Decominoes

A decomino, or 10-omino, is a polyomino of order 10, that is, a polygon in the plane made of 10 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 4,655 different free decominoes (the free decominoes comprise 195 with holes and 4,460 without holes). When reflections are considered distinct, there are 9,189 one-sided decominoes. When rotations are also considered distinct, there are 36,446 fixed decominoes.

The 4,655 free decominoes can be classified according to their symmetry groups:

Unlike both octominoes and nonominoes, no decomino has rotational symmetry of order 4.

195 decominoes have holes. This makes it trivial to prove that the complete set of decominoes cannot be packed into a rectangle, and that not all decominoes can be tiled.

The 4,460 decominos without holes comprise 44,600 unit squares. Thus, the largest square that can be tiled with distinct decominoes is at most 210 units on a side (210 squared is 44,100). Such a square containing 4,410 decominoes was constructed by Livio Zucca.

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