A260320 -id:A260320 - cp's OEIS frontend

A000170 Number of ways of placing n nonattacking queens on an n X n board.

1, 1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184, 14772512, 95815104, 666090624, 4968057848, 39029188884, 314666222712, 2691008701644, 24233937684440, 227514171973736, 2207893435808352, 22317699616364044, 234907967154122528

Offset: 0

N. J. A. Sloane

For n > 3, a(n) is the number of maximum independent vertex sets in the n X n queen graph. - Eric W. Weisstein, Jun 20 2017
Number of nodes on level n of the backtrack tree for the n queens problem (a(n) = A319284(n, n)). - Peter Luschny, Sep 18 2018
Number of permutations of [1...n] such that |p(j)-p(i)| != j-i for iXiangyu Chen, Dec 24 2020
M. Simkin shows that the number of ways to place n mutually nonattacking queens on an n X n chessboard is ((1 +/- o(1))*n*exp(-c))^n, where c = 1.942 +/- 0.003. These are approximately (0.143*n)^n configurations. - Peter Luschny, Oct 07 2021

			a(2) = a(3) = 0, since on 2 X 2 and 3 X 3 chessboards there are no solutions.
.
a(4) = 2:
  +---------+ +---------+
  | . . Q . | | . Q . . |
  | Q . . . | | . . . Q |
  | . . . Q | | Q . . . |
  | . Q . . | | . . Q . |
  +---------+ +---------+
a(5) = 10:
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
  | . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
  | . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
  | . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
  | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |
  | Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
  | Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
  | . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
  | . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
  | . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
  | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
a(6) = 4:
  +-------------+ +-------------+ +-------------+ +-------------+
  | . . . . Q . | | . . . Q . . | | . . Q . . . | | . Q . . . . |
  | . . Q . . . | | Q . . . . . | | . . . . . Q | | . . . Q . . |
  | Q . . . . . | | . . . . Q . | | . Q . . . . | | . . . . . Q |
  | . . . . . Q | | . Q . . . . | | . . . . Q . | | Q . . . . . |
  | . . . Q . . | | . . . . . Q | | Q . . . . . | | . . Q . . . |
  | . Q . . . . | | . . Q . . . | | . . . Q . . | | . . . . Q . |
  +-------------+ +-------------+ +-------------+ +-------------+
- _Hugo Pfoertner_, Mar 17 2019

M. Gardner, The Unexpected Hanging, pp. 190-2, Simon & Shuster NY 1969
Jieh Hsiang, Yuh-Pyng Shieh and Yao-Chiang Chen, The cyclic complete mappings counting problems, in Problems and Problem Sets for ATP, volume 02-10 of DIKU technical reports, G. Sutcliffe, J. Pelletier and C. Suttner, eds., 2002.
D. E. Knuth, The Art of Computer Programming, Volume 4, Pre-fascicle 5B, Introduction to Backtracking, 7.2.2. Backtrack programming. 2018.
M. Kraitchik, The Problem of The Queens, Mathematical Recreations, 2nd ed., New York, Dover, 1953, pp. 247-256.
Massimo Nocentini, "An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation", PhD Thesis, University of Florence, 2019. See Ex. 67.
W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York, Dover, 1987, pp. 166-172 (The Eight Queens Problem).
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
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).
R. J. Walker, An enumerative technique for a class of combinatorial problems, pp. 91-94 of Proc. Sympos. Applied Math., vol. 10, Amer. Math. Soc., 1960.
M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.

Jordan Bell and Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309, Issue 1, Jan 06 2009, Pages 1-31.
D. Bill, Durango Bill's The N-Queens Problem
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656.
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656. [Annotated scanned copy]
Candida Bowtell and Peter Keevash, The n-queens problem, arXiv:2109.08083 [math.CO] 2021.
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
V. Chvatal, All solutions to the problem of eight queens
V. Chvatal, All solutions to the problem of eight queens [Cached copy, pdf format, with permission]
Gheorghe Coserea, Solutions for n=8.
Gheorghe Coserea, Solutions for n=9.
Gheorghe Coserea, MiniZinc model for generating solutions.
Matteo Fischetti and Domenico Salvagnin, Chasing First Queens by Integer Programming, 2018.
Matteo Fischetti and Domenico Salvagnin, Finding First and Most-Beautiful Queens by Integer Programming, arXiv:1907.08246 [cs.DS], 2019.
J. Freeman, A neural network solution to the n-queens problem, The Mathematica J., 3 (No. 3, 1993), 52-56.
Ian P. Gent, Christopher Jefferson and Peter Nightingale, Complexity of n-Queens Completion, Journal of Artificial Intelligence Research 59 (2017), see p 816.
Eric Grigoryan, Investigation of the Regularities in the Formation of Solutions n-Queens Problem, Modeling of Artificial Intelligence, 2018, 5(1), 3-21.
E. Grigoryan, Linear algorithm for solution n-Queens Completion problem, arXiv:1912.05935 [cs.AI], 2019.
James Grime and Brady Haran, The 8 Queen Problem, Numberphile video (2015).
Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.
Kenji Kise, Takahiro Katagiri, Hiroki Honda and Toshitsugu Yuba, Solving the 24-queens Problem using MPI on a PC Cluster, Technical Report UEC-IS-2004-6, Graduate School of Information Systems, The University of Electro-Communications (2004).
D. E. Knuth, Donald Knuth's 24th Annual Christmas Lecture: Dancing Links, Stanfordonline, Video published on YouTube, Dec 12, 2018.
W. Kosters, n-Queens bibliography
Vaclav Kotesovec, Non-attacking chess pieces, Sixth edition, 795 pages, Feb 02 2013 (minor update Mar 29 2016).
Zur Luria, New bounds on the number of n-queens configurations, arXiv:1705.05225 [math.CO], 2017.
Zur Luria, Michael Simkin, A lower bound for the n-queens problem, arXiv:2105.11431 [math.CO], 2021.
E. Masehian, H. Akbaripour and N. Mohabbati-Kalejahi, Solving the n Queens Problem using a Tuned Hybrid Imperialist Competitive Algorithm, 2013.
E. Masehian, H. Akbaripour and N. Mohabbati-Kalejahi, Landscape analysis and efficient metaheuristics for solving the n-queens problem, Computational Optimization and Applications, 2013; DOI 10.1007/s10589-013-9578-z.
Nasrin Mohabbati-Kalejahi, Hossein Akbaripour and Ellips Masehian, Basic and Hybrid Imperialist Competitive Algorithms for Solving the Non-attacking and Non-dominating n -Queens Problems, Studies in Computational Intelligence Volume 577, 2015, pp 79-96. DOI 10.1007/978-3-319-11271-8_6.
Ralph Morrison and Noah Speeter, The Gonality of Queen's Graphs, arXiv:2312.04686 [math.CO], 2023.
Parth Nobel, Akshay Agrawal and Stephen Boyd, Computing tighter bounds on the n-queens constant via Newton’s method, arXiv:2112.03336 [math.CO], 2021.
J. Pope and D. Sonnier, A linear solution to the n-Queens problem using vector spaces, Journal of Computing Sciences in Colleges, Volume 29 Issue 5, May 2014 Pages 77-83.
T. B. Preußer and M. R. Engelhardt, Putting Queens in Carry Chains, No. 27, Journal of Signal Processing Systems, Volume 88, Issue 2, August 2017. (The title refers to the fact that the article discusses the case n = 27.)
Thomas B. Preußer, Bernd Nägel and Rainer G. Spallek, Putting Queens in Carry Chains, Slides, HIPEAC WRC'09.
E. M. Reingold, Letter to N. J. A. Sloane, Dec 27 1973
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Werner Florian Samayoa, Maria Liz Crespo, Sergio Carrato, Agustin Silva, and Andres Cicuttin, HyperFPGA: An Experimental Testbed for Heterogeneous Supercomputing, 2023.
Michael Simkin, The number of n-queens configurations, arXiv:2107.13460 [math.CO], 2021.
Wenxi Wang, Muhammad Usman, Alyas Almaawi, Kaiyuan Wang, Kuldeep S. Meel and Sarfraz Khurshid, A Study of Symmetry Breaking Predicates and Model Counting, National University of Singapore (2020).
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
Eric Weisstein's World of Mathematics, Queen Graph
Eric Weisstein's World of Mathematics, Queens Problem
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Wikipedia, Eight Queens Puzzle
Cheng Zhang and Jianpeng Ma, Counting Solutions for the N-queens and Latin Square Problems by Efficient Monte Carlo Simulations, arXiv:0808.4003 [cond-mat.stat-mech], 2008.

Cf. A002562, A065256.
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A099152, A006717, A051906, A319284 (backtrack trees).
Cf. A260318, A260319, A260320.
Main diagonal of A348129.

Strong conjecture: there is a constant c around 2.54 such that a(n) is asymptotic to n!/c^n; weak conjecture: lim_{n -> infinity} (1/n) * log(n!/a(n)) = constant = 0.90.... - Benoit Cloitre, Nov 10 2002
Lim_{n->infinity} a(n)^(1/n)/n = exp(-A359441) = 0.1431301... [Simkin 2021]. - Vaclav Kotesovec, Jan 01 2023
a(n) = 8 * A260320(n) + 4 * A260319(n) + 2 * A260318(n) for n >= 2 (see Kraitchik reference). - Jason Bard, Aug 12 2025

Terms for n=21-23 computed by Sylvain PION (Sylvain.Pion(AT)sophia.inria.fr) and Joel-Yann FOURRE (Joel-Yann.Fourre(AT)ens.fr).
a(24) from Kenji KISE (kis(AT)is.uec.ac.jp), Sep 01 2004
a(25) from Objectweb ProActive INRIA Team (proactive(AT)objectweb.org), Jun 11 2005 [Communicated by Alexandre Di Costanzo (Alexandre.Di_Costanzo(AT)sophia.inria.fr)]. This calculation took about 53 years of CPU time.
a(25) has been confirmed by the NTU 25Queen Project at National Taiwan University and Ming Chuan University, led by Yuh-Pyng (Arping) Shieh, Jul 26 2005. This computation took 26613 days CPU time.
The NQueens-at-Home web site gives a different value for a(24), 226732487925864. Thanks to Goran Fagerstrom for pointing this out. I do not know which value is correct. I have therefore created a new entry, A140393, which gives the NQueens-at-home version of the sequence. - N. J. A. Sloane, Jun 18 2008
It now appears that this sequence (A000170) is correct and A140393 is wrong. - N. J. A. Sloane, Nov 08 2008
Added a(26) as calculated by Queens(AT)TUD [http://queens.inf.tu-dresden.de/]. - Thomas B. Preußer, Jul 11 2009
Added a(27) as calculated by the Q27 Project [https://github.com/preusser/q27]. - Thomas B. Preußer, Sep 23 2016
a(0) = 1 prepended by Joerg Arndt, Sep 16 2018

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

Original entry on oeis.org

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

N. J. A. Sloane, Jul 22 2015

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.

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

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]

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

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

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

cp's OEIS Frontend

A000170 Number of ways of placing n nonattacking queens on an n X n board.

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A260318 Number of doubly symmetric characteristic solutions to the n-queens problem.

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

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A261597 Triangular array T(n, k) read by rows (n >= 1, 1 <= k <= n): row n gives the lexicographically earliest asymmetric characteristic solution to the n queens problem, or n zeros if no asymmetric characteristic solution exists. The k-th queen is placed in square (k, T(n, k)).

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