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Polyknight

Figure formed by knights moves on a grid

Summary

Figure formed by knights moves on a grid

knight]] in which doubling back is allowed. It is a [[polyform]] with square cells which are not necessarily connected, comparable to the [[polyking]]. Alternatively, it can be interpreted as a connected subset of the vertices of a [[knight's graph]], a graph formed by connecting pairs of lattice squares that are a knight's move apart.<ref>{{citation

Enumeration of polyknights

Free, one-sided, and fixed polyknights

Three common ways of distinguishing polyominoes for enumeration can also be extended to polyknights:

free polyknights 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 polyknights are distinct when none is a translation or rotation of another (pieces that cannot be flipped over).
fixed polyknights are distinct when none is a translation of another (pieces that can be neither flipped nor rotated).

The following table shows the numbers of polyknights of various types with n cells.

n	free	one-sided	fixed
1	1	1	1
2	1	2	4
3	6	8	28
4	35	68	234
5	290	550	2,162
6	2,680	5,328	20,972
7	26,379	52,484	209,608
8	267,598	534,793	2,135,572
9	2,758,016	5,513,338	22,049,959
10	28,749,456	57,494,308	229,939,414
OEIS

|File:Pentaknights.png|The 290 free pentaknights. |File:Hexaknights.png|The 2,680 free hexaknights.

Notes

References

Redelmeier, D. Hugh. (1981). "Counting polyominoes: yet another attack". Discrete Mathematics.

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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