Polyamonds are obtained by joining equilateral triangles. There is only one mono-, dia- and triamond, 3 tetriamonds, 4 pentiamonds, 12 hexiamonds and 24 hetiamonds. With 12 tiles the Hexiamonds are best suited for a laying game.



Solubility of Hexiamond Figures

Each figure and each Hexiamond token consists of 2 different “types” of triangles: Triangles with point up and triangles with point down. In 10 of the 12 hexiamonds, 3 triangles point upwards and 3 point downwards. For the other two hexiamonds shown below, the ratio of the two types of triangles is 4 to 2.

If you add the triangles with point upwards and the triangles with point downwards for a figure consisting of all 12 hexiamonds, the difference between the two sums may only be 0, 4 or -4. Otherwise no solution can exist. In my solution program Hexia I called this value parity.

This figure consists of 39 triangles with point upwards and 33 triangles with point downwards. The difference (parity) is 6. This figure cannot be laid with the twelve Hexiamond pieces.


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