| It helps to understand some of the features of
the magic T-hexagon if we begin by obtaining formulae for the numbers of
rows and number of triangles in a T-hexagon of side n. Neither are hard
to find!
The following table gives the number of rows in a T-hexagon.
| Side |
# Rows |
| 1 |
2 |
| 2 |
4 |
| 3 |
6 |
| n |
2n |
To find the formula for the number of triangles in a T-hexagon of
side n, we can start with looking at the first few such hexagons.
| Side |
# Triangles |
| 1 |
3 + 3 = 6 |
| 2 |
5 + 7 + 7 + 5 = 24 |
| 3 |
7 + 9 + 11 + 11 + 9 + 7 = 54 |
| n |
6n2 |
The formula can be found by using the well-known fact that the first
n odd numbers add to n2. By the way, don't you find it
somewhat intriguing that the 'area' in terms of unit shapes of a hexagon
is 6 times the square of number of unit shapes along a side! This means that for the n-sided
T-hexagon, the numbers 1 to 6n2 have to be fitted into 2n
rows. These total 3n2(6n2 + 1), and thus the total
of the numbers in every row has to be:
3n(6n2 + 1)/2
Because both 3 and 6n2 + 1 are odd, only when n is even
can this be a whole number, which it has to be as the sum of whole
numbers. Thus only
when n is even can the magic T-hexagon be contemplated ... let alone
constructed. For example, when n = 1, we know that the numbers 1, 2, 3,
4, 5 and 6 cannot be fitted into the simple T-hexagon of side 1 Some
ideas about the T-hexagons of side 2, 4, 6 and 8 are given on the pages linked
to this one at the top left side. Anti-Magic T-Hexagons
When we first found that magic T-hexagons can only exist for even
values of n, it felt unsatisfactory that the T-hexagons of odd order had
no such properties. It's not that T-hexagons of odd order can't be
constructed ... it was just that they cannot be magic. But it that
strictly true? We think not, and so borrowed the idea of anti-magic
from the realm of magic squares and an interesting parallel. |
