The On-Line Encyclopedia of Integer Sequences (OEIS)

A006075 revision #53

A006075

Minimal number of knights needed to cover an n X n board.
(Formerly M3224)

1, 4, 4, 4, 5, 8, 10, 12, 14, 16, 21, 24, 28, 32, 36, 40, 46, 52, 57, 62

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

How many knights are needed to occupy or attack every square of an n X n board?

Also known as the domination number of the n X n knight graph. - Eric W. Weisstein, May 27 2016

Upper bounds for the terms after a(20) = 62 are as follows: 68, 75, 82, 88, 96, 102, ... (see Frank Rubin's web site).

The value a(15) = 37 given by Jackson and Pargas is wrong. A simulated annealing-based program I wrote found several complete coverages of a 15 X 15 board with 36 knights. - John Danaher (jsd(AT)mit.edu), Oct 24 2000

REFERENCES

David C. Fisher, On the N X N Knight Cover Problem, Ars Combinatoria 69 (2003), 255-274.

M. Gardner, Mathematical Magic Show. Random House, NY, 1978, p. 194.

Anderson H. Jackson and Roy P. Pargas, Solutions to the N x N Knights Cover Problem, J. Recreat. Math., Vol. 23(4), 1991, 255-267.

Bernard Lemaire, Knights Covers on N X N Chessboards, J. Recreat. Math., Vol. 31-2, 2003, 87-99.

Frank Rubin, Improved knight coverings, Ars Combinatoria 69 (2003), 185-196.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Watkins, Across the Board: The Mathematics of Chessboard Problems (2004), p. 97.

LINKS

Table of n, a(n) for n=1..20.

J. Danaher, Results for 15 X 15 board.

Lee Morgenstern, Knight Domination. [Much material, including optimality proofs for the values given in this entry]

Frank Rubin, Contest Center Web Site, Knight Coverings for Large Chessboards. [Much material, including many illustrations]

Frank Rubin, Illustration of three 52-knight coverings of an 18 X 18 board. (see Frank Rubin's web site, from which this is taken, for many further examples)

Eric Weisstein's World of Mathematics, Domination Number.

Eric Weisstein's World of Mathematics, Knight Graph.

Eric Weisstein's World of Mathematics, Knights Problem.

EXAMPLE

Illustrations for a(3) = 4, a(4) = 4, a(5) = 5 (o = empty square, X = knight):

ooo .. oooo .. ooooo

oXo .. oXXo .. ooXoo

XXX .. oXXo .. oXXXo

...... oooo .. ooXoo

.............. ooooo

CROSSREFS

A006076 gives number of inequivalent ways to cover the board using a(n) knights, A103315 gives total number.