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Given 4 each of these pieces ... And this triangular grid: In how plenty of ways deserve to you to fill the hexagon with the 12 pieces?First number the triangle of the hexagon favor this Define a spoke to it is in a two-triangle diamondwith one point at the center, and also the other suggest on the leaf of the hexagon.There room 6 the them, consist of of triangles(3-9 G) (7-8 B) (11-10 R) (14-15 R) (22-16 G) (18-17 B).(The letter R G or B signify the colors that will certainly fill those places.)Then classify the remedies by the variety of spokes occupied:0, 2, 4, or 6.It is not possible to to fill 1, 3, or 5, since the central triangles either belong to a spoke, or are filled in pairs.The total variety of solutions is 12.If you overlook the colors, then the number is just 6.In the center three columns below, the last two solutionsare rotations of the ones above them, however with the colors changed.Once you choose the spokes, every little thing else is forced,or virtually so in the situation of 0 spokes. Zero Spokes ( 0 ) ( 2 services )These resemble donuts, v 9 diamonds approximately the perimeter,and 3 in the center.

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2 directions for the inner ring make various solutions.The very first one has all red together, all green together, all blue together.The other has three of each shade 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 22,23B1,2R 3,4R 5,11G 6,7R 8,15G 9,10B 12,19G 13,14B 16,17R 18,24G 20,21B 22,23BTwo the opposite Spokes ( 2o ) (3 solutions)These room 60- or 120-degree rotations of every other, yet with different shade spokes.The an initial has 2 blue spokes, the 2nd green, the 3rd red.1,2R 3,4R 5,11G 6,13G 7,8B
9,10B 12,19G 14,20G 15,16B 17,18B 21,22R 23,24R1,2R 3,9G 4,5B 6,7R 8,15G 10,17G 11,12B 13,14B 16,22G 18,19R 20,21B 23,24R1,7G 2,3B 4,5B 6,13G 8,9R 10,11R 12,19G 14,15R 16,17R 18,24G 20,21B 22,23BTwo surrounding Spokes ( 2a ) (3 solutions)These room 60-degree rotations of each other, v different shade spokes.1,2R 3,9G 4,5B 6,7R 8,15G 10,11R 12,19G 13,14B 16,17R 18,24G 20,21B 22,23B1,7G 2,3B 4,5B 6,13G 8,9R 10,11R 12,19G 14,20G 15,16B 17,18B 21,22R 23,24R1,2R 3,4R 5,11G 6,7R 8,15G 9,10B 12,19G 13,14B 16,22G 17,18B 20,21B 23,24RFour Spokes ( 4 ) (3 solutions)These are additionally 60-degree rotations of each other. They room the complements the the ahead set.The an initial has two environment-friendly spokes, the 2nd 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 23,24R3,9G 7,8B 10,11R 14,15R 1,2R 4,5B 6,13G 12,19G 16,17R 18,24G 20,21B 22,23B3,9G 7,8B 17,18B 10,11R 1,2R 4,5B 6,13G 12,19G 14,20G 15,16B 21,22R 23,24RSix spokes: ( 6 ) (1 solution)Opposite the 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