Pentomino

A pentomino is a geometric shape composed of five (Greek pente) identical squares, connected orthogonally. Compare this to a domino (two squares), tetromino (four squares), or polyomino (any number of squares).

There are twelve different pentominoes, and they are named after letters of the alphabet. (NOTE: The mirror image of a pentomino does not count as a different pentomino.)

If you allow mirror images to count as different pentominos, this brings the total to 18. The ones lettered T, V, I, X, U, and W have mirror images that are equivalent after rotation. This matters in some computer games, where mirror image moves are not allowed, such as Tetris-clones and Rampart.

Considering rotations of multiples of 90 degrees only, we have the following symmetry categories:

• L, N, P, F and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image.
• Z can be oriented in 4 ways: 2 by rotation, and 2 more for the mirror image.
• T, V, U and W can be oriented in 4 ways by rotation.
• I can be oriented in 2 ways by rotation.
• X can be oriented in only one way.

For 2D figures in general there is one more category: being orientable in 2 ways, which are each other's mirror image, for example a swastika. There is no pentomino in this category.

For example, the eight possible orientations of the Y pentomino are as follows:

A standard pentomino puzzle is to arrange a set of the twelve possible shapes into a rectangles without holes: 3x20, 4x15, 5x12, 6x10. There are 2,339 solutions for the 6x10 rectangle, 1,010 solutions for 5x12, 368 solutions for 4x15 and just two solutions for the 3x20 rectangle (not counting the three trivial variations of every solution, obtained by rotation and taking the mirror image). Click here for example solutions.

Pentominoes are prominently featured in a subplot of the novel Imperial Earth by Arthur C. Clarke.