Equilateral Triangles

If you color equilateral triangles with n colors at the three sides you get (n^3+2n)/3 one-sided and (n^3+3n^2+2n)/6 two-sided distinct pieces. In 1929 MacMahon suggested the pieces and in 1971 Philpott asked for sets, which can fill a triangle. Possible values for n are 1,2,24 and a finite set of numbers above 5000. So far I haven't seen a solution for n=24, but here is one.

If you'd like to solve a smaller puzzle by hand, you can choose a version with 24 pieces and three different edge types from Gamepuzzles.

Colors One-sided Pieces Two-sided Pieces
Number Examples Number Examples
8 176 combined triangle plus rhomb, symmetric 4-4-9-4-4 9 hexagon
9 249 symmetric 8-5-8-5-8-5 hexagon
16 1376 symmetric 8-8-39-8-8-39 hexagon
24 4624 size 68 triangle

8 colors, 176 one-sided pieces:

9 different edge types, 249 one-sided pieces:

16 colors, 1396 one-sided pieces:

24 color, 4624 one-sided pieces: