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Magic Hexagon

The Magic T-Hexagon of Side 4

Getting Started

We can follow the same preliminaries as we went through for the side 2 configuration. Given that there are 2n = 8 rows, and that there are 6n2 = 96 triangles in this 4-sided configuration, the total in each row has to be:

96. 97 / 2 . 8 = 582

The same type of  the purple and blue triangles can be located, but this time only 6 of the 8 rows are taken up by the surrounds.

Thus the purple and yellow regions must add to 582 * 2 = 1164. The purple and blue regions still have the same totals and we can use this fact to say that the lower limit for the inner 2-sided T-hexagon  = 24.25/2 = 300, while the upper limit = 1164 - 24.25/4 = 1014.
If all of these inner totals can be accommodated in magic T-hexagons of side 4, this means that there will be at least 714 of them. But it appears that this is an impossibly large target to aim for. The problem occurs in Row 3 where there are 6 blues and 5 yellows. If the upper limit of 1014 is used, the average yellow is 1014/24 = 42.25, while the average blue is 12.5. Taking average values for the blue and yellow triangles, this leaves 582 - 6*12.5 - 5*42.25 = 295.75 for the two mustard triangles. This is far more than can be made by just two numbers in the mustard cells. We calculated that the average yellow needs to be around 26.5, and the average blue around 44.5. It means that the highest that the yellow region can go is 26.5*24 = 636.

These are the figures that we experimented with to get the ball rolling.