-108n^2+90n-27+{5\over2n-1}$

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

[edit] 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 $6 n 2$ triangles. The sum of all these numbers is given by:

${S}={6n^2(6n^2 + 1)\over 2}$

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

$M={S\over R}={3n^2(6n^2+1)\over 2n}$

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 13 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,