Polyomino

A polyomino is a polyform with the square as its base form. It is constructed by placing a number of identical squares in distinct locations on the plane, keeping the shape connected, and in such a way that at least one edge of each square coincides with an edge of one of the other squares. Polyominoes with from 1 to 6 squares are called respectively monominoes, dominoes, trominoes (or triominoes), tetrominoes, pentominoes and hexominoes. Related to polyominoes are polyiamonds (formed from equilateral triangles), polyhexes (formed from regular hexagons), and other polyforms.

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In some contexts, the definition of a polyomino is relaxed or refined. Sometimes it is requested that polyominoes are simply connected, while on other occasions may have holes (in other words, regions which are not tiled with squares but which are unconnected to the exterior of the polyomino). Sometimes polyominoes are generalised to three or more dimensions by aggregating cubes or hypercubes.

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Polyominoes have been used in popular puzzles since the late 19th century, but were first studied systematically by Solomon W. Golomb and were popularized by Martin Gardner. The game Tetris is based on tetrominoes.

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Polyominoes are a source of combinatorial problems, the first being to enumerate polyominoes for various sizes. No formula has been found except for special classes of polyominoes. However, a number of estimates are known, and there are algorithms for counting them.

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Polyform: In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specifi...

Square: A square as a geometric shape is described and illustrated at square (geometry). There are other concepts derived from it:...

Plane: Plane may mean:...