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Magic Hexagons

The Magic T-Hexagon of Side 2

Getting Started

Before showing how a magic T-hexagon of side 2 can be made, a couple of observations about some constraints in its construction are worth making.

Given that there are 2n = 4 rows, and that there are 6n2 = 24 triangles in this configuration, the total in each row has to be:

24. 25 / 2 . 4 = 75

Now consider the two following diagrams. In the first, the purple triangle is surrounded by 3 rows of 75, and thus the total of all the numbers in the purple triangle is 75.
The same is true of the blue triangle in the second diagram. Given that the purple and blue triangles have a 1-sided T-hexagon in common, we can see that, in the diagram below, the purple and yellow parts and the blue and yellow parts both add to 75.

It is also useful to note that the remaining purple and blue triangles must add to the same total.

Characterising the Hexagons

It was found useful to characterise the T-hexagons by the total of the yellow part - that is the inside T-hexagon of side n - 1. For example, if the numbers 1 - 6 are used in this hexagon (giving a total of 21), then the numbers in the purple and and blue parts must both add to 54, giving the yellow + purple total = 75. Thus the smallest total in the yellow hexagon that can be contemplated is 21.

To find the largest total that can be accommodated in the yellow part, we need to put numbers as small as possible into the purple and blue parts. For this, the numbers 1 - 6 cannot be used, because they add to 21 and thus cannot be shared between purple and blue triangles that have the same total. Thus the numbers 1, 2, 3, 4, 5 and 7 need to be tried. This gives 11 each to the purple and blue, and the remaining yellow part has to have a total of 64. Thus the largest total in the yellow hexagon that can be contemplated is 64.

The next page gives some examples of magic T-hexagons.