Polyknight
Figure formed by knights moves on a grid
title: "Polyknight" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polyforms"] description: "Figure formed by knights moves on a grid" topic_path: "general/polyforms" source: "https://en.wikipedia.org/wiki/Polyknight" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Figure formed by knights moves on a grid ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/01/Tetraknights.png" caption="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.{{citation"] ::
| last1 = Aleksandrowicz | first1 = Gadi | last2 = Barequet | first2 = Gill | editor1-last = Atallah | editor1-first = Mikhail J. | editor2-last = Li | editor2-first = Xiang-Yang | editor3-last = Zhu | editor3-first = Binhai | contribution = Parallel enumeration of lattice animals | doi = 10.1007/978-3-642-21204-8_13 | pages = 90–99 | publisher = Springer | series = Lecture Notes in Computer Science | title = Frontiers in Algorithmics and Algorithmic Aspects in Information and Management - Joint International Conference, FAW-AAIM 2011, Jinhua, China, May 28-31, 2011. Proceedings | volume = 6681 | year = 2011| isbn = 978-3-642-21203-1
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. ::data[format=table]
| 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 | |||
| :: |
|title=Free polyknights |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.
::callout[type=info title="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. ::