Games - Polyomino

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	Games - 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.

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 (polycube)or hypercubes.

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.

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.

Free, one-sided, and fixed polyominoes

There are three common ways of defining distinct polyominoes:

free polyominoes must be different under translation, rotation, and reflection
one-sided polyominoes must be different under translation and rotation within in the plane
fixed polyominoes must be different only under translation.

The dihedral group

D_4

is the group of symmetries (symmetry group) of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the x-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its symmetric images under the symmetries of

D_4

. However, those images are not necessarily distinct: the more symmetry a free polyomino has, the less distinct fixed versions there are. Therefore, a free polyomino which is invariant under some or all non-trivial symmetries of

D_4

may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group

D_4

.

Possible symmetries (with the least number of squares needed in a polyomino):

8 fixed polyominoes for each free polyomino:
*no symmetry (4)
4 fixed polyominoes for each free polyomino:
*symmetry with respect to one of the grid line directions (4)
*symmetry with respect to a diagonal line (3)
*2-fold rotational symmetry (4)
2 fixed polyominoes for each free polyomino:
*symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry (2)
*symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry (7)
*4-fold rotational symmetry (8)
1 fixed polyomino for each free polyomino:
*all symmetry of the square (1)

Special polyominos

A straight polyomino is a polyomino with all of the squares in a line.

A T-polyomino is a polyomino in the shape of a T, with three squares in the crossbar at the top of the T.

An L-polyomino is a polyomino in the shape of an L, formed by adding a single square to a straight polyomino.

A square polyomino is a polyomino in the shape of a square.

A skew polyomino is formed by joining two straight polyominos of the same length, so that the right end square of one is directly above the left end square of the other.

Number of polyominoes

We call n the number of squares, and

A_n

the number of fixed polyominoes with n squares (possibly with holes). An enumeration gives the following table:

The number of free polyominoes without holes is given by ; the number of one-sided polyominoes is given by .

As of 2004, Iwan Jensen has enumerated the fixed polyominoes up to n=56:

A_{56}

is approximately 6.915×10³¹. Free polyominoes have been enumerated up to n=28. See the external links for tables containing the known results.

Algorithms for enumeration of fixed polyominoes

Inductive exhaustive search

The most obvious method of enumerating the polyominoes, and also one of the slowest, is inductive exhaustive search. Given a list of polyominoes of area n, take each polyomino in turn, embed it in an n×n square, surround that square with a collar of cells to create an (n+2)×(n+2) square. For each vacant cell in that square that is adjacent to at least one occupied cell, fill the cell and strike out a bounding row of vacant cells and a bounding column of vacant cells. The resulting (n+1)×(n+1) square contains a candidate polyomino of area n+1. If this configuration has not been encountered before, it is added to the list of polyominoes of area n+1. Comparison with the polyominoes of area n+1 already seen must take account of position and symmetry, depending on whether fixed or free polyominoes are being counted. Position can be accounted for by translating the candidate polyomino to the top left corner of the (n+1)×(n+1) square. In order to compute the number of fixed polyominoes, rotations and reflections must also be accounted for.

[ Visit the complete Wikipedia entry for Polyomino ]

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