Pentomino

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. 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. 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. 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. 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. The F, L, N, P, Y, and Z pentominoes are chiral; adding their reflections (F′, L′, N′, Q, Y′, Z′) 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.