# Polyking

The 22 free tetrakings

A polyking (or 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 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][2]

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

## Notes

1. ^ a b Golomb, Solomon W. (1994). Polyominoes (2nd ed.). Princeton, New Jersey: Princeton University Press. ISBN 0-691-02444-8.
2. ^ Redelmeier, D. Hugh (1981). "Counting polyominoes: yet another attack". Discrete Mathematics 36: 191–203. doi:10.1016/0012-365X(81)90237-5.