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Pentomino

From Academic Kids

A pentomino is a geometric shape composed of five (Greek πέντε / 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. (The mirror image of a pentomino does not count as a different pentomino.)

Missing image
Pentonimos.png
Image:Pentonimos.png

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. The F-pentomino is often referred to as the R-pentomino as well, notably in reference to Conway's Game of Life.

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:

Missing image
Pentonimoy.png
Image:Pentonimoy.png

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. There is a subdivision into two subsets of six pentomino's which each have a 6x5 solution. This provides eight 6x10 solutions, and also eight solutions of 5x12.

Remembering the mnemonic "UXPILNFTWYTZV" will enable you to quickly complete the 3x20 rectangle to the amazement of people who have tried it, possibly for hours:

uuxpppllllftttwwzvvv
uxxxpplnnffftwwyzzzv
uuxiiiiinnnftwyyyyzv

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

"Pentominoes" was registered as a trademark by Solomon W. Golomb (#1008964 USPTO 1975 April 15), but this trademark is no longer in effect as of 1982.

There is also a board game of skill based on it, called pentominoes.

The game is played on a 8x8 grid by two or three players. Players take turns in placing pentominoes on the board so that they do not overlap with existing tiles and no tile is used more than once. The objective is to be the last player to place a tile on the board.

The two-player version has been weakly solved; it is a first-player win.

Pentominoes, and similar shapes, are also the basis of a number of other tiling patterns and puzzles.

See also

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