Polyomino

A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a type of tiling, where each square is a tile. Polyominoes are named according to the number of squares they contain.

Specifically, a polyomino composed of n squares is called an n-omino. Common examples include the domino (2 squares), the tromino (3 squares), the tetromino (4 squares), the pentomino (5 squares), and the hexomino (6 squares).

Polyominoes can be considered either free, fixed, or one-sided. Free polyominoes are distinct regardless of rotations or reflections. Fixed polyominoes are distinct under rotations but not reflections. One-sided polyominoes are distinct under reflections but not rotations. When discussing polyominoes, the term often implicitly refers to free polyominoes unless otherwise specified.

Polyominoes are studied in recreational mathematics and have connections to fields like combinatorics and tiling problems. A key question in polyomino study is determining the number of n-ominoes for a given n. These numbers grow rapidly as n increases. Tiling a given region with polyominoes is a classic combinatorial problem.