A260319 Number of singly symmetric characteristic solutions to the n-queens problem.

0, 0, 0, 0, 0, 1, 2, 1, 4, 3, 12, 18, 32, 105, 310, 734, 2006, 4526, 11046, 36035, 93740, 312673, 895310, 2917938, 8532332, 28929567

Offset: 1

Author

N. J. A. Sloane, Jul 22 2015

Keywords

• nonn
• more

Comments

The problem of placing eight queens on a chessboard so that no one of them can take any other in a single move is a particular case of the more general problem: On a square array of n X n cells place n objects, one on each of n different cells, in such a way that no two of them lie on the same row, column, or diagonal.

There are no centrosymmetric solutions for n < 6, if by "centrosymmetric" we exclude "doubly symmetric" cases; and there is just one complete set for n = 6: 246135, 362514, 415263, 531642.

On the ordinary chessboard of 8 X 8 cells there are a total of 92 solutions, consisting of 11 sets of equivalent ordinary solutions and one set of equivalent symmetric solutions. There are no doubly symmetric solutions in this case. These sets may be generated in the ordinary case by 15863724, 16837425, 24683175, 2571384, 25741863, 26174835, 26831475, 27368514, 27581463, 35841726, 36258174 and in the symmetric case by 35281746.

References

• Maurice Kraitchik: Mathematical Recreations. Mineola, NY: Dover, 2nd ed. 1953, p. 247-255 (The Problem of the Queens).

Links

• P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
• M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47.
• M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1923, 64 pages. See p. 47. [Incomplete annotated scan of title page and pages 18-51]

Crossrefs

A002562 = A260318 + A260319 + A260320, A000170 = 2*A260318 + 4*A260319 + 8*A260320 (n>1).

Formula

a(n) = -A000170(n)/4 + 2*A002562(n) - 3*A260318(n)/2, n > 1. - R. J. Mathar, Jul 24 2015

Extensions

Name edited (by inserting "singly", since "doubly symmetric" solutions are symmetric but not counted here) by Don Knuth, Mar 25 2022