pentomino

A pentomino is a polyomino consisting of five unit squares connected edge-to-edge. Like other polyominoes, pentominoes are studied up to translation, rotation, and reflection. In the standard classification of free pentominoes, there are 12 distinct shapes.

These shapes are traditionally named after letters they resemble: F, I, L, N, P, T, U, V, W, X, Y, and Z. Many pentominoes are asymmetric, and some have distinct mirror images (chirality) while others are symmetric.

Counting variants, there are three commonly used categories: free pentominoes (rotations and reflections considered the same) number 12; one-sided pentominoes (reflections counted as distinct) number 18; fixed pentominoes (translations, rotations, and reflections all counted separately) number 63.

Historically, pentominoes were popularized by Solomon W. Golomb in the mid-20th century as part of the study of polyominoes. The term pentomino combines penta-, meaning five, with domino, referring to the square unit. Pentominoes appear in recreational mathematics, puzzle design, and computational tiling problems, offering rich combinatorial challenges.

A central theme in pentomino puzzles is tiling. A famous challenge is to tile a 6-by-10 rectangle using one of each of the twelve pentominoes. Other rectangle sizes, such as 5-by-12 and 4-by-15, can also be tiled with the complete set of pentominoes, illustrating the diversity of possible tilings.