A pentomino (or 5-omino) is a polyomino of order 5; that is, a polygon in the plane made of 5 equal-sized squares connected edge to edge. The term is derived from the Greek word for '5' and "domino". When rotations and reflections are not considered to be distinct shapes, there are 12 different free pentominoes. When reflections are considered distinct, there are 18 one-sided pentominoes. When rotations are also considered distinct, there are 63 fixed pentominoes. Pentomino tiling puzzles and games are popular in recreational mathematics. Usually, video games such as Tetris imitations and Rampart consider mirror reflections to be distinct, and thus use the full set of 18 one-sided pentominoes. (Tetris itself uses 4-square shapes called tetrominos.) Each of the twelve pentominoes satisfies the Conway criterion; hence, every pentomino is capable of tiling the plane. Each chiral pentomino can tile the plane without being reflected.
History
The earliest puzzle containing a complete set of pentominoes appeared in Henry Dudeney's book The Canterbury Puzzles, published in 1907. The earliest tilings of rectangles with a complete set of pentominoes appeared in the Problemist Fairy Chess Supplement in 1935, and further tiling problems were explored in the PFCS, and its successor, the Fairy Chess Review. Pentominoes were formally defined by American professor Solomon W. Golomb starting in 1953 and later in his 1965 book Polyominoes: Puzzles, Patterns, Problems, and Packings. They were introduced to the general public by Martin Gardner in his October 1965 Mathematical Games column in Scientific American. Golomb coined the term "pentomino" from the Ancient Greek πέντε / pénte, "five", and the -omino of domino, fancifully interpreting the "d-" of "domino" as if it were a form of the Greek prefix "di-" (two). Golomb named the 12 free pentominoes after letters of the Latin alphabet that they resemble, using the mnemonic FILiPiNo along with the end of the alphabet (TUVWXYZ). Japanese master craftsman Saburo Oguro also popularized a wooden set of pentominoes based on the Chinese Zodiac symbols, which led to the wide-scale proliferation of polyomino puzzles. John Horton Conway proposed an alternate labeling scheme for pentominoes, using O instead of I, Q instead of L, R instead of F, and S instead of N. The resemblance to the letters is more strained, especially for the O pentomino, but this scheme has the advantage of using 12 consecutive letters of the alphabet. It is used by convention in discussing Conway's Game of Life, where, for example, one speaks of the R-pentomino instead of the F-pentomino.
Symmetry F, L, N, P, and Y can be oriented in 8 ways: 4 by rotation, and 4 more for the mirror image. Their symmetry group consists only of the identity mapping. T, and U can be oriented in 4 ways by rotation. They have an axis of reflection aligned with the gridlines. Their symmetry group has two elements, the identity and the reflection in a line parallel to the sides of the squares. V and W also can be oriented in 4 ways by rotation. They have an axis of reflection symmetry at 45° to the gridlines. Their symmetry group has two elements, the identity and a diagonal reflection. Z can be oriented in 4 ways: 2 by rotation, and 2 more for the mirror image. It has point symmetry, also known as rotational symmetry of order 2. Its symmetry group has two elements, the identity and the 180° rotation. I can be oriented in 2 ways by rotation. It has two axes of reflection symmetry, both aligned with the gridlines. Its symmetry group has four elements, the identity, two reflections and the 180° rotation. It is the dihedral group of order 2, also known as the Klein four-group. X can be oriented in only one way. It has four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4. Its symmetry group, the dihedral group of order 4, has eight elements. The F, L, N, P, Y, and Z pentominoes are chiral; adding their reflections (F′, J, N′, Q, Y′, S) brings the number of one-sided pentominoes to 18. If rotations are also considered distinct, then the pentominoes from the first category count eightfold, the ones from the next three categories (T, U, V, W, Z) count fourfold, I counts twice, and X counts only once. This results in 5×8 + 5×4 + 2 + 1 = 63 fixed pentominoes. The eight possible orientations of the F, L, N, P, and Y pentominoes, and the four possible orientations of the T, U, V, W, and Z pentominoes are illustrated:
For 2D figures in general there are two more categories:
Being orientable in 2 ways by a rotation of 90°, with two axes of reflection symmetry, both aligned with the diagonals. This type of symmetry requires at least a heptomino. Being orientable in 2 ways, which are each other's mirror images, for example a swastika. This type of symmetry requires at least an octomino.
Games
Tiling puzzle (2D)
A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box must have an area of 60 units. Possible sizes are 6×10, 5×12, 4×15 and 3×20. The 6×10 case was first solved in 1960 by married couple Colin Brian Haselgrove and Jenifer Haselgrove. There are exactly 2,339 solutions, excluding trivial variations obtained by rotation and reflection of the whole rectangle but including rotation and reflection of a subset of pentominoes (which sometimes provides an additional solution in a simple way). The 5×12 box has 1010 solutions, the 4×15 box has 368 solutions, and the 3×20 box has just 2 solutions (one is shown in the figure, and the other one can be obtained from the solution shown by rotating, as a whole, the block consisting of the L, N, F, T, W, Y, and Z pentominoes). A somewhat easier (more symmetrical) puzzle, the 8×8 rectangle with a 2×2 hole in the center, was solved by Dana Scott as far back as 1958. There are 65 solutions. Scott's algorithm was one of the first applications of a backtracking computer program. Variations of this puzzle allow the four holes to be placed in any position. One of the external links uses this rule. Efficient algorithms have been described to solve such problems, for instance by Donald Knuth. Running on modern hardware, these pentomino puzzles can now be solved in mere seconds.
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