Martin Gardner popularized the polyiamonds via Scientific American. Below is a attempt to further promote these shapes via cartoons and art work.
There are 153 tilings of the order 5 sphinx. It takes 25 of the order 1 sphinx to tile the order 5 sphinx. The 12 aspects of the order 1 sphinx tile are color coded. The blacked out areas show the sphinx tiles that change form one solution to the next. This implies that there are at least two different tilings possible for each black shape. Tiling #13 shows the smallest possible area of change and Tiling # 4 shows the second smallest area of change. Tiling # 63 and # 64 show that all tiles change positions at least once. Tilings 36,42,81,93,96,119,133,136 split the S5 in two parts.
For this sequence of the 153 tilings all the shapes that change have edge joined tiles. All adjacent tilings share at least 4 common tiles that do not change.
Polyiamond Division of the H2 Hexagon
Number shapes on a triangular grid divided into equal polyiamond areas containing equal sums give a polyiamond magic constant. The maximum number of tilings is refered to as a Most Perfect Tiling. 
Only 11 of the 12 hexiamonds can tile the H2 hexagon. The shape resulting from a frame shift of the hexagon permits only 6 tilings.
Magic Square Material
- https://oeis.org/A303295 OEIS A303295
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