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A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior.

Polyominoes are classified according to how many cells they have:

Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity.[1] Many results with the pieces of 1 to 6 squares were first published in Fairy Chess Review between the years 1937 to 1957, under the name of “dissection problems.” The name polyomino was invented by Solomon W. Golomb in 1953 and it was popularized by Martin Gardner.[2]

Related to polyominoes are polyiamonds, formed from equilateral triangles; polyhexes, formed from regular hexagons; and other plane polyforms. Polyominoes have been generalised to higher dimensions by joining cubes to form polycubes, or hypercubes to form polyhypercubes.

Like many puzzles in recreational mathematics, polyominoes raise many combinatorial problems. The most basic is enumerating polyominoes of a given size. No formula has been found except for special classes of polyominoes. A number of estimates are known, and there are algorithms for calculating them.

Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only simply connected polyominoes.[3]


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