A pentomino (or 5-omino) is a polyomino of order 5: five equal-sized squares connected edge to edge. If you don’t count rotations and reflections as different, there are exactly 12 free pentominoes; count reflections as distinct and you get 18 one-sided pentominoes; count every rotation and reflection separately and you get 63 fixed pentominoes. The 12 free pieces cover 60 unit squares, so they tile rectangles such as 6×10, 5×12, 4×15, or 3×20.
How many solutions?
The classic 6×10 rectangle has 2,339 distinct solutions if you treat the whole rectangle as fixed (so you don’t double-count by rotating or reflecting the entire board). If you do count rotating and reflecting the whole board as different, that gives 9,356 tilings (2,339 × 4). Other rectangles: 5×12 has 1,010 solutions, 4×15 has 368, and 3×20 has only 2. An easier variant is the 8×8 square with a 2×2 hole in the center—solved by Dana Scott in 1958—which has 65 solutions. The 6×10 count was first found by Colin Brian and Jenifer Haselgrove in 1960.
History
The earliest puzzle using a full set of pentominoes appeared in Henry Dudeney’s The Canterbury Puzzles (1907). Rectangle-tiling problems with the full set showed up in the Problemist Fairy Chess Supplement in 1935. Solomon W. Golomb formally defined pentominoes from 1953 onward and wrote the definitive treatment in his 1965 book Polyominoes: Puzzles, Patterns, Problems, and Packings. He coined the name from Ancient Greek pénte (“five”) and the “-omino” of domino, and named the 12 free pentominoes after letters they resemble: FILiPiNO plus TUVWXYZ. Martin Gardner introduced them to a wide audience in his October 1965 “Mathematical Games” column in Scientific American.
Symmetry and puzzles
F, L, N, P, and Y have no reflection symmetry, so each can be placed in 8 ways (4 rotations × 2 reflections). T, U, V, W, and Z have 4 distinct orientations; I has 2; the X pentomino has only 1. You can’t fill an 8×8 square with the 12 pentominoes alone—that needs 64 squares and the set has 60. With four cells left empty (e.g. a symmetric 2×2 hole), many patterns are possible. Each of the 12 pentominoes satisfies the Conway criterion and can tile the plane by itself.
Pentominoes have inspired board games (e.g. Golomb’s Game, and the French game Blokus uses pentominoes among other polyominoes), video games (e.g. Tetris was inspired by pentomino puzzles but uses 4-square tetrominoes), and fiction—Arthur C. Clarke and Blue Balliett have featured them. They remain a staple of recreational mathematics.
Sources: Pentomino (Wikipedia)