Hexamino

A hexamino is a Polyomino composed of six of the same squares dimension. If the Rotation and the reflection are not considered, then there are 35 possible different tiles.

The figure shows all the possible hexaminos, coloured according to their group of symmetry:

20 hexaminos, in black, is without Symétrie.
6 hexaminos, in red, has an axis of symmetry according to the lines of the grid. their group of symmetry has two elements, the identity and the reflection according to a line parallel at the sides of the grid.
2 hexaminos, in green, has an axis of symmetry to 45° compared to the lines of the grid. Their group of symmetry has two elements: identity and the diagonal reflection.
5 hexaminos, in blue, has a symmetry according to point, such a known like a Symétrie of rotation of order 2. Their group of symmetry has two elements: identity and the rotation of 180°.
2 hexaminos, crimson, has two axes of symmetry, aligned according to the lines of the grid. Their group of symmetry has four elements.

If the reflections of the hexaminos were distinct, then the first and the fourth categories would double in the face, which would give a surplus of 25 hexaminos for a total of 60 hexaminos different.

Paving

Although a whole of 35 hexaminos contains 210 squares, it is not possible to pave a Rectangle without hole, contrary to the Pentamino S. a proof, simple, called upon the parity. If the hexaminos are deposited on a chess-board, then 11 cover an even number of squares black (either 2 white and 4 black, or vice versa) and 24 cover an odd number of squares black (3 white and 3 black). In all, an even number of squares black squares will be covered, it does not matter arrangement. However, a rectangle of 210 squares contains 105 squares black and 105 squares white.

There are other simple figures of 210 squares that the hexaminos can pave without hole. For example, a square 15 × 15 with a rectangle 3 × 5 in less in the center makes 210 squares. Having 106 squares white and 104 squares black (or the reverse), its paving is possible according to the proof of the parity mentioned herebefore. A solution exists: to see.

Owners of the cube

Among the hexaminos, eleven of them are owners making it possible to make a cube. They are illustrated opposite, with same colorings by group of symmetry.

Sources

Page maintained by Jürgen Köller, which included symmetry and paving
a page on the polyominos
owners of the cube

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