The 22 free tetrakings

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 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.

nfreeone-sidedfixed
1111
2224
35620
42234110
594166638
65249913832
73,0315,93123,592
818,77037,196147,941
9118,133235,456940,982
10758,3811,514,6186,053,180
114,915,6529,826,17739,299,408
1232,149,29664,284,947257,105,146
OEISA030222A030233A006770

Notes

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