Given four each of these pieces … and this triangular grid:
In how many ways can you fill the hexagon
with the 12 pieces?
First number the triangles of the hexagon like this
Define a spoke to be a two-triangle diamond
with one point at the center,
and the other point on the edge of the hexagon.
There are 6 of them, consisting of triangles
(3-9 G) (7-8 B) (11-10 R) (14-15 R) (22-16 G) (18-17 B).
(The letters R G or B denote the colors that will fill those places.)
Then classify the solutions by the number of spokes occupied:
0, 2, 4, or 6.
It is not possible to fill 1, 3, or 5, because the central triangles either
belong to a spoke, or are filled in pairs.
The total number of solutions is 12.
If you disregard the colors, then the number is just 6.
In the middle three columns below, the last two solutions
are rotations of the ones above them, but with the colors changed.
Once you choose the spokes, everything else is forced,
or nearly so in the case of 0 spokes.
Zero Spokes ( 0 ) ( 2 solutions )
These resemble donuts, with 9 diamonds around the perimeter,
and 3 in the center. Two directions for the inner ring make different solutions.
The first one has all red together, all green together, all blue together.
The other has three of each color together and the fourth one opposite.
1,2R 3,4R 5,11G 6,7R 8,9R 10,17G 12,19G 13,14B 15,16B 18,24G 20,21B 22,23B
1,2R 3,4R 5,11G 6,7R 8,15G 9,10B 12,19G 13,14B 16,17R 18,24G 20,21B 22,23B
Two Opposite Spokes ( 2o ) (3 solutions)
These are 60- or 120-degree rotations of each other, but with different color spokes.
The first has two blue spokes, the second green, the third red.
1,2R 3,4R 5,11G 6,13G 7,8B 9,10B 12,19G 14,20G 15,16B 17,18B 21,22R 23,24R
1,2R 3,9G 4,5B 6,7R 8,15G 10,17G 11,12B 13,14B 16,22G 18,19R 20,21B 23,24R
1,7G 2,3B 4,5B 6,13G 8,9R 10,11R 12,19G 14,15R 16,17R 18,24G 20,21B 22,23B
Two Adjacent Spokes ( 2a ) (3 solutions)
These are 60-degree rotations of each other, with different color spokes.
1,2R 3,9G 4,5B 6,7R 8,15G 10,11R 12,19G 13,14B 16,17R 18,24G 20,21B 22,23B
1,7G 2,3B 4,5B 6,13G 8,9R 10,11R 12,19G 14,20G 15,16B 17,18B 21,22R 23,24R
1,2R 3,4R 5,11G 6,7R 8,15G 9,10B 12,19G 13,14B 16,22G 17,18B 20,21B 23,24R
Four Spokes ( 4 ) (3 solutions)
These are also 60-degree rotations of each other.
They are the complements of the previous set.
The first has two green spokes,
the second two red, the third two blue spokes.
3,9G 10,11R 16,22G 17,18B 1,2R 4,5B 6,7R 8,15G 12,19G 13,14B 20,21B 23,24R
3,9G 7,8B 10,11R 14,15R 1,2R 4,5B 6,13G 12,19G 16,17R 18,24G 20,21B 22,23B
3,9G 7,8B 17,18B 10,11R 1,2R 4,5B 6,13G 12,19G 14,20G 15,16B 21,22R 23,24R
Six spokes: ( 6 ) (1 solution)
Opposite of the donuts.
1,2R 3,9G 4,5B 6,13G 7,8B 10,11R 12,19G 14,15R 16,22G 17,18B 20,21B 23,24R