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A pseudo-polyomino, also called a polyking, polyplet or hinged polyomino, is a plane geometric figure formed by joining one or more equal
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 geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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 trian ...
with square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
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 popu ...
es are a subset of the polykings.
The name "polyking" refers to the king
King is a royal title given to a male monarch. A king is an Absolute monarchy, absolute monarch if he holds unrestricted Government, governmental power or exercises full sovereignty over a nation. Conversely, he is a Constitutional monarchy, ...
in chess
Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...
. 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 semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
, rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
, reflection or glide reflection
In geometry, a glide reflection or transflection is a geometric transformation that consists of a reflection across a hyperplane and a translation ("glide") in a direction parallel to that hyperplane, combined into a single transformation. Bec ...
) 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