This is definitely a situation where a little bit of algebra helps.
Suppose that we let S stand for the common total of the sides and the
inner hexagon, and T for the total of the numbers in the white
triangles. Then:
3 sides + inner hexagon + white triangles = 4(sum of the numbers 1
- 6)
3S + S + T = 4 × 21
4S + T = 4 × 21
S + T/4 = 21
This means that T has to be divisible by 4. Thus the least that T can
be is 4 and the most is 16. We give solutions for these two extreme
values below ... and leave it to you to find solutions for the other two
possibilities of 8 and 12.
Note: A quick way to find a solution to this problem is one
that can also be used to help finding magic hexagons. It is based on
fixing the triangles that add to T and then working round the outside in
a systematic way. 
If
the T-totals are made and the sides are correct, then the inner hexagon
just has to fall into place. |
