Magic hexagon
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 integer
s 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. 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).
Although there are no normal magical hexagons with order greater than 3, certain abnormal ones do exist. In this case, abnormal means starting the sequence of numbers other than with 1. Arsen Zahray discovered these order 4 and 5 hexagons:
The order 4 hexagon starts with 3 and ends with 39, its rows summing to 111. The order 5 hexagon starts with 6 and ends with 66 and sums to 244.
An order 6 hexagon can be seen below. It was created by Louis Hoelbling, October 11, 2004:
It starts with 21, ends with 111, and its sum is 546.
The largest magic hexagon so far was discovered using simulated annealing by Arsen Zahray on 22 March 2006:
It starts with 2, ends with 128 and its sum is 635.
However, a slightly larger, order 8 magic hexagon was generated by Louis K. Hoelbling on February 5, 2006:
It starts with -84 and ends with 84, and its sum is 0.
The magic constant M of a normal magic hexagon can be determined as follows. The numbers in the hexagon are consecutive, and run from 1 to (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:
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.
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
triangles. The sum of all these numbers is given by:
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
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,
To find out more about magic T-hexagons, visit Hexagonia or the Hall of Hexagons.
Centered hexagonal number
A centered hexagonal number, or hex number, is a centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot in a hexagonal lattice....
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
Magic constant
The magic constant or magic sum of a magic square is the sum of numbers in any row, column, and diagonal of the magic square. For example, the magic square shown below has a magic constant of 15....
. A normal magic hexagon contains the consecutive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s 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. Meng also gave a less intricate constructive proof.
 |  |
| Order 1 M = 1 | Order n=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).
Although there are no normal magical hexagons with order greater than 3, certain abnormal ones do exist. In this case, abnormal means starting the sequence of numbers other than with 1. Arsen Zahray discovered these order 4 and 5 hexagons:
 |  |
| Order 4 M = 111 | Order 5 M = 244 |
The order 4 hexagon starts with 3 and ends with 39, its rows summing to 111. The order 5 hexagon starts with 6 and ends with 66 and sums to 244.
An order 6 hexagon can be seen below. It was created by Louis Hoelbling, October 11, 2004:
It starts with 21, ends with 111, and its sum is 546.
The largest magic hexagon so far was discovered using simulated annealing by Arsen Zahray on 22 March 2006:
It starts with 2, ends with 128 and its sum is 635.
However, a slightly larger, order 8 magic hexagon was generated by Louis K. Hoelbling on February 5, 2006:
It starts with -84 and ends with 84, and its sum is 0.
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, and run from 1 to (3n^2-3n+1). Hence their sum is a triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...
, 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:
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.
Another type of magic hexagon
Hexagons can also be constructed with triangles, as the following diagrams show.| 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
triangles. The sum of all these numbers is given by: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
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,
To find out more about magic T-hexagons, visit Hexagonia or the Hall of Hexagons.