Explicit Solutions to the N-Queens Problem for all N Bo B e r n h a r d s s o n D e p a r t m e n t of A u t o m a t i c C o n t r o l L a n d I n s t i t u t e of Technology P.O. Box 118 S-22100 L u n d S W E D E N bob@control.lth.se The n-queens problem is often used as a benchmark problem for AI research and in combinatorial optimization. An example is the recent article [1] in this magazine that presented a polynomial time algorithm for finding a solution. Several CPU-hours were spent finding solutions for some n up to 500,000. I would like to draw the readers' attention to the less known fact that explicit solutions for this problem exist for all n ~ 4, see below. This result was published 1969 in [2]. The construction is quite simple and requires no combinatorial search or computer time whatsoever. To find one solution to the n-queens problem is therefore completely trivial. The n-Queens problem is there= fore a bad benchmark problem. A lot of CPU-hours can be saved in the future noting this. A verification of the construction below is given in [2]. Let (i, j') denote the square on row i and columnj. Depending on n there are three eases: (A) n even but not of the form 6k + 2: Place queens on elements (B) n even but not of the form 6k: Place queens on elements (~, 1 + [2(~ - 1) + n/2 - 1 modulo n]) (n + 1 - j, n - [2(./- 1) + n/2 - 1 modulo n]), j = 1,2,...n/2 (C) n odd: Use ease A or ease B on n-1 and extend with a queen at (n,n) It could finally be noted that the related problem of finding a/l solutions to the n-queens problem is non-trivial. It was introducced by Gauss with n = 8. References [1] Sosic, R. and Gu, J. (1990): A Polynomial Time Algorithm for the N-Queens Problem, SIGART Bulletin, 1, p. 7-11. [2] Hoffman, E.J., Loessi, J.C. and Moore, R.C. (1969): Constructions for the Solution of the m Queens Problem, Mathematics Magazine, p. 66-72. (j,2j) (n/2 + j, 2j - 1)2 j = 1, 2 , . . . , n/2 S I G A R T B u l l e t i n , Vol. 2, No. 2
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Explicit solutions to the N -queens problem for all N
Bernhardsson, Bo
Follow ACM SIGART Bulletin
, Volume 2 (2)
Association for Computing Machinery – Feb 1, 1991
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