Pseudo-polyomino

Pseudo-polyomino

Pseudo-polyomino

Geometric shapes formed from squares

A pseudo-polyomino, also called a polyking, polyplet or hinged polyomino, is a plane geometric figure formed by joining one or more equal squares edge-to-edge or corner-to-corner at 90°. It is a polyform with square cells. The polyominoes are a subset of the polykings.

The 22 free tetrakings

The name "polyking" refers to the king in chess. 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.[1]

Enumeration of polykings

10 congruent mutilated chessboards 7x7 constructed with the 94 pseudo-pentominoes, or pentaplets

Free, one-sided, and fixed polykings

There are three common ways of distinguishing polyominoes and polykings for enumeration:[1]

  • free polykings are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) 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.

More information n, free ...
Free polykings
  • The 94 free pentakings.
    The 94 free pentakings.
  • The 524 free hexakings.
    The 524 free hexakings.
  • The 3,031 free heptakings.
    The 3,031 free heptakings.

Notes

  1. [1]
    Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8.
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