A pseudo-polyomino, also called a polyking, polyplet or hinged polyomino, is a plane geometric figure formed by joining one or more equal

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

s edge-to-edge or corner-to-corner at 90°. It is a

polyform In recreational mathematics, a polyform is a plane figure or solid compound 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 ...

with

cells. The

polyomino 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. Polyominoes have been used in pop ...

es are a subset of the polykings. The name "polyking" refers to the

king King is the title given to a male monarch in a variety of contexts. The female equivalent is queen regnant, queen, which title is also given to the queen consort, consort of a king. *In the context of prehistory, antiquity and contempora ...

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...

. The ''n''-kings are the ''n''-square shapes which could be occupied by a king on an infinite chessboard in the course of legal moves. Golomb uses the term ''pseudo-polyomino'' referring to kingwise-connected sets of squares.

Enumeration of polykings

Free, one-sided, and fixed polykings

There are three common ways of distinguishing polyominoes and polykings for enumeration: *''free'' polykings are distinct when none is a rigid transformation (

translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...

rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflecti ...

) of another (pieces that can be picked up and flipped over). *''one-sided'' polykings are distinct when none is a translation or rotation of another (pieces that cannot be flipped over). *''fixed'' polykings are distinct when none is a translation of another (pieces that can be neither flipped nor rotated). The following table shows the numbers of polykings of various types with ''n'' cells.

Notes

External links

* {{Polyforms Polyforms Recreational mathematics