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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[edit]

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

Free, one-sided, and fixed polykings[edit]

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.

n free one-sided fixed
1 1 1 1
2 2 2 4
3 5 6 20
4 22 34 110
5 94 166 638
6 524 991 3832
7 3,031 5,931 23,592
8 18,770 37,196 147,941
9 118,133 235,456 940,982
10 758,381 1,514,618 6,053,180
11 4,915,652 9,826,177 39,299,408
12 32,149,296 64,284,947 257,105,146
OEIS A030222 A030233 A006770


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

External links[edit]