Nominoes

Nominoes are plane figures formed by joining a finite number of equal-sized squares edge to edge. In many sources they are called polyominoes; the term nominoes appears in puzzle communities as an alternative name or synonym. In formal mathematics the term polyomino is preferred, while nominoes is common in recreational contexts.

An nomino of order n consists of n unit squares connected edge to edge so that the union is connected. Shapes are typically considered up to rigid motion. Depending on conventions, rotations and reflections may be treated as identical (free nominoes), distinct but allowed (one-sided nominoes), or distinct without identifying any motions (fixed nominoes).

Examples include monomino (n = 1), domino (n = 2), tromino (n = 3), tetromino (n = 4), and pentomino (n = 5). The set of pentominoes is especially well known, consisting of 12 distinct free shapes.

Nominoes feature in tiling and packing problems, puzzle design, and computational enumeration. Many nominoes can tile the plane in multiple ways; classic puzzles include tiling rectangles with all pentominoes, such as the 6-by-10 rectangle using each pentomino exactly once. The study of nominoes intersects combinatorics, geometry, and algorithm design, with various counting, symmetry, and tiling questions explored for small and large orders.

See also: polyomino, pentomino, tiling, combinatorial enumeration.