A002564 Number of different ways one can attack all squares on an n X n chessboard using the minimum number of queens.

1, 4, 1, 12, 186, 4, 86, 4860, 114, 8, 2, 8, 288, 4632, 205832, 2968, 124, 16, 84

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

Number of distinct solutions to minimum dominating set on queens' graph Q(n). See A002563 for non-isomorphic solutions.

For same problem, but with non-attacking queens, see A002568. - Vaclav Kotesovec, Sep 07 2012

In other words, number of minimum dominating sets in the n X n queen graph. - Eric W. Weisstein, Dec 31 2017

For n > 2, also the number of minimal edge covers in the n X n queen graph. - Eric W. Weisstein, Dec 09 2024

a(20) >= 4152. - Eric W. Weisstein, Jul 28 2025

References

• W. Ahrens, Mathematische Unterhaltungen und Spiele, second edition (1910), Vol. 1, p. 301.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andy Huchala, Python program.
• Matthew D. Kearse and Peter B. Gibbons, Computational Methods and New Results for Chessboard Problems, Australasian Journal of Combinatorics 23 (2001), 253-284.
• Mia Müßig, Julia code to compute the sequence
• M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49.
• M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 49. [Incomplete annotated scan of title page and pages 18-51]
• Eric Weisstein's World of Mathematics, Minimal Edge Cover.
• Eric Weisstein's World of Mathematics, Minimum Dominating Set.
• Eric Weisstein's World of Mathematics, Queen Graph.
• Eric W. Weisstein, Symmetrically inequivalent configurations for n = 1 to 7
• Eric W. Weisstein, Symmetrically inequivalent configurations for n = 8
• Eric W. Weisstein, Symmetrically inequivalent configurations for n = 9 to 13
• Eric W. Weisstein, Symmetrically inequivalent configurations for n = 14
• Eric W. Weisstein, Symmetrically inequivalent configurations for n = 16
• Eric W. Weisstein, Symmetrically inequivalent configurations for n = 17 to 19

Crossrefs

Cf. A002563, A002568, A182333, A286883.

A075458 gives number of queens required. - Sean A. Irvine, Apr 05 2014

Extensions

New name of the sequence from Vaclav Kotesovec, Sep 07 2012

a(9)-a(10) from Vaclav Kotesovec, Sep 07 2012

a(11) from Svyatoslav Starkov, Sep 16 2013

a(12)-a(13) from Sean A. Irvine, Apr 07 2014

Definition edited by N. J. A. Sloane, Dec 25 2017 at the suggestion of Brendan McKay.

a(14) from Andy Huchala, Mar 13 2024

a(15)-a(19) from Mia Muessig, Oct 04 2024