n-Queens bibliography

324 references

This page, maintained by Walter Kosters from Universiteit Leiden, contains literature related to the n-queens problem. It has been recently updated by Pieter Bas Donkersteeg. Currently there are 324 references.

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Author	Title	Year	Journal/Proceedings	Reftype	DOI/URL
Abramson, B. & Yung, M.	Divide and Conquer under Global Constraints: A Solution to the $n$-Queens Problem [Abstract] [BibTeX]	1989	Journal of Parallel and Distributed Computing Vol. 6, pp. 649-662	article	DOI
Abstract: Configuring $n$ mutually nonattacking Queens on an $nn$ chessboard is a classical problem that was first posed over a century ago. Over the past few decades, this problem has become important to computer scientists by serving as the standard example of a globally constrained problem which is solvable using backtracking search methods. A related problem, placing the $n$-Queens on a toroidal board, has been discussed in detail by Poyla and Chandra. Their work focused on characterizing the solvable cases and finding solutions which arrange the Queens in a regular pattern. This paper describes a new divide-and-conquer algorithm that solves both problems and investigates the relationship between them. The connection between the solutions of the two problems illustrates an important, but frequently overlooked, method of algorithm design: detailed combinatorial analysis of an overconstrained variation can reveal solutions to the corresponding original problem. The solution is an example of solving a globally constrained problem using the divide-and-conquer technique, rather than the usual backtracking algorithm. The former is much faster in both sequential and parallel environments.
BibTeX: @article{Abramson1989, author = {B. Abramson and M.M. Yung}, title = {Divide and Conquer under Global Constraints: A Solution to the $n$-Queens Problem}, journal = {Journal of Parallel and Distributed Computing}, year = {1989}, volume = {6}, pages = {649-662}, doi = {http://dx.doi.org/10.1016/0743-7315(89)90011-7} }
Abramson, B. & Yung, M.	Construction Through Decomposition: A Divide-and-Conquer Algorithm for the $n$-Queens Problem [BibTeX]	1986	Proceedings of 1986 ACM Fall Joint Computer Conference, pp. 620-628	inproceedings
BibTeX: @inproceedings{Abramson1986, author = {B. Abramson and M.M. Yung}, title = {Construction Through Decomposition: A Divide-and-Conquer Algorithm for the $n$-Queens Problem}, booktitle = {Proceedings of 1986 ACM Fall Joint Computer Conference}, year = {1986}, pages = {620-628} }
Ahrens, W.	Encyklopädie der Mathematischen Wissenschaften, Erster Band in Zwei Teilen. Zweiter Teil [BibTeX]	1902		book
BibTeX: @book{Ahrens1902, author = {W. Ahrens}, title = {Encyklopädie der Mathematischen Wissenschaften, Erster Band in Zwei Teilen. Zweiter Teil}, publisher = {B. G. Teubner}, year = {1902} }
Ahrens, W.	Mathematische Unterhaltungen und Spiele [BibTeX]	1901		book
BibTeX: @book{Ahrens1901, author = {W. Ahrens}, title = {Mathematische Unterhaltungen und Spiele}, publisher = {B.G. Teubner}, year = {1901} }
Alavi, Y., Lick, D. & Liu, J.	Strongly Diagonal Latin Squares and Permutation Cubes [BibTeX]	1994	Proceedings of the Twenty-ﬁfth Southeastern International Conference on Combinatorics, Graph Theory and Computing, pp. 65–70	inproceedings
BibTeX: @inproceedings{Alavi1994, author = {Y. Alavi and D.R. Lick and J. Liu}, title = {Strongly Diagonal Latin Squares and Permutation Cubes}, booktitle = {Proceedings of the Twenty-ﬁfth Southeastern International Conference on Combinatorics, Graph Theory and Computing}, year = {1994}, pages = {65–70} }
Allison, L., Yee, C. & McGaughey., M.	Three-Dimensional Queens Problems [Abstract] [BibTeX]	1989	(89/130)	techreport	URL
Abstract: The two-dimensional $N$-queens problem is generalised to three dimensions and to $N^2$-queens. There are non-toroidal and toroidal variants. A computer search has been carried out for (non-toroidal) solutions up to $N=14$. We conjecture that toroidal solutions exist iff the smallest factor of $N$ is greater than 7.
BibTeX: @techreport{Allison1988, author = {L. Allison and C.N. Yee and M. McGaughey.}, title = {Three-Dimensional Queens Problems}, year = {1989}, number = {89/130}, url = {http://www.csse.monash.edu.au/~lloyd/tildeAlgDS/Recn/Queens3D/} }
Alvis, D. & Kinyon, M.	Birkhoff's Theorem for Panstochastic Matrices [BibTeX]	2001	The American Mathematical Monthly Vol. 108(1), pp. 28-37	article	DOI
BibTeX: @article{Alvis20001, author = {D. Alvis and M. Kinyon}, title = {Birkhoff's Theorem for Panstochastic Matrices}, journal = {The American Mathematical Monthly}, year = {2001}, volume = {108(1)}, pages = {28-37}, doi = {http://dx.doi.org/10.2307/2695673} }
Ambrus, G. & Barát, J.	A Contribution to Queens Graphs: A Substitution Method [Abstract] [BibTeX]	2006	Discrete Mathematics Vol. 306, pp. 1105-1114	article	DOI
Abstract: A graph $G$ is a queens graph if the vertices of $G$ can be mapped to queens on the chessboard such that two vertices are adjacent if and only if the corresponding queens attack each other, i.e. they are in horizontal, vertical or diagonal position. We prove a conjecture of Beineke, Broere and Henning that the Cartesian product of an odd cycle and a path is a queens graph. We show that the same does not hold for two odd cycles. The representation of the Cartesian product of an odd cycle and an even cycle remains an open problem. We also prove constructively that any finite subgraph of the rectangular grid or the hexagonal grid is a queens graph. Using a small computer search we solve another conjecture of the authors mentioned above, saying that $K_3,4$ minus an edge is a minimal non-queens graph.
BibTeX: @article{Ambrus2006, author = {G. Ambrus and J. Barát}, title = {A Contribution to Queens Graphs: A Substitution Method}, journal = {Discrete Mathematics}, year = {2006}, volume = {306}, pages = {1105-1114}, doi = {http://dx.doi.org/10.1016/j.disc.2006.03.002} }
Andrews, W.	Magic Squares and Cubes [BibTeX]	1960		book
BibTeX: @book{Andrews1960, author = {W.S. Andrews}, title = {Magic Squares and Cubes}, publisher = {Dover Publications Inc., NewYork}, year = {1960}, edition = {2nd} }
Atkin, A., Hay, L. & Larson, R.	Enumeration and Construction of Pandiagonal Latin Squares of Primeorder [Abstract] [BibTeX]	1983	Computers and Mathematics with Applications Vol. 9, pp. 267-292	article	DOI
Abstract: A complete enumeration and algebraic description is given of all pandiagonal Latin squares of order $leq 13$. For $n = 5$, 7 and 11 there are (up to equivalence) exactly the $n-3$ cyclic squares. For $n = 13$ there are 12,386 inequivalent squares; of these 10 are cyclic (in all directions) and 1560 are semi-cyclic (cyclic in a single direction). Systematic methods are given for constructing semi-cyclic pandiagonal Latin squares of any prime order $> 11$.
BibTeX: @article{Atkin1983, author = {A.O.L. Atkin and L. Hay and R.G. Larson}, title = {Enumeration and Construction of Pandiagonal Latin Squares of Primeorder}, journal = {Computers and Mathematics with Applications}, year = {1983}, volume = {9}, pages = {267-292}, doi = {http://dx.doi.org/10.1016/0898-1221(83)90130-X} }
Ball, W.	Mathematical Recreations and Essays [BibTeX]	1892		book	URL
BibTeX: @book{Ball1892, author = {W.W.R. Ball}, title = {Mathematical Recreations and Essays}, publisher = {Macmillan and Co., London}, year = {1892}, url = {http://www.gutenberg.org/etext/26839} }
Barr, J. & Rao, S.	The $n$-Queens Problem in Higher Dimensions [BibTeX]	2006	Elemente der Mathematik Vol. 61, pp. 133-137	article	URL
BibTeX: @article{Barr2006a, author = {J. Barr and S. Rao}, title = {The $n$-Queens Problem in Higher Dimensions}, journal = {Elemente der Mathematik}, year = {2006}, volume = {61}, pages = {133-137}, url = {http://www.ems-ph.org/journals/show_pdf.php?issn=0013-6018&vol=61&iss=4&rank=1} }
Barwell, B.	Solution to Problem 811 [BibTeX]	1980	Journal of Recreational Mathematics Vol. 13, pp. 61	article
BibTeX: @article{Barwell1980, author = {B. Barwell}, title = {Solution to Problem 811}, journal = {Journal of Recreational Mathematics}, year = {1980}, volume = {13}, pages = {61} }
Beasley, J.	The Mathematics of Games [BibTeX]	1989	Recreations in Mathematics, volume 5	incollection
BibTeX: @incollection{Beasley1989, author = {J.D. Beasley}, title = {The Mathematics of Games}, booktitle = {Recreations in Mathematics, volume 5}, publisher = {The Clarendon Press - Oxford University Press}, year = {1989} }
Behmann, H.	Das gesamte Schachbrett unter Beachtung der Regeln des Achtköniginnenproblems zu Besetzen [BibTeX]	1910	Mathematisch-Naturwissenschaftliche Blätter. Organ des Arnstädter Verbandes mathematischer und naturwissenschaftlicher Vereine an Deutschen Hochschulen Vol. 8, pp. 87-89	article
BibTeX: @article{Behmann1910, author = {H. Behmann}, title = {Das gesamte Schachbrett unter Beachtung der Regeln des Achtköniginnenproblems zu Besetzen}, journal = {Mathematisch-Naturwissenschaftliche Blätter. Organ des Arnstädter Verbandes mathematischer und naturwissenschaftlicher Vereine an Deutschen Hochschulen}, year = {1910}, volume = {8}, pages = {87-89} }
Beineke, L., Broere, I. & Henning, M.	Queens Graphs [Abstract] [BibTeX]	1999	Discrete Mathematics Vol. 206, pp. 63-75	article	DOI
Abstract: The queens graph of a $(0,1)$-matrix $A$ is the graph whose vertices correspond to the 1's in $A$ and in which two vertices are adjacent if and only if some diagonal or line of $A$ contains the corresponding 1's. A basic question is the determination of which graphs are queens graphs. We establish that a complete block graph is a queens graph if and only if it does not contain $K_1,5$ as an induced subgraph. A similar result is shown to hold for trees and cacti. Every grid graph is shown to be a queens graph, as are the graphs $K_nP_m$ and $C_2nP_m$ for all integers $n,mgeq 2$. We show that a complete multipartite graph is a queens graph if and only if it is a complete graph or an induced subgraph of $K_4,4$, $K_1,3,3$, $K_2,2,2$ or $K_1,1,2,2$. It is also shown that $K_3,4−e$ is not a queens graph.
BibTeX: @article{Beineke1999, author = {L.W. Beineke and I. Broere and M.A. Henning}, title = {Queens Graphs}, journal = {Discrete Mathematics}, year = {1999}, volume = {206}, pages = {63-75}, doi = {http://dx.doi.org/10.1016/S0012-365X(98)00392-6} }
Bell, J.	An Introduction to SDR's and Latin Squares [Abstract] [BibTeX]	2005	Morehead Electronic Journal of Applicable Mathematics Vol. 4(MATH-2005-03)	article	URL
Abstract: In this paper we study systems of distinct representatives (SDR's) and Latin squares, considering SDR's especially in their application to constructing Latin squares. We give proofs of several important elementary results for SDR's and Latin squares, in particular Hall's marriage theorem and lower bounds for the number of Latin squares of each order, and state several other results, such as necessary and sufficient conditions for having a common SDR for two families. We consider some of the applications of Latin squares both in pure mathematics, for instance as the multiplication table for quasigroups, and in applications, such as analyzing crops for differences in fertility and susceptibility to insect attack. We also present a brief history of the study of Latin squares and SDR's.
BibTeX: @article{Bell2005, author = {J. Bell}, title = {An Introduction to SDR's and Latin Squares}, journal = {Morehead Electronic Journal of Applicable Mathematics}, year = {2005}, volume = {4}, number = {MATH-2005-03}, url = {http://www.moreheadstate.edu/mejam/} }
Bell, J. & Stevens, B.	A Survey of Known Results and Research Areas for $n$-Queens [Abstract] [BibTeX]	2009	Discrete Mathematics Vol. 309, pp. 1-31	article	DOI
Abstract: In this paper we survey known results for the $n$-Queens problem of placing $n$ nonattacking Queens on an $nn$ chessboard and consider extensions of the problem, e.g. other board topologies and dimensions. For all solution constructions, we either give the construction, an outline of it, or a reference. In our analysis of the modular board, we give a simple result for finding the intersections of diagonals. We then investigate a number of open research areas for the problem, stating several existing and new conjectures. Along with the known results for $n$-Queens that we discuss, we also give a history of the problem. In particular, we note that the first proof that n nonattacking Queens can always be placed on an n×n board for $n > 3$ is by E. Pauls, rather than by W. Ahrens who is typically cited. We have attempted in this paper to discuss all the mathematical literature in all languages on the $n$-Queens problem. However, we look only briefly at computational approaches.
BibTeX: @article{Bell2009, author = {J. Bell and B. Stevens}, title = {A Survey of Known Results and Research Areas for $n$-Queens}, journal = {Discrete Mathematics}, year = {2009}, volume = {309}, pages = {1-31}, doi = {http://dx.doi.org/10.1016/j.disc.2007.12.043} }
Bell, J. & Stevens, B.	Results for the $n$-Queens Problem on the Möbius Board [BibTeX]	2008	Australasian Journal of Combinatorics Vol. 42, pp. 21-34	article
BibTeX: @article{Bell2008, author = {J. Bell and B Stevens}, title = {Results for the $n$-Queens Problem on the Möbius Board}, journal = {Australasian Journal of Combinatorics}, year = {2008}, volume = {42}, pages = {21-34} }
Bell, J. & Stevens, B.	Constructing Orthogonal Pandiagonal Latin Squares and Panmagic Squares from Modular $n$-Queens Solutions [Abstract] [BibTeX]	2007	Journal of Combinatorial Designs Vol. 15(3), pp. 221-234	article	DOI
Abstract: In this article, we show how to construct pairs of orthogonal pandiagonal Latin squares and panmagic squares from certain types of modular $n$-Queens solutions. We prove that when these modular $n$-Queens solutions are symmetric, the panmagic squares thus constructed will be associative, where for an $nn$ associative magic square $A = (a_ij)$, for all $i$ and $j$ it holds that $a_ij + a_n-i-1,n-j-1 = c$ for a fixed $c$. We further show how to construct orthogonal Latin squares whose modular difference diagonals are Latin from any modular $n$-Queens solution. As well, we analyze constructing orthogonal pandiagonal Latin squares from particular classes of non-linear modular $n$-Queens solutions. These pandiagonal Latin squares are not row cyclic, giving a partial solution to a problem of Hedayat. 2007
BibTeX: @article{Bell2007a, author = {J. Bell and B. Stevens}, title = {Constructing Orthogonal Pandiagonal Latin Squares and Panmagic Squares from Modular $n$-Queens Solutions}, journal = {Journal of Combinatorial Designs}, year = {2007}, volume = {15(3)}, pages = {221-234}, doi = {http://dx.doi.org/10.1002/jcd.20143} }
Bennett, B. & Potts, R.	Arrays and Brooks [BibTeX]	1967	Journal of the Australian Mathematical Society Vol. 7, pp. 23-31	article
BibTeX: @article{Bennett1967, author = {B.T. Bennett and R.B. Potts}, title = {Arrays and Brooks}, journal = {Journal of the Australian Mathematical Society}, year = {1967}, volume = {7}, pages = {23-31} }
Bennett, G.	The Eight Queens Problem (or Super Imposable Solutions for $88$ Boards) [BibTeX]	1910	The Messenger of Mathematics Vol. 39, pp. 19	article
BibTeX: @article{Bennett1910, author = {G.T. Bennett}, title = {The Eight Queens Problem (or Super Imposable Solutions for $88$ Boards)}, journal = {The Messenger of Mathematics}, year = {1910}, volume = {39}, pages = {19} }
Berge, C.	Graphes et Hypergraphes [BibTeX]	1970	Monographies Universitaires de Mathématiques, 37	incollection
BibTeX: @incollection{Berge1970, author = {C. Berge}, title = {Graphes et Hypergraphes}, booktitle = {Monographies Universitaires de Mathématiques, 37}, publisher = {Dunod, Paris}, year = {1970} }
Bernhardsson, B.	Explicit Solution to the $n$-Queens Problems for all $n$ [Abstract] [BibTeX]	1991	ACM SIGART Bulletin Vol. 2, pp. 7	article	DOI
Abstract: The $n$-queens problem is often used as a benchmark problem for AI research and in combinatorial optimization. An example is the recent article teSosic1990 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.
BibTeX: @article{Bernhardsson1991, author = {B. Bernhardsson}, title = {Explicit Solution to the $n$-Queens Problems for all $n$}, journal = {ACM SIGART Bulletin}, year = {1991}, volume = {2}, pages = {7}, doi = {http://dx.doi.org/10.1145/122319.122322} }
Bezzel, F.	Proposal of Eight Queens Problem [BibTeX]	1848	Berliner Schachzeitung Vol. 3, pp. 363	article
BibTeX: @article{Bezzel1848, author = {F.W.M. Bezzel}, title = {Proposal of Eight Queens Problem}, journal = {Berliner Schachzeitung}, year = {1848}, volume = {3}, pages = {363} }
Bitner, J. & Reingold, E.	Backtrack Programming Techniques [Abstract] [BibTeX]	1975	Communications of the ACM Vol. 18, pp. 651-656	article	DOI
Abstract: The purpose of this paper is twofold. First, a brief exposition of the general backtrack technique and its history is given. Second, it is shown how the use of macros can considerably shorten the computation time in many cases. In particular, this technique has allowed the solution of two previously open combinatorial problems, the computation of new terms in a well-known series, and the substantial reduction in computation time for the solution to another combinatorial problem. This article deals with the basics of backtracking.
BibTeX: @article{Bitner1975, author = {J.R. Bitner and E.M. Reingold}, title = {Backtrack Programming Techniques}, journal = {Communications of the ACM}, year = {1975}, volume = {18}, pages = {651-656}, doi = {http://dx.doi.org/10.1145/361219.361224} }
Blumenthal, L.	Discussions: An Extension of the Gauss Problem of Eight Queens [BibTeX]	1928	The American Mathematical Monthly Vol. 35(6), pp. 307-309	article	DOI
BibTeX: @article{Blumenthal1928, author = {L.M. Blumenthal}, title = {Discussions: An Extension of the Gauss Problem of Eight Queens}, journal = {The American Mathematical Monthly}, year = {1928}, volume = {35(6)}, pages = {307-309}, doi = {http://dx.doi.org/10.2307/2298678} }
Bode, J.-P. & Harborth, H.	Independent Chess pieces on Euclidean Boards [BibTeX]	2000	Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 33, pp. 209-223	article
BibTeX: @article{Bode2000, author = {J.-P. Bode and H. Harborth}, title = {Independent Chess pieces on Euclidean Boards}, journal = {Journal of Combinatorial Mathematics and Combinatorial Computing}, year = {2000}, volume = {33}, pages = {209-223} }
Bozinovski, A. & Bozinovski, S.	$n$-Queens Pattern Generation: An Insight into Space Complexity of a Backtracking Algorithm [Abstract] [BibTeX]	2004	ACM International Conference Proceeding Series; Proceedings of the 2004 International Symposium on Information and Communication Technologies, pp. 281-286	inproceedings
Abstract: It is proposed a method for tracking partial solutions while executing a backtracking algorithm. That enables observation of space requirements of a backtracking algorithm. To illustrate the method, the well known benchmark $n$-Queens problem is considered. Results of the experiments are shown and discussed.
BibTeX: @inproceedings{Bozinovski2004, author = {A. Bozinovski and S. Bozinovski}, title = {$n$-Queens Pattern Generation: An Insight into Space Complexity of a Backtracking Algorithm}, booktitle = {ACM International Conference Proceeding Series; Proceedings of the 2004 International Symposium on Information and Communication Technologies}, year = {2004}, pages = {281-286} }
Bratko, I.	Prolog Programming for Artificial Intelligence [BibTeX]	1986		book
BibTeX: @book{Bratko1986, author = {I. Bratko}, title = {Prolog Programming for Artificial Intelligence}, publisher = {Addison-Wesley}, year = {1986} }
Bruen, A. & Dixon, R.	The $n$-Queens Problem [Abstract] [BibTeX]	1975	Discrete Mathematics Vol. 12, pp. 393-395	article	DOI
Abstract: We present some new solutions to the problem of arranging n queens on an $n n$ chessboard with no two taking each other. Recent related work of other authors is also discussed.
BibTeX: @article{Bruen1975, author = {A. Bruen and R. Dixon}, title = {The $n$-Queens Problem}, journal = {Discrete Mathematics}, year = {1975}, volume = {12}, pages = {393-395}, doi = {http://dx.doi.org/10.1016/0012-365X(75)90079-5} }
Burger, A., Cockayne, E. & Mynhardt, C.	Domination and Irredundance in the Queens' Graph [Abstract] [BibTeX]	1997	Discrete Mathematics Vol. 163, pp. 47-66	article	DOI
Abstract: The vertices of the queens' graph $Q_n$ are the squares of an $n n$ chessboard and two squares are adjacent if a queen placed on one covers the other. It is shown that the domination number of $Q_n$ is at most $31n/54 + O(1)$, that $Q_n$ possesses minimal dominating sets of cardinality $5n/2 - O(1)$ and that the cardinality of any irredundant set of vertices of $Q_n$ ($n geq 9$) is at most $lfloor 6n+6-8n+n+1 .
BibTeX: @article{Burger1997, author = {A.P. Burger and E.J. Cockayne and C.M. Mynhardt}, title = {Domination and Irredundance in the Queens' Graph}, journal = {Discrete Mathematics}, year = {1997}, volume = {163}, pages = {47-66}, doi = {http://dx.doi.org/10.1016/0012-365X(95)00327-S} }
Burger, A. & Mynhardt, C.	An Improved Upper Bound for Queens Domination Numbers [Abstract] [BibTeX]	2003	Discrete Mathematics Vol. 266, pp. 119-131	article	DOI
Abstract: We consider the domination number of the queens graph $Q_n$ and show that if, for some fixed $k$, there is a dominating set of $Q_4k+1$ of a certain type with cardinality $2k+1$, then for any $n$ large enough, $Q_n)leq [(3k+5)/(6k+3)]+O(1)$. The same construction shows that for any $mgeq 1$ and $n=2(6m-1)(2k+1)-1$, $Q_n^t)leq [(2k+3)/(4k+2)]+O(1)$ where $Q_n^t$ is the toroidal $nn$ queens graph.
BibTeX: @article{Burger2003, author = {A.P. Burger and C.M. Mynhardt}, title = {An Improved Upper Bound for Queens Domination Numbers}, journal = {Discrete Mathematics}, year = {2003}, volume = {266}, pages = {119-131}, doi = {http://dx.doi.org/10.1016/S0012-365X(02)00802-6} }
Burger, A. & Mynhardt, C.	An Upper Bound for the Minimum Number of Queens Covering the $nn$ Chessboard [Abstract] [BibTeX]	2002	Discrete Applied Mathematics Vol. 121, pp. 51-60	article	DOI
Abstract: We show that the minimum number of queens required to cover the $nn$ chessboard is at most $815n+O(1)$.
BibTeX: @article{Burger2002, author = {A.P. Burger and C.M. Mynhardt}, title = {An Upper Bound for the Minimum Number of Queens Covering the $nn$ Chessboard}, journal = {Discrete Applied Mathematics}, year = {2002}, volume = {121}, pages = {51-60}, doi = {http://dx.doi.org/10.1016/S0166-218X(01)00244-X} }
Burger, A. & Mynhardt, C.	Properties of Dominating Sets of the Queens Graph $Q_4k+3$ [BibTeX]	2000	Utilitas Mathematica Vol. 57, pp. 237-253	article
BibTeX: @article{Burger2000, author = {A.P. Burger and C.M. Mynhardt}, title = {Properties of Dominating Sets of the Queens Graph $Q_4k+3$}, journal = {Utilitas Mathematica}, year = {2000}, volume = {57}, pages = {237-253} }
Burger, A. & Mynhardt, C.	Small Irredundance Numbers for Queens Graphs [BibTeX]	2000	Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 33, pp. 33-43	article
BibTeX: @article{Burger2000a, author = {A.P. Burger and C.M. Mynhardt}, title = {Small Irredundance Numbers for Queens Graphs}, journal = {Journal of Combinatorial Mathematics and Combinatorial Computing}, year = {2000}, volume = {33}, pages = {33-43} }
Burger, A. & Mynhardt, C.	Symmetry and Domination in Queens' Graphs [BibTeX]	2000	Bulletin of the Institute of Combinatorics and its Applications Vol. 29, pp. 11-24	article
BibTeX: @article{Burger2000b, author = {A.P. Burger and C.M. Mynhardt}, title = {Symmetry and Domination in Queens' Graphs}, journal = {Bulletin of the Institute of Combinatorics and its Applications}, year = {2000}, volume = {29}, pages = {11-24} }
Burger, A. & Mynhardt, C.	Queens on Hexagonal Boards [BibTeX]	1999	Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 31, pp. 97-111	article
BibTeX: @article{Burger1999, author = {A.P. Burger and C.M. Mynhardt}, title = {Queens on Hexagonal Boards}, journal = {Journal of Combinatorial Mathematics and Combinatorial Computing}, year = {1999}, volume = {31}, pages = {97-111} }
Burger, A., Mynhardt, C. & Cockayne, E.	Regular Solutions of the $n$-Queens Problem on the Torus [Abstract] [BibTeX]	2004	Utilitas Mathematica Vol. 65, pp. 219-230	article
Abstract: The $n$-queens problem on the torus is the problem of placing $n$ queens on an $nn$ chessboard drawn on the torus so that no two queens attack each other. This is known to be possible if and only if $n equiv pm 1 (mod 6)$. A solution to this problem is said to be regular if it places queens on all squares with co-ordinates $(x + a, kx + b)$ for some fixed integers $k neq 0$, $a$ and $b$. We determine the number of non-isometric regular solutions for each $n equiv pm 1 (mod 6)$.
BibTeX: @article{Burger2004, author = {A.P. Burger and C.M. Mynhardt and E.J. Cockayne}, title = {Regular Solutions of the $n$-Queens Problem on the Torus}, journal = {Utilitas Mathematica}, year = {2004}, volume = {65}, pages = {219-230} }
Burger, A., Mynhardt, C. & Cockayne, E.	Queens Graphs for Chessboards on the Torus [BibTeX]	2001	Australasian Journal of Combinatorics Vol. 24, pp. 231-246	article	URL
BibTeX: @article{Burger2001, author = {A.P. Burger and C.M. Mynhardt and E.J. Cockayne}, title = {Queens Graphs for Chessboards on the Torus}, journal = {Australasian Journal of Combinatorics}, year = {2001}, volume = {24}, pages = {231-246}, url = {http://ajc.maths.uq.edu.au/pdf/24/ajc-v24-p231.pdf} }
Burger, A., Mynhardt, C. & Cockayne, E.	Domination Numbers for the Queens' Graph [BibTeX]	1994	Bulletin of the Institute of Combinatorics and its Applications Vol. 10, pp. 73-82	article
BibTeX: @article{Burger1994, author = {A.P. Burger and C.M. Mynhardt and E.J. Cockayne}, title = {Domination Numbers for the Queens' Graph}, journal = {Bulletin of the Institute of Combinatorics and its Applications}, year = {1994}, volume = {10}, pages = {73-82} }
Bussey, W.	A Note on the Problem of the Eight Queens [BibTeX]	1922	The American Mathematical Monthly Vol. 29(7), pp. 252-253	article	DOI
BibTeX: @article{Bussey1922, author = {W.H. Bussey}, title = {A Note on the Problem of the Eight Queens}, journal = {The American Mathematical Monthly}, year = {1922}, volume = {29(7)}, pages = {252-253}, doi = {http://dx.doi.org/10.2307/2299223} }
Cadoli, M. & Schaerf, M.	Partial Solutions with Unique Completion [Abstract] [BibTeX]	2006	Vol. 4155Reasoning, Action and Interaction in AI Theories and Systems, pp. 101-115	inproceedings	DOI
Abstract: In this paper we investigate the computational complexity of combinatorial problems with givens, i.e., partial solutions, and where a unique solution is required. Examples for this article are taken from the games of Sudoku, $N$-queens and related games. We will show the computational complexity of many decision and search problems related to Sudoku, a number of similar games and their generalization. Furthermore, we propose a logical description of several such problems that can lead to a formulation in the language of Quantified Boolean Formulae (QBF) and, hence, their mechanization via a QBF solver. Some experiments on finding the minimum number of givens necessary/sufficient to guarantee uniqueness of solution are shown.
BibTeX: @inproceedings{Cadoli2006, author = {M. Cadoli and M. Schaerf}, title = {Partial Solutions with Unique Completion}, booktitle = {Reasoning, Action and Interaction in AI Theories and Systems}, publisher = {Spinger}, year = {2006}, volume = {4155}, pages = {101-115}, doi = {http://dx.doi.org/10.1007/11829263} }
Cairns, G.	Pillow Chess [BibTeX]	2002	Mathematics Magazine Vol. 75, pp. 173-186	article	URL
BibTeX: @article{Cairns2002, author = {G. Cairns}, title = {Pillow Chess}, journal = {Mathematics Magazine}, year = {2002}, volume = {75}, pages = {173-186}, url = {http://www.jstor.org/stable/3219240} }
Cairns, G.	Queens on Non-Square Tori [BibTeX]	2001	The Electronic Journal of Combinatorics Vol. 8(1)(N6), pp. 1-3	article	URL
BibTeX: @article{Cairns2001, author = {G. Cairns}, title = {Queens on Non-Square Tori}, journal = {The Electronic Journal of Combinatorics}, year = {2001}, volume = {8(1)}, number = {N6}, pages = {1-3}, url = {http://www.combinatorics.org/Volume_8/PDF/v8i1n6.pdf} }
Campbell, P.	Gauss and the Eight Queens Problem, A Study in Miniature of the Propagation of Historical Error [Abstract] [BibTeX]	1977	Historia Mathematica Vol. 4, pp. 397-404	article	DOI
Abstract: An 1874 article by J. W. L. Glaisher asserted that the eight queens problem of recreational mathematics originated in 1850 with Franz Nauck proposing it to Gauss, who then gave the complete solution. In fact the problem was first proposed two years earlier by Max Bezzel, proposed again by Nauck in a newspaper Gauss happened to read, and only partially solved by Gauss in a casual attempt. Glaisher had access to an accurate account of the history in German but perhaps could not read the language well; the error subsequently spread through the recreational mathematics literature.
BibTeX: @article{Campbell1977, author = {P.J. Campbell}, title = {Gauss and the Eight Queens Problem, A Study in Miniature of the Propagation of Historical Error}, journal = {Historia Mathematica}, year = {1977}, volume = {4}, pages = {397-404}, doi = {http://dx.doi.org/10.1016/0315-0860(77)90076-3} }
Carter, T. & Weakley, W.	The $n$-Queens Problem with Diagonal Constraints [BibTeX]	2005	Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 53, pp. 165-178	article
BibTeX: @article{Carter2005, author = {T.A. Carter and W.D. Weakley}, title = {The $n$-Queens Problem with Diagonal Constraints}, journal = {Journal of Combinatorial Mathematics and Combinatorial Computing}, year = {2005}, volume = {53}, pages = {165-178} }
Catalan, E.	Unknown [BibTeX]	1864	Nouvelles Annales de Mathématiques 216me, t. XIII, pp. 187	inproceedings
BibTeX: @inproceedings{Catalan1864, author = {E.C. Catalan}, title = {Unknown}, booktitle = {Nouvelles Annales de Mathématiques 216me, t. XIII}, year = {1864}, pages = {187} }
Chandra, A.	Independent Permutations, as Related to a Problem of Moser and a Theorem of Pólya [Abstract] [BibTeX]	1974	Journal of Combinatorial Theory, Series A Vol. 16, pp. 111-120	article	DOI
Abstract: We introduce the notion of a set of independent permutations on the domain $0, 1,ldots n-1$, and obtain bounds on the size of the largest such set. The results are applied to a problem proposed by Moser in which he asked for the largest number of nodes in a $d$-cube of side $n$ such that no $n$ of these nodes are collinear. Independent permutations are also related to the problem of placing $n$ non-capturing superqueens (chess queens with wrap-around capability) on an $n times n$ board. As a special case of this treatment we obtain Pólya's theorem that this problem can be solved if and only if $n$ is not a multiple of 2 or 3.
BibTeX: @article{Chandra1974, author = {A.K. Chandra}, title = {Independent Permutations, as Related to a Problem of Moser and a Theorem of Pólya}, journal = {Journal of Combinatorial Theory, Series A}, year = {1974}, volume = {16}, pages = {111-120}, doi = {http://dx.doi.org/10.1016/0097-3165(74)90076-4} }
Chatham, R.	The $N+k$ Queens Problem Page [BibTeX]	2009		misc	URL
BibTeX: @misc{Chatham, author = {R.D. Chatham}, title = {The $N+k$ Queens Problem Page}, year = {2009}, url = {http://people.moreheadstate.edu/fs/d.chatham/n+kqueens.html} }
Chatham, R.	Reflections on the $N + k$ Queens Problem [Abstract] [BibTeX]	2009	College Mathematics Journal Vol. 40, pp. 204-210	article	URL
Abstract: Given a regular chessboard, can you place eight queens on it, so that no two queens attack each other? More generally, given a square chessboard with $N$ rows and $N$ columns, can you place $N$ queens on it, so that no two queens attack each other? This puzzle, known as the $N$ queens problem, is old, and famous, and has an extensive history. Here we present a recently formulated elaboration, which we call the $N + k$ queens problem. We describe some of what is known about the $N + k$ queens problem, prove a few new results, and propose some open questions.
BibTeX: @article{Chatham2009b, author = {R.D. Chatham}, title = {Reflections on the $N + k$ Queens Problem}, journal = {College Mathematics Journal}, year = {2009}, volume = {40}, pages = {204-210}, url = {http://people.moreheadstate.edu/fs/d.chatham/cmj204-210.pdf} }
Chatham, R., Doyle, M., Fricke, G., Reitmann, J., Skaggs, R. & Wolff, M.	Independence and Domination Separation on Chessboard Graphs [Abstract] [BibTeX]	2009	Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 68, pp. 3-17	article	URL
Abstract: A legal placement of Queens is any placement of Queens on an order $N$ chessboard in which any two attacking Queens can be separated by a Pawn. The Queens independence separation number is the minimum number of Pawns which can be placed on an $n n$ board to result in a separated board on which a maximum of $m$ independent Queens can be placed. We prove that $N + k$ Queens can be separated by $k$ Pawns for large enough $N$ and provide some results on the number of fundamental solutions to this problem. We also introduce separation relative to other domination-related parameters for Queens, Rooks, and Bishops.
BibTeX: @article{Chatham2009first, author = {R.D. Chatham and M. Doyle and G.H. Fricke and J. Reitmann and R.D. Skaggs and M. Wolff}, title = {Independence and Domination Separation on Chessboard Graphs}, journal = {Journal of Combinatorial Mathematics and Combinatorial Computing}, year = {2009}, volume = {68}, pages = {3-17}, url = {http://people.moreheadstate.edu/fs/d.chatham/QueensSep2.pdf} }
Chatham, R., Doyle, M., Miller, J., Rogers, A., Skaggs, R. & Ward, J.	Algorithm Performance for Chessboard Separation Problems [Abstract] [BibTeX]	2009	Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 70	article	URL
Abstract: Chessboard separation problems are modifications to classic chessboard problems, such as the $N$ Queens Problem, in which obstacles are placed on the chessboard. This paper focuses on a variation known as the $N + k$ Queens Problem, in which $k$ Pawns and $N + k$ mutually non-attacking Queens are to be placed on an $N$-by-$N$ chessboard. Results are presented from performance studies examining the efficiency of sequential and parallel programs that count the number of solutions to the $N + k$ Queens Problem using traditional backtracking and dancing links. The use of Stochastic Local Search for determining existence of solutions is also presented. In addition, preliminary results are given for a similar problem, the $N +k$ Amazons.
BibTeX: @article{Chatham2009, author = {R.D. Chatham and M. Doyle and J.J. Miller and A.M. Rogers and R.D. Skaggs and J.A. Ward}, title = {Algorithm Performance for Chessboard Separation Problems}, journal = {Journal of Combinatorial Mathematics and Combinatorial Computing}, year = {2009}, volume = {70}, url = {http://people.moreheadstate.edu/fs/d.chatham/dlxMCCC.pdf} }
Chatham, R., Fricke, G. & Skaggs, R.	The Queens Separation Problem [Abstract] [BibTeX]	2006	Utilitas Mathematica Vol. 69, pp. 129-141	article	URL
Abstract: We define a legal placement of Queens to be any placement in which any two attacking Queens can be separated by a Pawn. The Queens separation number is defined to be equal to the minimum number of Pawns which can separate some legal placement of $m$ Queens on an order $n$ chess board. We prove that $n + 1$ Queens can be separated by 1 Pawn and conjecture that $n + k$ Queens can be separated by $k$ Pawns for large enough $n$. We also provide some results on the separation number of other chess pieces.
BibTeX: @article{Chatham2006, author = {R.D. Chatham and G.H. Fricke and R.D. Skaggs}, title = {The Queens Separation Problem}, journal = {Utilitas Mathematica}, year = {2006}, volume = {69}, pages = {129-141}, url = {http://people.moreheadstate.edu/fs/d.chatham/queenssep.pdf} }
Chen, J.-C.	An Efficient Non-Probabilistic Search Algorithm for the $n$-Queens Problem [Abstract] [BibTeX]	2007	Proceedings of the Third Conference on IASTED International Conference: Advances in Computer Science and Technology	inproceedings	URL
Abstract: We present a new heuristic search for the $n$-Queens problem, which is neither backtracking nor random search. This algorithm finds systematically a solution in linear time. Its speed is faster than the fastest algorithm known so far. On an ordinary personal computer, it can find a solution for 3000000 Queens in less than 5 seconds.
BibTeX: @inproceedings{Chen2007, author = {J.-C. Chen}, title = {An Efficient Non-Probabilistic Search Algorithm for the $n$-Queens Problem}, booktitle = {Proceedings of the Third Conference on IASTED International Conference: Advances in Computer Science and Technology}, year = {2007}, url = {http://portal.acm.org/citation.cfm?id=1322534} }
Chen, M.	The Maximum Number of Mutually Uncapturable Strong Queens [BibTeX]	1991	Journal of Qinghai Normal University (Natural Science) Vol. 1, pp. 9-12	article
BibTeX: @article{Chen1991, author = {M. Chen}, title = {The Maximum Number of Mutually Uncapturable Strong Queens}, journal = {Journal of Qinghai Normal University (Natural Science)}, year = {1991}, volume = {1}, pages = {9-12} }
Chen, M., Sun, R. & Zhu, J.	Partial $n$-Solutions to the Modular $n$-Queen Problem [BibTeX]	1992	Chinese Science Bulletin Vol. 37(17), pp. 1422-1425	article
BibTeX: @article{Chen1992, author = {M. Chen and R. Sun and J. Zhu}, title = {Partial $n$-Solutions to the Modular $n$-Queen Problem}, journal = {Chinese Science Bulletin}, year = {1992}, volume = {37(17)}, pages = {1422-1425} }
Chen, M., Sun, R. & Zhu, J.	Partial $n$-Solution to the Modular $n$-Queens Problem. II [BibTeX]	1992	Combinatorics and Graph Theory, Proceedings of the Spring School and International Conference on Combinatorics (SSICC '92), pp. 1-4	inproceedings
BibTeX: @inproceedings{Chen1992a, author = {M. Chen and R. Sun and J. Zhu}, title = {Partial $n$-Solution to the Modular $n$-Queens Problem. II}, booktitle = {Combinatorics and Graph Theory, Proceedings of the Spring School and International Conference on Combinatorics (SSICC '92)}, publisher = {World Scientific}, year = {1992}, pages = {1-4} }
Chvátal, V.	Colouring the Queen Graphs [BibTeX]	2005		misc	URL
BibTeX: @misc{Chvatal2005, author = {V. Chvátal}, title = {Colouring the Queen Graphs}, year = {2005}, url = {http://users.encs.concordia.ca/~chvatal/queengraphs.html} }
Clapp, R., Mudge, T. & Volz, R.	Solutions to the $n$-Queens Problem Using Tasking in Ada [BibTeX]	1986	ACM SIGPLAN Notices Vol. 21, pp. 99-110	article	DOI
BibTeX: @article{Clapp1986, author = {R.M. Clapp and T.N. Mudge and R.A. Volz}, title = {Solutions to the $n$-Queens Problem Using Tasking in Ada}, journal = {ACM SIGPLAN Notices}, year = {1986}, volume = {21}, pages = {99-110}, doi = {http://dx.doi.org/10.1145/15042.15046} }
Clark, D.	A Combinatorial Theorem on Circulant Matrices [BibTeX]	1985	The American Mathematical Monthly Vol. 92(10), pp. 725-729	article	DOI
BibTeX: @article{Clark1985, author = {D.S. Clark}, title = {A Combinatorial Theorem on Circulant Matrices}, journal = {The American Mathematical Monthly}, year = {1985}, volume = {92(10)}, pages = {725-729}, doi = {http://dx.doi.org/10.2307/2323225} }
Clark, D. & Shisha, O.	Invulnerable Queens on an Infinite Chessboard [BibTeX]	1989	Proceedings of the Third International Conference on Combinatorial Mathematics, pp. 133-139	inproceedings
BibTeX: @inproceedings{Clark1989, author = {D.S. Clark and O. Shisha}, title = {Invulnerable Queens on an Infinite Chessboard}, booktitle = {Proceedings of the Third International Conference on Combinatorial Mathematics}, year = {1989}, pages = {133-139} }
Clark, D. & Shisha, O.	Proof without Words: Inductive Construction of an infinite Chessboard with Maximal Placement of Nonattacking Queens [BibTeX]	1988	Mathematics Magazine Vol. 61, pp. 98	article	URL
BibTeX: @article{Clark1988, author = {D.S. Clark and O. Shisha}, title = {Proof without Words: Inductive Construction of an infinite Chessboard with Maximal Placement of Nonattacking Queens}, journal = {Mathematics Magazine}, year = {1988}, volume = {61}, pages = {98}, url = {http://www.jstor.org/stable/2690038} }
Cockayne, E.	Chessboard Domination Problems [Abstract] [BibTeX]	1990	Discrete Mathematics Vol. 86, pp. 13-20	article	DOI
Abstract: A graph may be formed from an $n n$ chessboard by taking the squares as the vertices and two vertices are adjacent if a chess piece situated on one square covers the other. In this paper we survey some recent results concerning domination parameters for certain graphs constructed in this way.
BibTeX: @article{Cockayne1990, author = {E.J. Cockayne}, title = {Chessboard Domination Problems}, journal = {Discrete Mathematics}, year = {1990}, volume = {86}, pages = {13-20}, doi = {http://dx.doi.org/10.1016/0012-365X(90)90344-H} }
Cockayne, E. & Hedetniemi, S.	On the Diagonal Queens Domination Problem [Abstract] [BibTeX]	1986	Journal of Combinatorial Theory, Series A Vol. 42, pp. 137-139	article	DOI
Abstract: It is shown that the problem of covering an $n times n$ chessboard with a minimum number of queens on a major diagonal is related to the number-theoretic function $r_3(n)$, the smallest number of integers in a subset of $1,n$ which must contain three terms in arithmetic progression.
BibTeX: @article{Cockayne1986, author = {E.J. Cockayne and S.T. Hedetniemi}, title = {On the Diagonal Queens Domination Problem}, journal = {Journal of Combinatorial Theory, Series A}, year = {1986}, volume = {42}, pages = {137-139}, doi = {http://dx.doi.org/10.1016/0097-3165(86)90012-9} }
Cockayne, E. & Mynhardt, C.	Properties of Queens Graphs and the Irredundance Number of $Q_7$ [BibTeX]	2001	Australasian Journal of Combinatorics Vol. 23, pp. 285-299	article	URL
BibTeX: @article{Cockayne2001, author = {E.J. Cockayne and C.M. Mynhardt}, title = {Properties of Queens Graphs and the Irredundance Number of $Q_7$}, journal = {Australasian Journal of Combinatorics}, year = {2001}, volume = {23}, pages = {285-299}, url = {http://ajc.maths.uq.edu.au/pdf/23/ajc-v23-p285.pdf} }
Cockayne, E. & Spencer, P.	On the Independent Queens Covering Problem [BibTeX]	1987	Graphs and Combinatorics Vol. 4, pp. 101-110	article	DOI
BibTeX: @article{Cockayne1987, author = {E.J. Cockayne and P.H. Spencer}, title = {On the Independent Queens Covering Problem}, journal = {Graphs and Combinatorics}, year = {1987}, volume = {4}, pages = {101-110}, doi = {http://dx.doi.org/10.1007/BF01864158} }
Colbourn, C. & Rosa, A.	Triple Systems [BibTeX]	1999		book
BibTeX: @book{Colbourn1999, author = {C.J. Colbourn and A. Rosa}, title = {Triple Systems}, publisher = {The Clarendon Press --- Oxford University Press}, year = {1999} }
Cournia, N.	Chessboard Domination on Programmable Graphics Hardware [Abstract] [BibTeX]	2006	Proceedings of the 44th Annual Southeast Regional Conference, pp. 62-67	inproceedings	DOI
Abstract: In this paper we present an algorithm to compute the minimum dominating number of a chessboard graph given any chess piece. We use the CPU to compute possible minimally dominating sets, which we then send to programmable graphics hardware to determine the set's domination. We find that the GPU accelerated algorithm performs better than a comparable CPU based algorithm for board sizes greater than 9. To our knowledge, this paper presents the first algorithm to determine the minimum domination number of a chessboard graph using the GPU.
BibTeX: @inproceedings{Cournia2006, author = {N. Cournia}, title = {Chessboard Domination on Programmable Graphics Hardware}, booktitle = {Proceedings of the 44th Annual Southeast Regional Conference}, year = {2006}, pages = {62-67}, doi = {http://dx.doi.org/10.1145/1185448.1185463} }
Crawford, K.	Solving the $n$-Queens Problem Using Genetic Algorithms [BibTeX]	1992	Proceedings of the 1992 ACM/SIGAPP Symposium on Applied Computing: Technological Challenges of the 1990's, pp. 1039-1047	inproceedings	DOI
BibTeX: @inproceedings{Crawford1992, author = {K.D. Crawford}, title = {Solving the $n$-Queens Problem Using Genetic Algorithms}, booktitle = {Proceedings of the 1992 ACM/SIGAPP Symposium on Applied Computing: Technological Challenges of the 1990's}, year = {1992}, pages = {1039-1047}, doi = {http://dx.doi.org/10.1145/130069.130128} }
Cull, P. & Pandey, R.	Isomorphism and the $n$-Queens Problem [Abstract] [BibTeX]	1994	ACM SIGCSE Bulletin Vol. 26, pp. 29-36	article	DOI
Abstract: The $n$-Queens problem is commonly used to teach the programming technique of backtrack search. The $n$-Queens problem may also be used to illustrate the important concept of isomorphism. Here we show how the $n$-Queens problem can be used as a vehicle to teach the concepts of isomorphism, transformation groups or generators, and equivalence classes. We indicate how these ideas can be used in a programming exercise. We include a bibliography of 29 papers.
BibTeX: @article{Cull1994, author = {P. Cull and R. Pandey}, title = {Isomorphism and the $n$-Queens Problem}, journal = {ACM SIGCSE Bulletin}, year = {1994}, volume = {26}, pages = {29-36}, doi = {http://dx.doi.org/10.1145/187387.187400} }
Cvetković, D.	Some Remarks on the Problem of $n$-Queens [BibTeX]	1969	Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. Vol. 274-301(290), pp. 100-102	article
BibTeX: @article{Cvetkovi'c1969, author = {D. Cvetković}, title = {Some Remarks on the Problem of $n$-Queens}, journal = {Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz.}, year = {1969}, volume = {274-301(290)}, pages = {100-102} }
Dean, D. & Parisi, G.	Statistical Mechanics of a Two-Dimensional System with Long-Range Interactions [Abstract] [BibTeX]	1998	Journal of Physics A: Mathematics and General Vol. 31, pp. 3949-3960	article	DOI
Abstract: We analyse the statistical physics of a two-dimensional lattice-based system with long-range interactions. The particles interact in a way analogous to the queens on a chess board. The long-range nature of the interaction gives the mathematics of the problem a simple geometric structure which simplifies both the analytic and numerical study of the system. We present some analytic calculations for the statics of the problem and we also perform Monte Carlo simulations which exhibit a dynamical transition between a high-temperature liquid regime and a low-temperature glassy regime exhibiting ageing in the two time-correlation functions.
BibTeX: @article{Dean1998, author = {D.S. Dean and G. Parisi}, title = {Statistical Mechanics of a Two-Dimensional System with Long-Range Interactions}, journal = {Journal of Physics A: Mathematics and General}, year = {1998}, volume = {31}, pages = {3949-3960}, doi = {http://dx.doi.org/10.1088/0305-4470/31/17/006} }
Del Manzano, H., Echevar(r)ia, C. & Steinberg, L.	Quantum Algorithm for $n$-Queens Problem [BibTeX]	2002	Computing Research Conference (CRC2002), Mayagüez, Puerto Rico	inproceedings	URL
BibTeX: @inproceedings{Manzano2002, author = {H.A. Del Manzano and C. Echevar(r)ia and L. Steinberg}, title = {Quantum Algorithm for $n$-Queens Problem}, booktitle = {Computing Research Conference (CRC2002), Mayagüez, Puerto Rico}, year = {2002}, url = {http://www.ece.uprm.edu/crc/crc2002/papers/DelManzano_Hector.pdf} }
Demirörs, O., Rafraf, N. & Tanik, M.	Obtaining $n$-Queens Solutions from Magic Squares and Constructing Magic Squares from $n$-Queens Solutions [BibTeX]	1992	Journal of Recreational Mathematics Vol. 24, pp. 272-280	article
BibTeX: @article{Demiroers1992, author = {O. Demirörs and N. Rafraf and M.M. Tanik}, title = {Obtaining $n$-Queens Solutions from Magic Squares and Constructing Magic Squares from $n$-Queens Solutions}, journal = {Journal of Recreational Mathematics}, year = {1992}, volume = {24}, pages = {272-280} }
Demirörs, O. & Tanik, M.	Peaceful Queens and Magic Squares [BibTeX]	1991	(91-CSE-7)	techreport
BibTeX: @techreport{Demiroers1991, author = {O. Demirörs and M.M. Tanik}, title = {Peaceful Queens and Magic Squares}, year = {1991}, number = {91-CSE-7} }
Dietrich, H. & Harborth, H.	Independence on Triangular Triangle Boards [Abstract] [BibTeX]	2005	Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft Vol. 54, pp. 73-87	article
Abstract: Triangular parts of the Euclidean triangle tessellation of the plane are considered as gameboards $T_n$. The independence number $n$ is the maximum number of non-attacking copies of a piece on $T_n$. For nine of the chess-like pieces $n$ is determined completely.
BibTeX: @article{Dietrich2005, author = {H. Dietrich and H. Harborth}, title = {Independence on Triangular Triangle Boards}, journal = {Abhandlungen der Braunschweigischen Wissenschaftlichen Gesellschaft}, year = {2005}, volume = {54}, pages = {73-87} }
Doyle, M., Rawe, B. & Rogers, A.	JDLX: Visualization of Dancing Links [Abstract] [BibTeX]	2008	Journal of Computing Sciences in Colleges Vol. 24, pp. 9-15	article
Abstract: Data structures courses have settled on a familiar canon of structures and algorithms, and this is reflected in the standard textbooks. It is often useful for instructors to enliven such courses by presenting data structures that are of more recent interest, ones that may simultaneously challenge students' understanding of algorithms and their skills in programming. Exact cover problems, exemplified by the newly popular Sudoku game as well as the classic 8-queens problem, may be efficiently solved by the DLX algorithm popularized by Knuth in 2000, and this can provide a good capstone experience in a data structures course. The DLX algorithm operates by recursion on circular multiply linked lists. Because the pointer mechanics of the DLX algorithm is quite complicated, visualization techniques are called for. As the choreography of ``dancing links" in DLX is highly visual anyway, this is very natural. In this paper we review best practices in algorithmic visualization for learners, and then describe a Java-based visualization of DLX applied to $N$-Queens. We also present some preliminary results that indicate that it is effective in enhancing student learning.
BibTeX: @article{Doyle2008, author = {M. Doyle and B. Rawe and A. Rogers}, title = {JDLX: Visualization of Dancing Links}, journal = {Journal of Computing Sciences in Colleges}, year = {2008}, volume = {24}, pages = {9-15} }
Draa, A., Meshoul, S., Talbi, H. & Batouche, M.	A Quantum-Inspired Differential Evolution Algorithm for Solving the N-Queens Problem [Abstract] [BibTeX]	2010	The International Arab Journal of Information Technology Vol. 7, pp. 21-27	article	URL
Abstract: In this paper, a quantum-inspired differential evolution algorithm for solving the N-queens problem is presented. The N-queens problem aims at placing N queens on an NxN chessboard, in such a way that no queen could capture any of the others. The proposed algorithm is a novel hybridization between differential evolution algorithms and quantum computing principles. Accordingly, differential evolution algorithms have been enhanced by the adoption of some quantum concepts such as quantum bits and states superposition. The use of the quantum interference has allowed this hybrid approach to have a remarkable efficiency and good results.
BibTeX: @article{Draa2010, author = {A. Draa and S. Meshoul and H. Talbi and M. Batouche}, title = {A Quantum-Inspired Differential Evolution Algorithm for Solving the N-Queens Problem}, journal = {The International Arab Journal of Information Technology}, year = {2010}, volume = {7}, pages = {21--27}, url = {http://www.ccis2k.org/iajit/PDF/vol.7,no.1/4.pdf} }
Draa, A., Talbi, H. & Batouche, M.	A Quantum-inspired Genetic Algorithm for Solving the $N$-Queens Problem [BibTeX]	2005	Proceedings of the 7th International Symposium on Programming and Systems (ISPS’2005), pp. 145-152	inproceedings
BibTeX: @inproceedings{Draa2005, author = {A. Draa and H. Talbi and M. Batouche}, title = {A Quantum-inspired Genetic Algorithm for Solving the $N$-Queens Problem}, booktitle = {Proceedings of the 7th International Symposium on Programming and Systems (ISPS’2005)}, year = {2005}, pages = {145-152} }
Dudeney, H.	Amusements in Mathematics [BibTeX]	1917		book	URL
BibTeX: @book{Dudeney1917, author = {H.E. Dudeney}, title = {Amusements in Mathematics}, publisher = {Thomas Nelson & Sons, Limited}, year = {1917}, url = {http://www.gutenberg.org/etext/16713} }
Durango Bill	The $N$-Queens Problem [BibTeX]			misc	URL
BibTeX: @misc{Durango, author = {Durango Bill}, title = {The $N$-Queens Problem}, url = {http://www.durangobill.com/N_Queens.html} }
Eiben, A., Raué, P.-E. & Ruttkay, Z.	GA-easy and GA-hard Constraint Satisfaction Problems [Abstract] [BibTeX]	1995	Vol. 923Proceedings of the ECAI-94 Workshop on Constraint Processing, pp. 267-283	inproceedings	DOI
Abstract: In this paper we discuss the possibilities of applying genetic algorithms (GA) for solving constraint satisfaction problems (CSP). We point out how the greediness of deterministic classical CSP solving techniques can be counterbalanced by the random mechanisms of GAs. We tested our ideas by running experiments on four different CSPs: $N$-queens, graph 3-colouring, the traffic lights and the Zebra problem. Three of the problems have proven to be GA-easy, and even for the GA-hard one the performance of the GA could be boosted by techniques familiar in classical methods. Thus GAs are promising tools for solving CSPs. In the discussion, we address the issues of non-solvable CSPs and the generation of all the solutions.
BibTeX: @inproceedings{Eiben1995, author = {A.E. Eiben and P.-E. Raué and Zs. Ruttkay}, title = {GA-easy and GA-hard Constraint Satisfaction Problems}, booktitle = {Proceedings of the ECAI-94 Workshop on Constraint Processing}, publisher = {Springer-Verlag}, year = {1995}, volume = {923}, pages = {267-283}, doi = {http://dx.doi.org/10.1007/3-540-59479-5_30} }
Eiben, A., Raué, P.-E. & Ruttkay, Z.	Solving Constraint Satisfaction Problems Using Genetic Algorithms [Abstract] [BibTeX]	1994	Vol. 2Proceedings of the 1st IEEE World Conference on Computational Intelligence, pp. 542-547	inproceedings	DOI
Abstract: This article discusses the applicability of genetic algorithms (GAs) to solve constraint satisfaction problems (CSPs). We discuss the requirements and possibilities of defining so-called heuristic GAs (HGAs), which can be expected to be effective and efficient methods to solve CSPs since they adopt heuristics used in classical CSP solving search techniques. We present and analyse experimental results gained by testing different heuristic GAs on the $N$-queens problem and on the graph 3-colouring problem
BibTeX: @inproceedings{Eiben1994, author = {A.E. Eiben and P.-E. Raué and Zs. Ruttkay}, title = {Solving Constraint Satisfaction Problems Using Genetic Algorithms}, booktitle = {Proceedings of the 1st IEEE World Conference on Computational Intelligence}, publisher = {IEEE Service Center}, year = {1994}, volume = {2}, pages = {542-547}, doi = {http://dx.doi.org/10.1109/ICEC.1994.350002} }
Eickenscheidt, B.	Das $n$-Damen-Problem auf dem Zylinderbrett [BibTeX]	1980	feenschach Vol. 50, pp. 382-385	article
BibTeX: @article{Eickenscheidt1980, author = {B. Eickenscheidt}, title = {Das $n$-Damen-Problem auf dem Zylinderbrett}, journal = {feenschach}, year = {1980}, volume = {50}, pages = {382-385} }
El-Qawasmeh, E. & Al-Noubani, K.	Reducing the Time Complexity of the $N$-Queens Problem [Abstract] [BibTeX]	2005	International Journal on Artificial Intelligence Tools Vol. 14, pp. 545-557	article	DOI
Abstract: This paper presents a fast algorithm for solving the $n$-queens problem. The basic idea of this algorithm is to use pre-computed solutions in 75% of the cases, while the remaining cases are solved by calling the Sosic's algorithm. The novelty of this algorithm is in the observation that these pre-computable cases exhibit a modular nature. In addition, the pre-computed solutions run 100 times faster than Sosic's algorithm in most cases.
BibTeX: @article{El-Qawasmeh2005, author = {E. El-Qawasmeh and K. Al-Noubani}, title = {Reducing the Time Complexity of the $N$-Queens Problem}, journal = {International Journal on Artificial Intelligence Tools}, year = {2005}, volume = {14}, pages = {545-557}, doi = {http://dx.doi.org/10.1142/S0218213005002247} }
El-Qawasmeh, E. & Al-Noubani, K.	A Polynomial Time Algorithm for the $N$-Queens Problems [BibTeX]	2004	Proceedings of the IASTED International Conference on Neural Networks and Computational Intelligence (NCI 2004), pp. 191-195	inproceedings
BibTeX: @inproceedings{El-Qawasmeh2004, author = {E. El-Qawasmeh and K. Al-Noubani}, title = {A Polynomial Time Algorithm for the $N$-Queens Problems}, booktitle = {Proceedings of the IASTED International Conference on Neural Networks and Computational Intelligence (NCI 2004)}, year = {2004}, pages = {191-195} }
Engelhardt, M.	A Group-based Search for Solutions of the $n$-Queens Problem [Abstract] [BibTeX]	2007	Discrete Mathematics Vol. 307, pp. 2535-2551	article	DOI
Abstract: The $n$-Queens problem is a well-known problem in mathematics, yet a full search for $n$-Queens solutions has been tackled until now using only simple algorithms (with the exception of the Rivin–Zabih algorithm). In this article, we discuss optimizations that mainly rely on group actions on the set of $n$-Queens solutions. Most of our arguments deal with the case of toroidal Queens; at the end, the application to the regular $n$-Queens problem is discussed, and also the Rivin–Zabih algorithm.
BibTeX: @article{Engelhardt2007, author = {M.R. Engelhardt}, title = {A Group-based Search for Solutions of the $n$-Queens Problem}, journal = {Discrete Mathematics}, year = {2007}, volume = {307}, pages = {2535-2551}, doi = {http://dx.doi.org/10.1016/j.disc.2007.01.007} }
Erbas, C., Rafraf, N. & Tanik, M.	Magic Squares Constructing by the Uniform Step Method Provide Solutions to the $n$-Queens Problem [BibTeX]	1991	(91-CSE-25)	techreport
BibTeX: @techreport{Erbas1991, author = {C. Erbas and N. Rafraf and M.M. Tanik}, title = {Magic Squares Constructing by the Uniform Step Method Provide Solutions to the $n$-Queens Problem}, year = {1991}, number = {91-CSE-25} }
Erbas, C., Sarkeshik, S. & Tanik, M.	Different Perspectives of the $n$-Queens Problem [Abstract] [BibTeX]	1992	CSC '92: Proceedings of the 1992 ACM Annual Conference on Communications, pp. 99-108	inproceedings	DOI
Abstract: The $N$-Queens problem is a commonly used example in computer science. There are numerous approaches proposed to solve the problem. We introduce several definitions of the problem, and review some of the algorithms. We classify the algorithms for the $N$-Queens problem into 3 categories. The first category comprises the algorithms generating all the solutions for a given $N$. The algorithms in the second category are desinged to generate only the fundamental solutions~teTopor1982. The algorithms in the last category generate only one or several solutions but not necessarily all of them.
BibTeX: @inproceedings{Erbas1992, author = {C. Erbas and S. Sarkeshik and M.M. Tanik}, title = {Different Perspectives of the $n$-Queens Problem}, booktitle = {CSC '92: Proceedings of the 1992 ACM Annual Conference on Communications}, year = {1992}, pages = {99-108}, doi = {http://dx.doi.org/10.1145/131214.131227} }
Erbas, C., Sarkeshik, S. & Tanik, M.	Algorithmic and Constructive Approaches to the $n$-Queens Problem [BibTeX]	1991	(91-CSE-31)	techreport
BibTeX: @techreport{Erbas1991a, author = {C. Erbas and S. Sarkeshik and M.M. Tanik}, title = {Algorithmic and Constructive Approaches to the $n$-Queens Problem}, year = {1991}, number = {91-CSE-31} }
Erbas, C. & Tanik, M.	Generating Solutions to the $n$-Queens Problem Using $2$-Circulants [BibTeX]	1995	Mathematics Magazine Vol. 68, pp. 343-356	article	URL
BibTeX: @article{Erbas1995a, author = {C. Erbas and M.M. Tanik}, title = {Generating Solutions to the $n$-Queens Problem Using $2$-Circulants}, journal = {Mathematics Magazine}, year = {1995}, volume = {68}, pages = {343-356}, url = {http://www.jstor.org/stable/2690923} }
Erbas, C. & Tanik, M.	Parallel Memory Allocation and Data Alignment in SIMD Machines [Abstract] [BibTeX]	1994	Parallel Algorithms and Applications Vol. 4, pp. 139-151	article	DOI
Abstract: In this paper, we introduce a memory storage scheme allowing conflict-free parallel access to rows, columns, square blocks, distributed blocks, and positive and negative diagonals of two dimensional arrays. Unlike the existing schemes, the proposed scheme can be used for an arbitrary number of memory modules and an arbitrary size of matrices. We develop a systematic procedure for the memory allocation based on a placement matrix constructed using circulant matrices. We, also, analyze the data alignment requirements of the proposed scheme, and demonstrate that the data vectors read from memory modules can be aligned for the processors using a set of shift, flip, and shuffle operations, which can be implemented by a data manipulation network.
BibTeX: @article{Erbas1994, author = {C. Erbas and M.M. Tanik}, title = {Parallel Memory Allocation and Data Alignment in SIMD Machines}, journal = {Parallel Algorithms and Applications}, year = {1994}, volume = {4}, pages = {139-151}, doi = {http://dx.doi.org/10.1080/10637199408915460} }
Erbas, C. & Tanik, M.	Storage Schemes for Parallel Memory Systems and the $n$-Queens Problem [BibTeX]	1992	Vol. 43Proceedings of the 15th Anniversary of the ASME ETCE Confererence, Computer Applications Symposium, pp. 115-120	inproceedings
BibTeX: @inproceedings{Erbas1992a, author = {C. Erbas and M.M. Tanik}, title = {Storage Schemes for Parallel Memory Systems and the $n$-Queens Problem}, booktitle = {Proceedings of the 15th Anniversary of the ASME ETCE Confererence, Computer Applications Symposium}, year = {1992}, volume = {43}, pages = {115-120} }
Erbas, C. & Tanik, M.	$n$-Queens Problem and its Algorithms [BibTeX]	1991	(91-CSE-8)	techreport
BibTeX: @techreport{Erbas1991b, author = {C. Erbas and M.M. Tanik}, title = {$n$-Queens Problem and its Algorithms}, year = {1991}, number = {91-CSE-8} }
Erbas, C. & Tanik, M.	$n$-Queens Problem and its Connection to the Polygons [BibTeX]	1991	(91-CSE-21)	techreport
BibTeX: @techreport{Erbas1991c, author = {C. Erbas and M.M. Tanik}, title = {$n$-Queens Problem and its Connection to the Polygons}, year = {1991}, number = {91-CSE-21} }
Erbas, C., Tanik, M. & Aliyazicioglu, Z.	Linear Congruence Equations for the Solutions of the $n$-Queens Problem [Abstract] [BibTeX]	1992	Information Processing Letters Vol. 41, pp. 301-306	article	DOI
Abstract: We demonstrate a method using linear congruence equations to generate solutions to the $N$-Queens problem. There are only a few papers in the literature generating solutions for every $N$. Our method generates solutions for every $N$, and the number of solutions produced by our method is larger than the number of solutions given in these papers.
BibTeX: @article{Erbas1992b, author = {C. Erbas and M.M. Tanik and Z. Aliyazicioglu}, title = {Linear Congruence Equations for the Solutions of the $n$-Queens Problem}, journal = {Information Processing Letters}, year = {1992}, volume = {41}, pages = {301-306}, doi = {http://dx.doi.org/10.1016/0020-0190(92)90156-P} }
Erbas, C., Tanik, M. & Aliyazicioglu, Z.	A Note on Falkowskis $n$-Queens Solutions [BibTeX]	1992	(92-CSE-14)	techreport
BibTeX: @techreport{Erbas1992c, author = {C. Erbas and M.M. Tanik and Z. Aliyazicioglu}, title = {A Note on Falkowskis $n$-Queens Solutions}, year = {1992}, number = {92-CSE-14} }
Erbas, C., Tanik, M. & Nair, V.	A Circulant Matrix Based Approach to Storage Schemes for Parallel Memory Systems [Abstract] [BibTeX]	1993	Proceedings of the Fifth IEEE Symposium on Parallel and Distributed Processing, pp. 92-99	inproceedings	DOI
Abstract: We introduce a memory storage scheme allowing conflict-free parallel access to rows, columns, square blocks, distributed blocks, and positive and negative diagonals of two dimensional arrays. Unlike the existing schemes, the proposed scheme can be used for an arbitrary number of memory modules and an arbitrary size of the arrays. We develop a systematic procedure for the memory allocation based on a placement matrix constructed using circulant matrices
BibTeX: @inproceedings{Erbas1993, author = {C. Erbas and M.M. Tanik and V.S.S. Nair}, title = {A Circulant Matrix Based Approach to Storage Schemes for Parallel Memory Systems}, booktitle = {Proceedings of the Fifth IEEE Symposium on Parallel and Distributed Processing}, year = {1993}, pages = {92-99}, doi = {http://dx.doi.org/10.1109/SPDP.1993.395546} }
Erdem, E. & Lifschitz, V.	Tight Logic Programs [Abstract] [BibTeX]	2003	Theory and Practice of Logic Programming Vol. 3, pp. 499-518	article	DOI
Abstract: This note is about the relationship between two theories of negation as failure --- one based on program completion, the other based on stable models, or answer sets. François Fages showed that if a logic program satisfies a certain syntactic condition, which is now called ‘tightness,’ then its stable models can be characterized as the models of its completion. We extend the definition of tightness and Fages' theorem to programs with nested expressions in the bodies of rules, and study tight logic programs containing the definition of the transitive closure of a predicate.
BibTeX: @article{Erdem2003, author = {E. Erdem and V. Lifschitz}, title = {Tight Logic Programs}, journal = {Theory and Practice of Logic Programming}, year = {2003}, volume = {3}, pages = {499-518}, doi = {http://dx.doi.org/10.1017/S1471068403001765} }
Falkowski, B.-J. & Schmitz, L.	A Note on the Queen's Problem [BibTeX]	1986	Information Processing Letters Vol. 23, pp. 39-46	article	DOI
BibTeX: @article{Falkowski1986, author = {B.-J. Falkowski and L. Schmitz}, title = {A Note on the Queen's Problem}, journal = {Information Processing Letters}, year = {1986}, volume = {23}, pages = {39-46}, doi = {http://dx.doi.org/10.1016/0020-0190(86)90128-6} }
Fillmore, J. & Williamson, S.	On Backtracking: A Combinatorial Description of the Algorithm [Abstract] [BibTeX]	1974	SIAM Journal on Computing Vol. 3, pp. 41-55	article	DOI
Abstract: A basic algorithm for solving many discrete problems is the so-called ``backtracking" algorithm. The basic problem is that of generating the elements of a subset $S_0 $ of a finite set in an efficient manner. If a group $G$ acts on $S_0 $, then one might wish to obtain only nonisomorphic elements of $S_0 $. In this paper the basic backtracking algorithm is described in terms of chains of partitions on the set $S$. The corresponding isomorph rejection problem is described in terms of $G$-invariant chains of partitions on $S$. Examples and flow charts are given.
BibTeX: @article{Fillmore1974, author = {J.P. Fillmore and S.G. Williamson}, title = {On Backtracking: A Combinatorial Description of the Algorithm}, journal = {SIAM Journal on Computing}, year = {1974}, volume = {3}, pages = {41-55}, doi = {http://dx.doi.org/10.1137/0203004} }
Finch, S.	Encyclopedia of Mathematics and its Applications [BibTeX]	2003	Vol. 94	inbook
BibTeX: @inbook{Finch2003, author = {S.R. Finch}, title = {Encyclopedia of Mathematics and its Applications}, publisher = {Cambridge University Press}, year = {2003}, volume = {94} }
Foley, J.	Manchester Dataflow Machine: Preliminary Benchmark Test Evaluation [Abstract] [BibTeX]	1987	(UMCS-87-11-2)	techreport	URL
Abstract: The Manchester Dataflow Hardware is supported by a Software compiler for the SISAL language and a number of programs have been written to act as Benchmark tests for the hardware. The Benchmark set used contains a wide range of programs including numerical algorithms, sorting, graph colouring and $n$ Queens algorithms plus others. All programs are compiled using a range of optimisations, including function inlining and vectorisation. The resulting statistics, obtained both by simulation and hardware are presented.
BibTeX: @techreport{Foley1987, author = {J. Foley}, title = {Manchester Dataflow Machine: Preliminary Benchmark Test Evaluation}, year = {1987}, number = {UMCS-87-11-2}, url = {http://intranet.cs.man.ac.uk/Intranet_subweb/library/cstechrep/Abstracts/UMCS-87-11-2.html} }
Foulds, L. & Johnston, D.	An Application of Graph Theory and Integer Programming: Chessboard Nonattacking Puzzles [BibTeX]	1984	Mathematics Magazine Vol. 57(3), pp. 95-104	article	URL
BibTeX: @article{Foulds1984, author = {L.R. Foulds and D.G. Johnston}, title = {An Application of Graph Theory and Integer Programming: Chessboard Nonattacking Puzzles}, journal = {Mathematics Magazine}, year = {1984}, volume = {57(3)}, pages = {95-104}, url = {http://www.jstor.org/stable/2689591} }
Franel, J.	$n$-Queens solution [BibTeX]	1894	L'Intermédiaire des Mathématiciens Vol. 11, pp. 140-141	article
BibTeX: @article{Franel1894, author = {J. Franel}, title = {$n$-Queens solution}, journal = {L'Intermédiaire des Mathématiciens}, year = {1894}, volume = {11}, pages = {140-141} }
Gómez(-Aiza), R., Montellano(-Ballesteros), J. & Strausz, R.	On the Modular $n$-Queen Problem in Higher Dimensions [Abstract] [BibTeX]	2004		misc	URL
Abstract: The modular $n$-queen problem in higher dimensions was introduced by Nudelman teNudelman1995. He showed that for a complete solution to exist in the $d$-dimensional modular $n$-chessboard, it is necessary that $n, (2d-1)!) = 1$, and that it is sufficient that $n, (2d-1)!) = 1$. He conjectured that the last condition is also necessary and showed that this is indeed the case for the class of linear solutions. In this notes, we observe that the conjecture is true for the larger class of polynomial solutions, which are solutions we present as a natural generalization of the bidimensional solutions developed by Kløve teKlove1977. We also generalize constructions of bidimensional solutions developed also by Kløve teKlove1981.
BibTeX: @misc{Gomez2004, author = {R. Gómez(-Aiza) and J.J. Montellano(-Ballesteros) and R. Strausz}, title = {On the Modular $n$-Queen Problem in Higher Dimensions}, year = {2004}, url = {http://www.liacs.nl/home/kosters/nqueens/papers/gomez2004.pdf} }
Gao, Q. & Hou, S.	Junior Researcher: A Discovery System that can solve the Queens Problems on a Constant Computational Complexity [Abstract] [BibTeX]	1990	Information Technology, 1990. Next Decade in Information Technology, Proceedings of the 5th Jerusalem Conference on (Cat. No.90TH0326-9), pp. 345-347	inproceedings	DOI
Abstract: An approach that uses the discovery system Junior Researcher to solve the $n$-Queens problems ($n geq 4$) is proposed. The functions, structure and features of Junior Researcher are described. A constant-complexity algorithm for solving the problem is then given.
BibTeX: @inproceedings{Gao1990, author = {Q.S. Gao and S.J. Hou}, title = {Junior Researcher: A Discovery System that can solve the Queens Problems on a Constant Computational Complexity}, booktitle = {Information Technology, 1990. Next Decade in Information Technology, Proceedings of the 5th Jerusalem Conference on (Cat. No.90TH0326-9)}, year = {1990}, pages = {345-347}, doi = {http://dx.doi.org/10.1109/JCIT.1990.128303} }
Gardner, M.	Chess Queens and Maximum Unattacked Cells [Abstract] [BibTeX]	1999	Math Horizons Vol. 7, pp. 12-16	article
Abstract: There is now an enormous literature on the old classic task of placing eight queens on a chessboard so that no queen attacks another. There are twelve solutions, not counting trivial rotations and reflections. The task naturally generalizes to enumerating the number of solutions for $n$ non-attacking queens on an $nn$ board.
BibTeX: @article{Gardner1999, author = {M. Gardner}, title = {Chess Queens and Maximum Unattacked Cells}, journal = {Math Horizons}, year = {1999}, volume = {7}, pages = {12-16} }
Gardner, M.	Fractal Music, Hypercards and More Mathematical Recreations from Scientific American Magazin [BibTeX]	1991		book
BibTeX: @book{Gardner1991, author = {M. Gardner}, title = {Fractal Music, Hypercards and More Mathematical Recreations from Scientific American Magazin}, publisher = {Freeman}, year = {1991} }
Gardner, M.	Wheels, Life, and Other Mathematical Amusements [BibTeX]	1983		book
BibTeX: @book{Gardner1983, author = {M. Gardner}, title = {Wheels, Life, and Other Mathematical Amusements}, publisher = {Freeman}, year = {1983} }
Gardner, M.	Patterns in Primes are a Clue to the Strong Law of Small Numbers [BibTeX]	1980	Scientific American Vol. 243, pp. 18-28	article
BibTeX: @article{Gardner1980, author = {M. Gardner}, title = {Patterns in Primes are a Clue to the Strong Law of Small Numbers}, journal = {Scientific American}, year = {1980}, volume = {243}, pages = {18-28} }
Gardner, M.	Mathematical Games [BibTeX]	1972	Scientific American Vol. 227, pp. 176-182	article
BibTeX: @article{Gardner1972, author = {Martin Gardner}, title = {Mathematical Games}, journal = {Scientific American}, year = {1972}, volume = {227}, pages = {176-182} }
Gardner, M.	The Unexpected Hanging and Other Mathematical Diversions [BibTeX]	1968		book
BibTeX: @book{Gardner1968, author = {M. Gardner}, title = {The Unexpected Hanging and Other Mathematical Diversions}, publisher = {Simon & Schuster}, year = {1968} }
Garey, M. & Johnson, D.	Computers and Intractability: A Guide to the Theory of NP-Completeness [BibTeX]	1979		book
BibTeX: @book{Garey1983, author = {M.R. Garey and D.S. Johnson}, title = {Computers and Intractability: A Guide to the Theory of NP-Completeness}, publisher = {W. H. Freeman and Co., San Fransisco, CA}, year = {1979} }
Garner, C. & Herzberg, A.	On McCarty's Queen Squares [BibTeX]	1981	The American Mathematical Monthly Vol. 88(8), pp. 612-613	article	DOI
BibTeX: @article{Garner1981, author = {C.W.L. Garner and A.M. Herzberg}, title = {On McCarty's Queen Squares}, journal = {The American Mathematical Monthly}, year = {1981}, volume = {88(8)}, pages = {612-613}, doi = {http://dx.doi.org/10.2307/2320511} }
Gauss, C.	Werke Band XII [BibTeX]	1973		book
BibTeX: @book{Gauss1973, author = {C.F. Gauss}, title = {Werke Band XII}, publisher = {George Olms Verlag, Hildesheim}, year = {1973} }
Gibbons, P. & Webb, J.	Some New Results for the Queens Domination Problem [BibTeX]	1997	Australasian Journal of Combinatorics Vol. 15, pp. 145-160	article	URL
BibTeX: @article{Gibbons1996, author = {P.B. Gibbons and J.A. Webb}, title = {Some New Results for the Queens Domination Problem}, journal = {Australasian Journal of Combinatorics}, year = {1997}, volume = {15}, pages = {145-160}, url = {http://ajc.maths.uq.edu.au/pdf/15/ajc-v15-p145.pdf} }
Gik, E.	Shakhmaty i Matematika (BibliotechkaKvant) [BibTeX]	1983	Vol. 24	book
BibTeX: @book{Gik1983, author = {E.Y. Gik}, title = {Shakhmaty i Matematika (BibliotechkaKvant)}, publisher = {Nauka, Moscow}, year = {1983}, volume = {24} }
Gik, E.	Matematika na shakhmatnoi doske (Nauchno-populiarnaiaseriia) [BibTeX]	1976		book
BibTeX: @book{Gik1976, author = {E.Y. Gik}, title = {Matematika na shakhmatnoi doske (Nauchno-populiarnaiaseriia)}, publisher = {Nauka, Moscow}, year = {1976} }
Ginsburg, J.	Gauss's Arithmetization of the Problem of $n$-Queens [BibTeX]	1939	Scripta Mathematica Vol. 5, pp. 63-66	article
BibTeX: @article{Ginsburg1939, author = {Ginsburg, J.}, title = {Gauss's Arithmetization of the Problem of $n$-Queens}, journal = {Scripta Mathematica}, year = {1939}, volume = {5}, pages = {63-66} }
Glaisher, J.	On the Problem of the Eight Queens [BibTeX]	1874	Edinburgh Philosophical Magazine Vol. 4(48), pp. 457-467	article
BibTeX: @article{Glaisher1874, author = {J.W.L. Glaisher}, title = {On the Problem of the Eight Queens}, journal = {Edinburgh Philosophical Magazine}, year = {1874}, volume = {4(48)}, pages = {457-467} }
Goldsby, M.	Solving the ``$N <= 8$-Queens" Problem with CSP and Modula-2 [BibTeX]	1987	SIGPLAN Notices Vol. 22, pp. 43-52	article	DOI
BibTeX: @article{Goldsby1987, author = {M.E. Goldsby}, title = {Solving the ``$N <= 8$-Queens" Problem with CSP and Modula-2}, journal = {SIGPLAN Notices}, year = {1987}, volume = {22}, pages = {43-52}, doi = {http://dx.doi.org/10.1145/24686.24689} }
Golomb, S.	Sphere Packing, Coding Metrics and Chess Puzzles [BibTeX]	1970	Chapel Hill Conference on Combinatorial Mathematics and its Applications, pp. 176-189	inproceedings
BibTeX: @inproceedings{Golomb1970, author = {S.W. Golomb}, title = {Sphere Packing, Coding Metrics and Chess Puzzles}, booktitle = {Chapel Hill Conference on Combinatorial Mathematics and its Applications}, year = {1970}, pages = {176-189} }
Golomb, S. & Baumert, L.	Backtrack Programming [Abstract] [BibTeX]	1965	Journal of the ACM Vol. 12, pp. 516-524	article	DOI
Abstract: A widely used method of efficient search is examined in detail. This examiniation provides the opprtunity to formulate its scope and methods in their full generality. In addition to a general exposition of the basic process, some important refinements are indicated. Examples are given which illustrate the salient features of this searching process.
BibTeX: @article{Golomb1965, author = {S.W. Golomb and L.D. Baumert}, title = {Backtrack Programming}, journal = {Journal of the ACM}, year = {1965}, volume = {12}, pages = {516-524}, doi = {http://dx.doi.org/10.1145/321296.321300} }
Golomb, S. & Taylor, H.	Constructions and Properties of Costas Arrays [Abstract] [BibTeX]	1984	Proceedings of the IEEE Vol. 72, pp. 1143-1163	article	DOI
Abstract: A Costas array is an $nn$ array of dots and blanks with exactly one dot in each row and column, and with distinct vector differences between all pairs of dots. As a frequency-hop pattern for radar or sonar, a Costas array has an optimum ambiguity function, since any translation of the array parallel to the coordinate axes produces at most one out-of-phase coincidence. We conjecture that $nn$ Costas arrays exist for every positive integer $n$. Using various constructions due to L. Welch, A. Lempel, and the authors, Costas arrays are shown to exist when $n = p - 1$, $n = q - 2$, $n = q - 3$, and sometimes when $n = q - 4$ and $n = q - 5$, where $p$ is a prime number, and $q$ is any power of a prime number. All known Costas array constructions are listed for 271 values of $n$ up to 360. The first eight gaps in this table occur at $n = 32$, 33, 43, 48, 49, 53, 54, 63. (The examples for $n = 19$ and $n = 31$ were obtained by augmenting Welch's construction.) Let $C(n)$ denote the total number of $nn$ Costas arrays. Costas calculated $C(n)$ for $n leq 12$. Recently, John Robbins found $C(13) = 12828$. We exhibit all the arrays for $n leq 8$. From Welch's construction, $C(n) geq 2n$ for infinitely many $n$. Some Costas arrays can be sheared into ``honeycomb arrays.'' All known honeycomb arrays are exhibited, corresponding to $n = 1$, 3, 7, 9, 15, 21, 27, 45. Ten unsolved problems are listed.
BibTeX: @article{Golomb1984, author = {S.W. Golomb and H. Taylor}, title = {Constructions and Properties of Costas Arrays}, journal = {Proceedings of the IEEE}, year = {1984}, volume = {72}, pages = {1143-1163}, doi = {http://dx.doi.org/10.1109/PROC.1984.12994} }
Golombeck, H.	Golombeck's Encyclopedia of Chess [BibTeX]	1977		book
BibTeX: @book{Golombeck1977, author = {H. Golombeck}, title = {Golombeck's Encyclopedia of Chess}, publisher = {Crown Publishers, New York}, year = {1977} }
Gosset, T.	The Eight Queens Problem [BibTeX]	1914	Messenger of Mathematics Vol. 44, pp. 48	article
BibTeX: @article{Gosset1914, author = {T. Gosset}, title = {The Eight Queens Problem}, journal = {Messenger of Mathematics}, year = {1914}, volume = {44}, pages = {48} }
Gray, J.	Is Eight Enough? The Eight Queens Problem Re-examined [Abstract] [BibTeX]	1993	ACM SIGCSE Bulletin Vol. 25, pp. 39-44,51	article	DOI
Abstract: A detailed analysis of a classic backtracking problem, The Eight Queen Problem is presented. The paper addresses additional facets of the Eight Queen Problem that might be overlooked when casually generating a program solution. The author suggests that the extra time taken to fully analyze the problem will result in a better understanding of the problem which in turn will manifest itself in a better program solution.
BibTeX: @article{Gray1993, author = {J.S. Gray}, title = {Is Eight Enough? The Eight Queens Problem Re-examined}, journal = {ACM SIGCSE Bulletin}, year = {1993}, volume = {25}, pages = {39-44,51}, doi = {http://dx.doi.org/10.1145/165408.165423} }
Grinstead, C., Hahne, B. & Van Stone, D.	On the Queen Domination Problem [Abstract] [BibTeX]	1990	Discrete Mathematics Vol. 86, pp. 21-26	article	DOI
Abstract: A configuration of queens on an $m m$ chessboard is said to dominate the board if every square either contains a queen or is attacked by a queen. The configuration is said to be non-attacking if no queen attacks another queen. Let $f(m)$ and $g(m)$ equal the minimum number of queens and the minimum number of non-attacking queens, respectively, needed to dominate an $m m$ chessboard. We prove that: 1. $f(m)leq1423m+O(1)$, and 2. $g(m)leq23m+O(1)$. These are the best upper bounds known at the present time for these functions.
BibTeX: @article{Grinstead1990, author = {C.M. Grinstead and B. Hahne and D. Van Stone}, title = {On the Queen Domination Problem}, journal = {Discrete Mathematics}, year = {1990}, volume = {86}, pages = {21-26}, doi = {http://dx.doi.org/10.1016/0012-365X(90)90345-I} }
Gu, J.	On a General Framework for Large-scale Constraint-Based Optimization [Abstract] [BibTeX]	1991	ACM SIGART Bulletin Vol. 2, pp. 8	article	DOI
Abstract: The explicit solution for the $n$-queens problem, mentioned in a letter from Bo Bernhardsson teBernhardsson1991, is basically Pauls's solution analyzed by Ahrens (See reference teAhrens1901 of our previous article in SIGART October issue 1990). The result was in public domain long before 1918 (not 1969). We also mentioned its weakness, namely: The class of solutions provided by analytical methods is very restricted, as Ahrens pointed out in teAhrens1901. They can only provide one solution for the $n$-queens problem and can not provide any solution (much better explicit solutions for the $n$-queens problem exist). This is not the case for search methods which can find, in principle, any solution. This distinction is crucial for practical applications of the $n$-queens problem.
BibTeX: @article{Gu1991, author = {J. Gu}, title = {On a General Framework for Large-scale Constraint-Based Optimization}, journal = {ACM SIGART Bulletin}, year = {1991}, volume = {2}, pages = {8}, doi = {http://dx.doi.org/10.1145/122319.122323} }
Gunther, S.	Zur Mathematisches Theorie des Schachbretts [BibTeX]	1874	Archiv der Mathematik und Physik Vol. 56, pp. 281-292	article
BibTeX: @article{Gunther1874, author = {S. Gunther}, title = {Zur Mathematisches Theorie des Schachbretts}, journal = {Archiv der Mathematik und Physik}, year = {1874}, volume = {56}, pages = {281-292} }
Guy, R.	Unsolved Problems in Number Theory [BibTeX]	1994		book
BibTeX: @book{Guy1994, author = {R.K. Guy}, title = {Unsolved Problems in Number Theory}, publisher = {Springer-Verlag}, year = {1994} }
Gómez, R.	On the $d$-Dimensional Modular $n$-Queen Problem [BibTeX]	1997	School: University of Maryland at College Park	mastersthesis
BibTeX: @mastersthesis{Gomez1997, author = {R. Gómez}, title = {On the $d$-Dimensional Modular $n$-Queen Problem}, school = {University of Maryland at College Park}, year = {1997} }
Han, J., Liu, J. & Cai, Q.	From Alife Agents to a Kingdom of $n$-Queens [Abstract] [BibTeX]	1999	Intelligent Agent Technology: Systems, Methodologies, and Tools, pp. 110-120	inproceedings	URL
Abstract: This paper presents a new approach to solving $n$-Queen problems, which involves a model of distributed autonomous agents with artificial life (ALife) and a method of representing $n$-Queen constraints in an agent environment. The distributed agents locally interact with their living environment, i.e., a chessboard, and execute their reactive behaviors by applying their behavioral rules for randomized motion, least-conflict position searching, and cooperating with other agents etc. The agent-based $n$-Queen problem solving system evolves through selection and contest according to the rule of Survival of the Fittest, in which some agents will die or be eaten if their moving strategies are less efficient than others. The experimental results have shown that this system is capable of solving large-scale $n$-Queen problems. This paper also provides a model of ALife agents for solving general CSPs.
BibTeX: @inproceedings{Jing1999, author = {J. Han and J. Liu and Q. Cai}, title = {From Alife Agents to a Kingdom of $n$-Queens}, booktitle = {Intelligent Agent Technology: Systems, Methodologies, and Tools}, year = {1999}, pages = {110-120}, url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.6158} }
Han, J., Liu, L. & Lu, T.	Evaluation of Declarative $n$-Queens Recursion: Deductive Database Approach [Abstract] [BibTeX]	1998	Information Sciences Vol. 105, pp. 69-100	article	DOI
Abstract: Can we evaluate a logic program declaratively? That is, can a logic program be evaluated correctly and efficiently, independent of query modes and rule/predicate ordering, finding a complete set of answers, and terminating properly? the answer could be ``yes'', at least for a good subclass of logic programs, based on our investigation and experimentation using a deductive database approach. In this paper, an $n$-queens problem, a classical logic program, is used as a running example to demonstrate the methodology. Our analysis shows that binding analysis and constraint exploration are two essential issues in the realization of declarative logic programming. The limitations of our methodology are also discussed in the paper.
BibTeX: @article{Han1998, author = {J. Han and L. Liu and T. Lu}, title = {Evaluation of Declarative $n$-Queens Recursion: Deductive Database Approach}, journal = {Information Sciences}, year = {1998}, volume = {105}, pages = {69-100}, doi = {http://dx.doi.org/10.1016/S0020-0255(97)10019-6} }
Hansche, B. & Vucenic, W.	On the $n$-Queens Problem [BibTeX]	1973	Notices of the American Mathematical Society Vol. 20, pp. 568	article
BibTeX: @article{Hansche1973, author = {B. Hansche and W. Vucenic}, title = {On the $n$-Queens Problem}, journal = {Notices of the American Mathematical Society}, year = {1973}, volume = {20}, pages = {568} }
Harborth, H., Kultan, V., Nyaradyova, K. & Spendelova, Z.	Independence on Triangular Hexagon Boards [BibTeX]	2003	Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing, pp. 215-222	inproceedings
BibTeX: @inproceedings{Harborth2003, author = {H. Harborth and V. Kultan and K. Nyaradyova and Z. Spendelova}, title = {Independence on Triangular Hexagon Boards}, booktitle = {Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing}, year = {2003}, pages = {215-222} }
Hayes, P.	A Problem of Chess Queens [BibTeX]	1992	Journal of Recreational Mathematics Vol. 24, pp. 264-271	article
BibTeX: @article{Hayes1992, author = {P. Hayes}, title = {A Problem of Chess Queens}, journal = {Journal of Recreational Mathematics}, year = {1992}, volume = {24}, pages = {264-271} }
Hedayat, A.	A Complete Solution to the Existence and Nonexistence of Knut Vik Designs and Orthogonal Knut Vik Designs. [Abstract] [BibTeX]	1977	Journal of Combinatorial Theory, Series A Vol. 22, pp. 331-337	article	DOI
Abstract: Hedayat and Federer (Ann. of Statist. 3 (1975), 445–-447) proved that Knut Vik designs do not exist for all even orders. They gave a simple algorithm for the construction of such designs for all other orders, except when the order of the design is divisible by 3. The existence of Knut Vik designs of orders divisible by 3 was left unsolved by these authors. It is shown here that Knut Vik designs do not also exist for all orders divisible by 3. An easy algorithm based on a result of Euler is provided for the construction of orthogonal Knut Vik designs for all orders not divisible by 2 or 3. Therefore, we can say that Knut Vik designs and orthogonal Knut Vik designs of order $n$ exist if and only if $n$ is not divisible by 2 or 3. The results are based on the concepts of a super diagonal and parallel super diagonals in an $n n$ array, which have been introduced and studied for the first time here. Other relevant results are also given.
BibTeX: @article{Hedayat1977, author = {A. Hedayat}, title = {A Complete Solution to the Existence and Nonexistence of Knut Vik Designs and Orthogonal Knut Vik Designs.}, journal = {Journal of Combinatorial Theory, Series A}, year = {1977}, volume = {22}, pages = {331-337}, doi = {http://dx.doi.org/10.1016/0097-3165(77)90007-3} }
Heden, O.	Maximal Partial Spreads and the Modular $n$-Queen Problem III [Abstract] [BibTeX]	2002	Discrete Mathematics Vol. 243, pp. 135-150	article	DOI
Abstract: Maximal partial spreads in $PG(3,q)$, $q=p^k$, $p$ odd prime and $qgeq 7$, are constructed for any integer $n$ in the interval $(q^2+1)/2+6leq nleq (5q^2+4q-1)/8$ in the case $q+1equiv 0,pm 2,pm 4,pm 6,pm 10, 12 (mod 24)$. In all these cases, maximal partial spreads of the size $(q^2+1)/2+n$ have also been constructed for some small values of the integer $n$. These values depend on $q$ and are mainly $n=3$ and $n=4$. Combining these results with previous results of the author and with that of others we can conclude that there exist maximal partial spreads in $PG(3,q)$, $q=p^k$ where $p$ is an odd prime and $qgeq 7$, of size $n$ for any integer $n$ in the interval $(q^2+1)/2+6leq n leq q^2-q+2$.
BibTeX: @article{Heden2002, author = {O. Heden}, title = {Maximal Partial Spreads and the Modular $n$-Queen Problem III}, journal = {Discrete Mathematics}, year = {2002}, volume = {243}, pages = {135-150}, doi = {http://dx.doi.org/10.1016/S0012-365X(00)00464-7} }
Heden, O.	Maximal Partial Spreads and the Modular $n$-Queen Problem. II [Abstract] [BibTeX]	1995	Discrete Mathematics Vol. 142, pp. 97-106	article	DOI
Abstract: We prove that if $q + 1 equiv 8 or 16 (mod 24)$ then, for any integer $n$ in the interval $(q^2 + 1)/2 + 3 leq n leq (5q^2 + 4q + 7)/8$, there is a maximal partial spread of size $n$ in $PG(3, q)$.
BibTeX: @article{Heden1995, author = {O. Heden}, title = {Maximal Partial Spreads and the Modular $n$-Queen Problem. II}, journal = {Discrete Mathematics}, year = {1995}, volume = {142}, pages = {97-106}, doi = {http://dx.doi.org/10.1016/0012-365X(94)00008-7} }
Heden, O.	Maximal Partial Spreads and the Modular $n$-Queen Problem [Abstract] [BibTeX]	1993	Discrete Mathematics Vol. 120, pp. 75-91	article	DOI
Abstract: We prove that for any integer n in the interval $(5q^2+4q-1)/8leq nleq q^2+q-2$ there is a maximal partial spread of size $n$ in $PG (3, q)$ where $q$ is odd and $q geq 7$. We also prove that there are maximal partial spreads of size $(q^2+3)/2$ when $q+1,24)=2$ or $4$ and of size $(q^2+5)/2$ when $q+1,24)=4$.
BibTeX: @article{Heden1993, author = {O. Heden}, title = {Maximal Partial Spreads and the Modular $n$-Queen Problem}, journal = {Discrete Mathematics}, year = {1993}, volume = {120}, pages = {75-91}, doi = {http://dx.doi.org/10.1016/0012-365X(93)90566-C} }
Heden, O.	On the Modular $n$-Queen Problem [Abstract] [BibTeX]	1992	Discrete Mathematics Vol. 102, pp. 155-161	article	DOI
Abstract: Let $M(n)$ denote the maximum number of queens on a modular chessboard such that no two attack each other. We prove that if 4 or 6 divides $n$ then $M(n) leq n-2$ and if $n, 24) = 8$ then $M(n)geq n - 2$. We also show that $M(24) = 22$.
BibTeX: @article{Heden1992, author = {O. Heden}, title = {On the Modular $n$-Queen Problem}, journal = {Discrete Mathematics}, year = {1992}, volume = {102}, pages = {155-161}, doi = {http://dx.doi.org/10.1016/0012-365X(92)90050-P} }
Hedetniemi, S., Hedetniemi, S. & Reynolds, R.	Domination in Graphs: Advanced Topics [BibTeX]	1998		book
BibTeX: @book{Hedetniemi1998, author = {S.M. Hedetniemi and S.T. Hedetniemi and R. Reynolds}, title = {Domination in Graphs: Advanced Topics}, publisher = {Marcel Dekker, New York}, year = {1998} }
Hernández, J. & Robert, L.	Figures of Constant Width on a Chessboard [BibTeX]	2005	The American Mathematical Monthly Vol. 112(1), pp. 42-50	article	URL
BibTeX: @article{Hern'andez2005, author = {J. Hernández and L. Robert}, title = {Figures of Constant Width on a Chessboard}, journal = {The American Mathematical Monthly}, year = {2005}, volume = {112(1)}, pages = {42-50}, url = {http://www.jstor.org/stable/2690038} }
Herzberg, A. & Garner, C.	Latin Queen Squares [BibTeX]	1981	Utilitas Mathematica Vol. 20, pp. 143-154	article
BibTeX: @article{Herzberg1981, author = {A.M. Herzberg and C.W.L. Garner}, title = {Latin Queen Squares}, journal = {Utilitas Mathematica}, year = {1981}, volume = {20}, pages = {143-154} }
Hitotomatu, H. & Noshita, K.	A Technique for Implementing Backtrack Algorithms and its Application [BibTeX]	1979	Information Processing Letters Vol. 8, pp. 174-175	article	DOI
BibTeX: @article{Hitotomatu1979, author = {H. Hitotomatu and K. Noshita}, title = {A Technique for Implementing Backtrack Algorithms and its Application}, journal = {Information Processing Letters}, year = {1979}, volume = {8}, pages = {174-175}, doi = {http://dx.doi.org/10.1016/0020-0190(79)90016-4} }
Hoffman, E., Loessi, J. & Moore, R.	Constructions for the Solution of the $m$-Queens Problem [BibTeX]	1969	Mathematics Magazine Vol. 42, pp. 66-72	article	URL
BibTeX: @article{Hoffman1969, author = {E.J. Hoffman and J.C. Loessi and R.C. Moore}, title = {Constructions for the Solution of the $m$-Queens Problem}, journal = {Mathematics Magazine}, year = {1969}, volume = {42}, pages = {66-72}, url = {http://www.jstor.org/stable/2689192} }
Hollander, D.	An Unexpected Two-Dimensional Space-Group Containing Seven of the Twelve Basic Solutions to the Eight Queens Problem [BibTeX]	1973	Journal of Recreational Mathematics Vol. 6(4), pp. 287-291	article
BibTeX: @article{Hollander1973, author = {D.H. Hollander}, title = {An Unexpected Two-Dimensional Space-Group Containing Seven of the Twelve Basic Solutions to the Eight Queens Problem}, journal = {Journal of Recreational Mathematics}, year = {1973}, volume = {6(4)}, pages = {287-291} }
Homaifar, A., Turner, J. & Ali, S.	The $n$-Queens Problem and Genetic Algorithms [Abstract] [BibTeX]	1992	Proceedings IEEE Southeast Conference, Volume 1, pp. 262-267	inproceedings	DOI
Abstract: The authors determined how well the operators of genetic algorithms handled very difficult combinatorial and constraint satisfaction problems. The $n$-Queens problem is a complex combinatorial problem. Genetic algorithms are efficient and robust search algorithms that can solve the $n$-Queens problem. To derive a problem, the genetic algorithm treats the problem as an ordering or sequencing problem and blindly traverses the search space to satisfy the large number of constraints posed by the inherent complexity of the problem. Results are presented for $N < 200$.
BibTeX: @inproceedings{Homaifar1992, author = {A. Homaifar and J. Turner and S. Ali}, title = {The $n$-Queens Problem and Genetic Algorithms}, booktitle = {Proceedings IEEE Southeast Conference, Volume 1}, year = {1992}, pages = {262-267}, doi = {http://dx.doi.org/10.1109/SECON.1992.202348} }
Hsiang, J., Hsu, D. & Shieh, Y.-P.	On the Hardness of Counting Problems of Complete Mappings [Abstract] [BibTeX]	2004	Discrete Mathematics Vol. 277, pp. 87-100	article	DOI
Abstract: A complete mapping of an algebraic structure $(G,+)$ is a bijection $f(x)$ of $G$ over $G$ such that $f(x)=x+h(x)$ for some bijection $h(x)$. A question often raised is, given an algebraic structure $G$, how many complete mappings of $G$ there are. In this paper we investigate a somewhat different problem. That is, how difficult it is to count the number of complete mappings of $G$. We show that for a closed structure, the counting problem is P-complete. For a closed structure with a left-identity and left-cancellation law, the counting problem is also P-complete. For an abelian group, on the other hand, the counting problem is beyond the P-class. Furthermore, the famous counting problems of $n$-queen and toroidal $n$-queen problems are both beyond the P-class.
BibTeX: @article{Hsiang2004, author = {J. Hsiang and D.F. Hsu and Y.-P. Shieh}, title = {On the Hardness of Counting Problems of Complete Mappings}, journal = {Discrete Mathematics}, year = {2004}, volume = {277}, pages = {87-100}, doi = {http://dx.doi.org/10.1016/S0012-365X(03)00176-6} }
Hsiang, J., Shieh, Y. & Chen, Y.	The Cyclic Complete Mappings Counting Problems [BibTeX]	2002	PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002	inproceedings	URL
BibTeX: @inproceedings{Hsiang2002, author = {J. Hsiang and Y. Shieh and Y. Chen}, title = {The Cyclic Complete Mappings Counting Problems}, booktitle = {PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002}, year = {2002}, url = {http://www.arping.idv.tw/cm/index.htm} }
Hu, X., Eberhart, R. & Shi, Y.	Swarm Intelligence for Permutation Optimization: A Case Study of $n$-Queens Problem [Abstract] [BibTeX]	2003	Proceedings IEEE Swarm Intelligence Symposium (SIS'03), pp. 243-246	inproceedings	DOI
Abstract: This paper introduces a modified particle swarm optimizer which deals with permutation problems. Particles are defined as permutations of a group of unique values. Velocity updates are redefined based on the similarity of two particles. Particles change their permutations with a random rate defined by their velocities. A mutation factor is introduced to prevent the current pBest from becoming stuck at local minima. Preliminary study on the $n$-queens problem shows that the modified PSO is promising in solving constraint satisfaction problems.
BibTeX: @inproceedings{Hu2003, author = {X. Hu and R.C. Eberhart and Y. Shi}, title = {Swarm Intelligence for Permutation Optimization: A Case Study of $n$-Queens Problem}, booktitle = {Proceedings IEEE Swarm Intelligence Symposium (SIS'03)}, year = {2003}, pages = {243-246}, doi = {http://dx.doi.org/10.1109/SIS.2003.1202275} }
Huff, G.	On Pairings of the First $2n$ Natural Numbers [BibTeX]	1973	Acta Arithmetica Vol. 23, pp. 117-126	article	URL
BibTeX: @article{Huff1973, author = {G.B. Huff}, title = {On Pairings of the First $2n$ Natural Numbers}, journal = {Acta Arithmetica}, year = {1973}, volume = {23}, pages = {117-126}, url = {http://matwbn.icm.edu.pl/ksiazki/aa/aa23/aa2322.pdf} }
Hukushima, K.	Extended Ensemble Monte Carlo Approach to Hardly Relaxing Problems [Abstract] [BibTeX]	2002	Computer Physics Communications Vol. 147, pp. 77-82	article	DOI
Abstract: A set of methods based on an idea of extended ensemble has been proposed for simulating hardly relaxing systems such as spin glasses. The multicanonical method, simulated tempering and exchange Monte Carlo are typical examples of this family. We briefly review extended ensemble Monte Carlo methods, particularly focusing on the exchange Monte Carlo method. Using the method, we study the number of solutions of the $N$ queens problem which is a kind of constraint-satisfaction problem. This problem is a typical example of hardly relaxing problems because there exist numerous solutions and energy barriers between them. Our numerical result supports the conjecture that the number of solutions is proportional to $N^N$ in the large $N$ limit. We also discuss the thermodynamic properties of the $N$ queens problem at finite temperatures introduced artificially.
BibTeX: @article{Hukushima2002, author = {K. Hukushima}, title = {Extended Ensemble Monte Carlo Approach to Hardly Relaxing Problems}, journal = {Computer Physics Communications}, year = {2002}, volume = {147}, pages = {77-82}, doi = {http://dx.doi.org/10.1016/S0010-4655(02)00207-2} }
Hwang, F. & Lih, K.	Latin Squares and Superqueens [Abstract] [BibTeX]	1983	Journal of Combinatorial Theory, Series A Vol. 34, pp. 110-114	article	DOI
Abstract: Let $L$ be a Latin square of order $n$ with entries from $0, 1, n-1$. In addition, $L$ is said to have the $(n, k)$ property if, in each right or left wrap around diagonal, the number of cells with entries smaller than $k$ is exactly $k$. It is established that a necessary and sufficient condition for the existence of Latin squares having the $(n, k)$ property is that of $(2\|n Rightarrow 2\| k)$ and $(3\|n Rightarrow 3\| k)$. Also, these Latin squares are related to a problem of placing nonattacking queens on a toroidal chessboard.
BibTeX: @article{Hwang1983, author = {F.K. Hwang and K.W. Lih}, title = {Latin Squares and Superqueens}, journal = {Journal of Combinatorial Theory, Series A}, year = {1983}, volume = {34}, pages = {110-114}, doi = {http://dx.doi.org/10.1016/0097-3165(83)90048-1} }
Iyer, M. & Menon, V.	On Coloring the $nn$ Chessboard [BibTeX]	1966	The American Mathematical Monthly Vol. 73(7), pp. 721-725	article	DOI
BibTeX: @article{Iyer1966, author = {M.R. Iyer and V.V. Menon}, title = {On Coloring the $nn$ Chessboard}, journal = {The American Mathematical Monthly}, year = {1966}, volume = {73(7)}, pages = {721-725}, doi = {http://dx.doi.org/10.2307/2313979} }
Küchmann, F.	Solving the Eight Queens Problem [BibTeX]	1997	MacTech Magazine: For Macintosh Programmers & Developers Vol. 13, pp. 20-27	article
BibTeX: @article{Kuchmann1997, author = {F.C. Küchmann}, title = {Solving the Eight Queens Problem}, journal = {MacTech Magazine: For Macintosh Programmers & Developers}, year = {1997}, volume = {13}, pages = {20-27} }
Kalé, L.	An Almost Perfect Heuristic for the $N$ Nonattacking Queens Problem [Abstract] [BibTeX]	1990	Information Processing Letters Vol. 34, pp. 173-178	article	DOI
Abstract: We present a heuristic technique for finding solutions to the $N$ nonattacking queens problem that is almost perfect in the sense that it finds a first solution without any backtracks in most cases. In addition to previously known variable-ordering heuristics and their extensions, it uses a value-ordering heuristic, which contributes dramatically to its success. Using these heuristics, solutions have been found for all values of $N$ between 4 and 1000.
BibTeX: @article{Kale1990, author = {L.V. Kalé}, title = {An Almost Perfect Heuristic for the $N$ Nonattacking Queens Problem}, journal = {Information Processing Letters}, year = {1990}, volume = {34}, pages = {173-178}, doi = {http://dx.doi.org/10.1016/0020-0190(90)90156-R} }
Katzman, M.	Counting Monomials [Abstract] [BibTeX]	2005	Journal of Algebraic Combinatorics Vol. 22, pp. 331-341	article	DOI
Abstract: This paper presents two enumeration techniques based on Hilbert functions. The paper illustrates these techniques by solving two chessboard problems.
BibTeX: @article{Katzman2005, author = {M. Katzman}, title = {Counting Monomials}, journal = {Journal of Algebraic Combinatorics}, year = {2005}, volume = {22}, pages = {331-341}, doi = {http://dx.doi.org/10.1007/s10801-005-4531-6} }
Kazarin, L., Kopylov, G. & Timofeev, E.	The Chromatic Number of a Special Class of Graphs [BibTeX]	1975	Vestnik Jaroslav Univ. Vyp. Vol. 9, pp. 37-46	article
BibTeX: @article{Kazarin1975, author = {L.S. Kazarin and G.N. Kopylov and E.A. Timofeev}, title = {The Chromatic Number of a Special Class of Graphs}, journal = {Vestnik Jaroslav Univ. Vyp.}, year = {1975}, volume = {9}, pages = {37-46} }
Kearse, M. & Gibbons, P.	A New Lower Bound on Upper Irredundance in the Queens' Graph [Abstract] [BibTeX]	2002	Discrete Mathematics Vol. 256, pp. 225-242	article	DOI
Abstract: The queens’ graph $Q_n$ has the squares of the $nn$ chessboard as its vertices, with two squares adjacent if they are in the same row, column, or diagonal. An irredundant set of queens has the property that each queen in the set attacks at least one square which is attacked by no other queen. $IR(Q_n)$ is the cardinality of the largest irredundant set of vertices in $Q_n$. Currently the best lower bound for $IR(Q_n)$ is $IR(Q_n)geq 2.5n-O(1)$, while the best upper bound is $IR(Q_n)leq lfloor 6n + 6 -8n +n + 1 for $ngeq 6$. Here the lower bound is improved to $IR(Q_n)geq 6n-O(n^2/3)$. In particular, it is shown for even $kgeq 6$ that $IR(Q_k^3)geq 6k^3-29k^2-O(k)$.
BibTeX: @article{Kearse2002, author = {M.D. Kearse and P.B. Gibbons}, title = {A New Lower Bound on Upper Irredundance in the Queens' Graph}, journal = {Discrete Mathematics}, year = {2002}, volume = {256}, pages = {225-242}, doi = {http://dx.doi.org/10.1016/S0012-365X(01)00467-8} }
Keating, J.	Hopfield Networks, Neural Data Structures and the Nine Flies Problem: Neural Network Programming Projects for Undergraduates [Abstract] [BibTeX]	1993	ACM SIGCSE Bulletin Vol. 25, pp. 33-37,40,60	article	DOI
Abstract: This paper describes two neural network programming projects suitable for undergraduate students who have already completed introductory courses in Programming and Data Structures. It briefly outlines the structure and operation of Hopfield Networks from a data structure stand-point and demonstrates how these type of neural networks may be used to solve interesting problems like Perelman's Nine Flies Problem. Although the Hopfield model is well defined mathematically, students do not have to be very familiar with the mathematics of the model in order to use it to solve problems. Students are actively encouraged to design modifications to their implementations in order to obtain faster or more accurate solutions. Additionally, students are also expected to compare the neural network's performance with traditional approaches, in order that they may appreciate the subtleties of both approaches. Sample results are provided from projects which have been completed during the last three-year period.
BibTeX: @article{Keating1993, author = {J.G. Keating}, title = {Hopfield Networks, Neural Data Structures and the Nine Flies Problem: Neural Network Programming Projects for Undergraduates}, journal = {ACM SIGCSE Bulletin}, year = {1993}, volume = {25}, pages = {33-37,40,60}, doi = {http://dx.doi.org/10.1145/164205.164224} }
Khan, S.	Modular $n$-Queen [BibTeX]	2003	Geombinatorics Vol. 12(4), pp. 217-221	article
BibTeX: @article{Khan2003, author = {S.U. Khan}, title = {Modular $n$-Queen}, journal = {Geombinatorics}, year = {2003}, volume = {12(4)}, pages = {217-221} }
Kim, S.	Problem 811 [BibTeX]	1979	Journal of Recreational Mathematics Vol. 12(1), pp. fply53	article
BibTeX: @article{Kim1979, author = {S. Kim}, title = {Problem 811}, journal = {Journal of Recreational Mathematics}, year = {1979}, volume = {12(1)}, pages = {fply53} }
Kise, K., Katagiri, T., Honda, H. & Yuba, T.	Solving the $n$-Queens Problem with a PG Cluster [Abstract] [BibTeX]	2004	IEICE Transactions on Information and Systems, Pt.1 (Japanese Edition)	article
Abstract: The $n$-Queens problem is to place N Queens of which no Queen can attack each other on an $nn$ chess board. This paper presents a sequential program which attains from 11% to 18% of improvement in the speed as compared with a present program. And by parallelizing using MPI and calculating using PC clusters, the number of solutions for the 24-Queens problem is calculated for the first time in the world. Main knowledge of this experience is as follows. 1) From 11% to 18% speed-up in a sequential program is attained by the optimization of memory reference and control structure, 2) A master-worker scheme is efffective in the parallelization, 3) The hyper-threading technology of Pentium4 processor attains 30% speed-up, 4) In the solution of a real problem, it is necessary to consider the efficiently as the whole system.
BibTeX: @article{Kise2004, author = {K. Kise and T. Katagiri and H. Honda and T. Yuba}, title = {Solving the $n$-Queens Problem with a PG Cluster}, journal = {IEICE Transactions on Information and Systems, Pt.1 (Japanese Edition)}, year = {2004} }
Kise, K., Katagiri, T., Honda, H. & Yuba, T.	Solving the 24-Queens Problem Using MPI on a PC Cluster [BibTeX]	2004	(UEC-IS-2004-6)	techreport
BibTeX: @techreport{Kise2004a, author = {K. Kise and T. Katagiri and H. Honda and T. Yuba}, title = {Solving the 24-Queens Problem Using MPI on a PC Cluster}, year = {2004}, number = {UEC-IS-2004-6} }
Klarner, D.	Queen Squares [BibTeX]	1979	Journal of Recreational Mathematics Vol. 12(3), pp. 177-178	article
BibTeX: @article{Klarner1979, author = {D.A. Klarner}, title = {Queen Squares}, journal = {Journal of Recreational Mathematics}, year = {1979}, volume = {12(3)}, pages = {177-178} }
Klarner, D.	The Problem of Reflecting Queens [BibTeX]	1967	American Mathematical Monthly Vol. 74(8), pp. 953-955	article	DOI
BibTeX: @article{Klarner1967, author = {D.A. Klarner}, title = {The Problem of Reflecting Queens}, journal = {American Mathematical Monthly}, year = {1967}, volume = {74(8)}, pages = {953-955}, doi = {http://dx.doi.org/10.2307/2315273} }
Kløve, T.	The Modular $n$-Queen Problem II [Abstract] [BibTeX]	1981	Discrete Mathematics Vol. 36, pp. 33-48	article	DOI
Abstract: We study classes of solutions to the modular $n$-queen problem. The main part of the paper is concerned with symmetric solutions (solutions invariant under 90 rotation). In the last section we study maximal partial solutions for those values of $n$ for which no solutions exist.
BibTeX: @article{Klove1981, author = {T. Kløve}, title = {The Modular $n$-Queen Problem II}, journal = {Discrete Mathematics}, year = {1981}, volume = {36}, pages = {33-48}, doi = {http://dx.doi.org/10.1016/0012-365X(81)90171-0} }
Kløve, T.	The Modular $n$-Queen Problem [Abstract] [BibTeX]	1977	Discrete Mathematics Vol. 19, pp. 289-291	article	DOI
Abstract: We show that the modular $n$-queen problem has a solution if and only if $n, 6) = 1$. We give a class of solutions for all these $n$.
BibTeX: @article{Klove1977, author = {T. Kløve}, title = {The Modular $n$-Queen Problem}, journal = {Discrete Mathematics}, year = {1977}, volume = {19}, pages = {289-291}, doi = {http://dx.doi.org/10.1016/0012-365X(77)90110-8} }
Knuth, D.	Dancing Links [BibTeX]	2000	Millennial Perspectives in Computer Science, pp. 187-214	inproceedings	URL
BibTeX: @inproceedings{Knuth2000, author = {D.E. Knuth}, title = {Dancing Links}, booktitle = {Millennial Perspectives in Computer Science}, publisher = {Palgrave}, year = {2000}, pages = {187-214}, url = {http://www-cs-faculty.stanford.edu/~knuth/papers/dancing-color.ps.gz} }
Koshy, T.	Elementary Number Theory with Applications [BibTeX]	2001		book
BibTeX: @book{Koshy2001, author = {T. Koshy}, title = {Elementary Number Theory with Applications}, publisher = {Harcourt Academic Press, San Diego}, year = {2001} }
Kotěšovec, V.	Mezi šachovnic a počtačem [BibTeX]	1996		misc	URL
BibTeX: @misc{Kotesovec1996, author = {V. Kotěšovec}, title = {Mezi šachovnic a počtačem}, year = {1996}, url = {http://web.iol.cz/vaclav.kotesovec/} }
Kovalenko, I.	Upper Bound on the Number of Complete Maps [BibTeX]	1996	Cybernetics and System Analysis Vol. 32, pp. 65-68	article	DOI
BibTeX: @article{Kovalenko1996, author = {I.N. Kovalenko}, title = {Upper Bound on the Number of Complete Maps}, journal = {Cybernetics and System Analysis}, year = {1996}, volume = {32}, pages = {65-68}, doi = {http://dx.doi.org/10.1007/BF02366583} }
Kraitchik, M.	Mathematical Recreations [BibTeX]	1942		book
BibTeX: @book{Kraitchik1942, author = {M. Kraitchik}, title = {Mathematical Recreations}, publisher = {W.W. Norton, New York}, year = {1942} }
Kreuzer, M. & Robbiano, L.	Computational Commutative Algebra. 2 [BibTeX]	2005		book
BibTeX: @book{Kreuzer2005, author = {M. Kreuzer and L. Robbiano}, title = {Computational Commutative Algebra. 2}, publisher = {Springer-Verlag, Berlin}, year = {2005} }
Kunde, M. & Gürtzig, K.	Efficient Sorting and Routing on Reconfigurable Meshes Using Restricted Bus Length [Abstract] [BibTeX]	1997	Proceedings of the 11th International Parallel Processing Symposium (IPPS1997), pp. 713-720	inproceedings	DOI
Abstract: Sorting and balanced routing problems for synchronous mesh-like processor networks with reconfigurable buses are considered. Induced by the argument that broadcasting along buses of arbitrary length within unit time seems rather non-realistic, we consider basic problems on reconfigurable meshes that can be solved efficiently even with restricted bus length.It is shown that on $r$-dimensional reconfigurable meshes of side length n with bus length bounded to a constant $l$ the $h-h$ sorting and routing problem can be solved within $hn+o(hrn)$ steps in any case and in $hn/2+o(hrn)$ steps with high probability, provided that $hl geq 4r$. This result is due to a data concentration method that is explained in the paper and it will hold even for certain very light loadings, i.e. with significantly less than one elements per processor on average. Extensions to two-dimensional reconfigurable meshes with diagonal links are considered.
BibTeX: @inproceedings{Kunde1997, author = {M. Kunde and K. Gürtzig}, title = {Efficient Sorting and Routing on Reconfigurable Meshes Using Restricted Bus Length}, booktitle = {Proceedings of the 11th International Parallel Processing Symposium (IPPS1997)}, year = {1997}, pages = {713-720}, doi = {http://dx.doi.org/10.1109/IPPS.1997.580985} }
Landau, E.	Über das Achtdamenproblem und seine Verallgemeinerung [BibTeX]	1896	Naturwiss. Wochenschrift Vol. 11, pp. 367-371	article
BibTeX: @article{Landau1896, author = {E. Landau}, title = {Über das Achtdamenproblem und seine Verallgemeinerung}, journal = {Naturwiss. Wochenschrift}, year = {1896}, volume = {11}, pages = {367-371} }
Laparewicz, A.	Królowe na Szachnownicy, Wektor [BibTeX]	1912	Mathematische-Physikalische Zeitschrift Vol. 1(6), pp. 326-336	article
BibTeX: @article{Laparewicz1912, author = {A. Laparewicz}, title = {Królowe na Szachnownicy, Wektor}, journal = {Mathematische-Physikalische Zeitschrift}, year = {1912}, volume = {1(6)}, pages = {326-336} }
Larson, L.	A Theorem About Primes Proved on a Chessboard [BibTeX]	1977	Mathematics Magazine Vol. 50, pp. 69-74	article	URL
BibTeX: @article{Larson1977, author = {L.C. Larson}, title = {A Theorem About Primes Proved on a Chessboard}, journal = {Mathematics Magazine}, year = {1977}, volume = {50}, pages = {69-74}, url = {http://www.jstor.org/stable/2689726} }
Laskar, R., McRae, A. & Wallis, C.	Domination in Triangulated Chessboard Graphs [BibTeX]	2003	Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing, pp. 107-123	inproceedings
BibTeX: @inproceedings{Laskar2003, author = {R. Laskar and A. McRae and C. Wallis}, title = {Domination in Triangulated Chessboard Graphs}, booktitle = {Proceedings of the Thirty-Fourth Southeastern International Conference on Combinatorics, Graph Theory and Computing}, year = {2003}, pages = {107-123} }
Laskar, R. & Wallis, C.	Chessboard Graphs, Related Designs, and Domination Parameters [Abstract] [BibTeX]	1999	Journal of Statistical Planning and Inference Vol. 76, pp. 285-294	article	DOI
Abstract: The graph-theoretic study of combinatorial chessboard problems can be extended to the study of line graphs of graphs of combinatorial designs. In particular, the determination of optimal placements of rooks on a chessboard corresponds to the determination of domination parameters of graphs of block designs. The determination of one such parameter, the independence number, is shown to follow directly from classical results in design theory. Additionally, the domination number of graphs of finite projective planes is discussed.
BibTeX: @article{Laskar1999, author = {R. Laskar and C. Wallis}, title = {Chessboard Graphs, Related Designs, and Domination Parameters}, journal = {Journal of Statistical Planning and Inference}, year = {1999}, volume = {76}, pages = {285-294}, doi = {http://dx.doi.org/10.1016/S0378-3758(98)00132-3} }
Le, M., Li, W. & Wang, E.	A Generalization of the $n$-Queen Problem [BibTeX]	1990	Journal of Systems Science and Mathematical Sciences Vol. 3(2), pp. 183-192	article
BibTeX: @article{Le1990, author = {M.H. Le and W. Li and E.T. Wang}, title = {A Generalization of the $n$-Queen Problem}, journal = {Journal of Systems Science and Mathematical Sciences}, year = {1990}, volume = {3(2)}, pages = {183-192} }
Le, M., Li, W. & Wang, E.	A Generalization of the $n$-Queen Problem [BibTeX]	1989	Journal of Systems Science and Mathematical Sciences Vol. 9(2), pp. 158-168	article
BibTeX: @article{Le1989, author = {M.H. Le and W. Li and E.T. Wang}, title = {A Generalization of the $n$-Queen Problem}, journal = {Journal of Systems Science and Mathematical Sciences}, year = {1989}, volume = {9(2)}, pages = {158-168} }
Le, T.-N. & Pham, C.-K.	A New $N$-Parallel Updating Method of the Hopfield-Type Neural Network for $n$-Queens Problem [Abstract] [BibTeX]	2005	Proceedings IEEE International Joint Conference on Neural Networks (IJCNN'05), pp. 788-791	inproceedings	URL
Abstract: In the previous $N$-parallel updating methods of the Hopfield-type neural network for $n$-Queens problem, $nn$ neurons have been grouped into $N$ groups. Each group composed of $N$ neurons which are located in a same horizontal line (column) or in a same diagonal line. However, these method did not give convergence results of 100% in all size of $N$. Also, they required a large convergence time steps. In our work, we propose a new $N$-parallel updating method of the Hopfield-type neural network for $n$-Queens problem, in which, a new grouping method for $N$ neurons composed in the same group has been adopted. As a result, simulation results of the proposed method show a best performance than the previous generally.
BibTeX: @inproceedings{Le2005, author = {T.-N. Le and C.-K. Pham}, title = {A New $N$-Parallel Updating Method of the Hopfield-Type Neural Network for $n$-Queens Problem}, booktitle = {Proceedings IEEE International Joint Conference on Neural Networks (IJCNN'05)}, year = {2005}, pages = {788-791}, url = {http://ieeexplore.ieee.org/servlet/opac?punumber=10421} }
Letavec, C. & Ruggiero, J.	The $n$-Queens Problem [BibTeX]	2002	INFORMS Transactions on Education Vol. 2	article	URL
BibTeX: @article{Letavec2002, author = {C. Letavec and J. Ruggiero}, title = {The $n$-Queens Problem}, journal = {INFORMS Transactions on Education}, year = {2002}, volume = {2}, url = {http://archive.ite.journal.informs.org/Vol2No3/LetavecRuggiero/LetavecRuggiero.pdf} }
Li, P., Guangxi, Z. & Xiao, L.	The Low-Density Parity-Check Codes Based on the $n$-Queen Problem [Abstract] [BibTeX]	2004	NRBC: Proceedings of the 2004 ACM Workshop on Next-Generation Residential Broadband Challenges, pp. 37-41	inproceedings	DOI
Abstract: This paper presents a new family of low-density parity-check (LDPC) code, the sparse parity-check matrix of which is constructed by self-defining non-diagonal identity sub-matrix that is a solution of the ``$n$n-queen problem". So this sub-matrix is called the $Q$-matrix and these LDPC codes are called the $Q$-matrixes LDPC codes. The $Q$-matrixes LDPC codes are good or very good codes with iterative decoding and their Tanner graphs are free of 4-lines cycle. Furthermore, they can be created in cycle form. Their encoding can be achieved in linear time. Especially, their code length and code rate can be flexible and quickly adjusted according to the practical situation, and the performance of high rate is also very good. The other unique excellence is that the large sparse parity-check matrixes of long $Q$-matrixes LDPC codes require very small storage space. The result of this paper is very significant not only for designing low complexity encoder, improving performance and reducing the complexity of the sum-product iterative decoding algorithm, but also for building practice system of encodable and decodable LDPC code.
BibTeX: @inproceedings{Li2004, author = {P. Li and Z. Guangxi and L. Xiao}, title = {The Low-Density Parity-Check Codes Based on the $n$-Queen Problem}, booktitle = {NRBC: Proceedings of the 2004 ACM Workshop on Next-Generation Residential Broadband Challenges}, publisher = {ACM Press}, year = {2004}, pages = {37-41}, doi = {http://dx.doi.org/10.1145/1026763.1026771} }
Lionnet, F.	Question 963 [BibTeX]	1869	Nouvelles Annales de Mathématiques Vol. 28, pp. 560	article
BibTeX: @article{Lionnet1869, author = {F.J.E. Lionnet}, title = {Question 963}, journal = {Nouvelles Annales de Mathématiques}, year = {1869}, volume = {28}, pages = {560} }
Lucas, E.	Récréations Mathématiques [BibTeX]	1973		book
BibTeX: @book{Lucas1973, author = {E. Lucas}, title = {Récréations Mathématiques}, publisher = {Librairie Scientifique et Technique Albert Blanchard, Paris}, year = {1973}, edition = {2nd (nouveau tirage)} }
Lucas, E.	Question 123 [BibTeX]	1894	L'Intermédiaire des Mathématiciens Vol. 11, pp. 67	article
BibTeX: @article{Lucas1894, author = {E. Lucas}, title = {Question 123}, journal = {L'Intermédiaire des Mathématiciens}, year = {1894}, volume = {11}, pages = {67} }
Madachy, J.	Mathematics on Vacation [BibTeX]	1966	, pp. 34-36	book
BibTeX: @book{Madachy1966, author = {J.S. Madachy}, title = {Mathematics on Vacation}, publisher = {Thomas Nelson and Sons Ltd.}, year = {1966}, pages = {34-36} }
Mandziuk, J.	Solving the $n$-Queens Problem with a Binary Hopfield-Type Network. Synchronous and Asynchronous Model [Abstract] [BibTeX]	1995	Biological Cybernetics Vol. 72, pp. 439-446	article	DOI
Abstract: The application of a discrete Hopfield-type neural network to solving the NP-Hard optimization problem --- the $N$-Queens Problem (NQP) --- is presented. The applied network is binary, and at every moment each neuron potential is equal to either 0 or 1. The network can be implemented in the asynchronous mode as well as in the synchronous one with n parallel running processors. In both cases the convergence rate is up to 100 and the experimental estimate of the average computational complexity is polynomial. Based on the computer simulation results and the theoretical analysis, the proper network parameters are established. The behaviour of the network is explained.
BibTeX: @article{Mandziuk1995, author = {J. Mandziuk}, title = {Solving the $n$-Queens Problem with a Binary Hopfield-Type Network. Synchronous and Asynchronous Model}, journal = {Biological Cybernetics}, year = {1995}, volume = {72}, pages = {439-446}, doi = {http://dx.doi.org/10.1007/BF00201419} }
Mandziuk, J. & Macukow, B.	A Neural Network Designed to Solve the $n$-Queens Problem [Abstract] [BibTeX]	1992	Biological Cybernetics Vol. 66, pp. 375-379	article	DOI
Abstract: In this paper we discuss the Hopfield neural network designed to solve the $N$-Queens Problem (NQP). Our network exhibits good performance in escaping from local minima of energy surface of the problem. Only in approximately 1% of trials it settles in a false stable state (local minimum of energy). Extenive simulations indicate that the network is efficient and less sensitive to changes of its initial energy (potentials of neurons). Two strategies employed to achieve the solution and results of computer simulation are presented. Some theoretical remarks about convergence of the network are added.
BibTeX: @article{Mandziuk1992, author = {J. Mandziuk and B. Macukow}, title = {A Neural Network Designed to Solve the $n$-Queens Problem}, journal = {Biological Cybernetics}, year = {1992}, volume = {66}, pages = {375-379}, doi = {http://dx.doi.org/10.1007/BF00203674} }
MathWorld	Queens Problem [BibTeX]	2009		misc	URL
BibTeX: @misc{MathWorld, author = {MathWorld}, title = {Queens Problem}, year = {2009}, url = {http://mathworld.wolfram.com/QueensProblem.html} }
McCarty, C.	Queen Squares [BibTeX]	1978	The American Mathematical Monthly Vol. 85(7), pp. 578-580	article	DOI
BibTeX: @article{McCarty1978, author = {C.P. McCarty}, title = {Queen Squares}, journal = {The American Mathematical Monthly}, year = {1978}, volume = {85(7)}, pages = {578-580}, doi = {http://dx.doi.org/10.2307/2320871} }
McKay, B., McLeod, J. & Wanless, I.	The Number of Transversals in a Latin Square [Abstract] [BibTeX]	2006	Designs, Codes and Cryptography Vol. 40, pp. 269-284	article	DOI
Abstract: A Latin Square of order $n$ is an $nn$ array of $n$ symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of $n$ entries, one selected from each row and each column of a Latin Square of order $n$ such that no two entries contain the same symbol. Define $T(n)$ to be the maximum number of transversals over all Latin squares of order $n$. We show that $b^n leq T(n) leq c^nnn!$ for $n geq 5$, where $b approx 1.719$ and $c approx 0.614$. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an $nn$ toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin Squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.
BibTeX: @article{McKay2006, author = {B.D. McKay and J.C. McLeod and I.M. Wanless}, title = {The Number of Transversals in a Latin Square}, journal = {Designs, Codes and Cryptography}, year = {2006}, volume = {40}, pages = {269-284}, doi = {http://dx.doi.org/10.1007/s10623-006-0012-8} }
Menon, V.	Problem E1782: Coloring a Chessboard [BibTeX]	1965	The American Mathematical Monthly Vol. 72(4), pp. 421	article	DOI
BibTeX: @article{Menon1965, author = {V.V. Menon}, title = {Problem E1782: Coloring a Chessboard}, journal = {The American Mathematical Monthly}, year = {1965}, volume = {72(4)}, pages = {421}, doi = {http://dx.doi.org/10.2307/2313512} }
Menon, V. & Goldberg, M.	Problem E1782: Coloring a Chessboard [BibTeX]	1966	The American Mathematical Monthly Vol. 73(6), pp. 670-671	article	DOI
BibTeX: @article{MenonGoldberg1966, author = {V.V. Menon and M. Goldberg}, title = {Problem E1782: Coloring a Chessboard}, journal = {The American Mathematical Monthly}, year = {1966}, volume = {73(6)}, pages = {670-671}, doi = {http://dx.doi.org/10.2307/2314824} }
Minton, S., Johnston, M., Philips, A. & Laird, P.	Minimizing Conflicts: A Heuristic Repair Method for Constraint Satisfaction and Scheduling Problems [Abstract] [BibTeX]	1992	Artificial Intelligence Vol. 58, pp. 161-205	article	DOI
Abstract: The paper describes a simple heuristic approach to solving large-scale constraint satisfaction and scheduling problems. In this approach one starts with an inconsistent assignment for a set of variables and searches through the space of possible repairs. The search can be guided by a value-ordering heuristic, the min-conflicts heuristic, that attempts to minimize the number of constraint violations after each step. The heuristic can be used with a variety of different search strategies. We demonstrate empirically that on the $n$-queens problem, a technique based on this approach performs orders of magnitude better than traditional backtracking techniques. We also describe a scheduling application where the approach has been used successfully. A theoretical analysis is presented both to explain why this method works well on certain types of problems and to predict when it is likely to be most effective.
BibTeX: @article{Minton1992, author = {S. Minton and M.D. Johnston and A.B. Philips and P. Laird}, title = {Minimizing Conflicts: A Heuristic Repair Method for Constraint Satisfaction and Scheduling Problems}, journal = {Artificial Intelligence}, year = {1992}, volume = {58}, pages = {161-205}, doi = {http://dx.doi.org/10.1016/0004-3702(92)90007-K} }
Miyamoto, K. & Nakajima, H.	Solving the $n$-Queens Problem on the Torus Using a Continuous-Dynamical-System Model of a Complex-Valued Neural Network of Phasor Type [Abstract] [BibTeX]	2006	(106)	techreport
Abstract: A method of solving the $n$-Queens problem on the Torus based on a complex-valued neural network of phasor type, which has its state variables on the unit circle in the complex plane, is considered. First, the positions of Queens on the chessboard are represented by the states of $N$ neurons, and a rule of updating the states are defined as a continuous dynamical system that minimizes an energy function of the states of neurons. To confirm the validity of this method, the stability of the solutions and the geometrical structure of the solution space are analyzed. The result of the analysis is investigated by numerical experiments, and it is found that the problem is solved well when $N$ is 5 and 7.
BibTeX: @techreport{Miyamoto2006, author = {K. Miyamoto and H. Nakajima}, title = {Solving the $n$-Queens Problem on the Torus Using a Continuous-Dynamical-System Model of a Complex-Valued Neural Network of Phasor Type}, year = {2006}, number = {106} }
Monsky, P.	Problem E3162: Superqueens [BibTeX]	1989	The American Mathematical Monthly Vol. 96(3), pp. 258-259	article	DOI
BibTeX: @article{Monsky1989, author = {P. Monsky}, title = {Problem E3162: Superqueens}, journal = {The American Mathematical Monthly}, year = {1989}, volume = {96(3)}, pages = {258-259}, doi = {http://dx.doi.org/10.2307/2325220} }
Monsky, P.	Problem E3162: Superqueens [BibTeX]	1986	The American Mathematical Monthly Vol. 93(7), pp. 566	article	DOI
BibTeX: @article{Monsky1986, author = {P. Monsky}, title = {Problem E3162: Superqueens}, journal = {The American Mathematical Monthly}, year = {1986}, volume = {93(7)}, pages = {566}, doi = {http://dx.doi.org/10.2307/2323039} }
Monsky, P.	Problem E2698: Superimposable Solutions [BibTeX]	1978	The American Mathematical Monthly Vol. 85(2), pp. 116-117	article	DOI
BibTeX: @article{Monsky1978, author = {P. Monsky}, title = {Problem E2698: Superimposable Solutions}, journal = {The American Mathematical Monthly}, year = {1978}, volume = {85(2)}, pages = {116-117}, doi = {http://dx.doi.org/10.2307/2321794} }
Monsky, P. & Goldstein, R.	Problem E2698: Toroidal $n$-Queens problem [BibTeX]	1979	The American Mathematical Monthly Vol. 86(4), pp. 309-310	article	URL
BibTeX: @article{Monsky1979, author = {P. Monsky and R.Z. Goldstein}, title = {Problem E2698: Toroidal $n$-Queens problem}, journal = {The American Mathematical Monthly}, year = {1979}, volume = {86(4)}, pages = {309-310}, url = {http://www.jstor.org/stable/2320763} }
Morris, P.	On the Density of Solutions in Equilibrium Points for the Queens Problem [BibTeX]	1992	Proceedings AAAI Conference on Artificial Intelligence AAAI-92	inproceedings	URL
BibTeX: @inproceedings{Morris1992, author = {P. Morris}, title = {On the Density of Solutions in Equilibrium Points for the Queens Problem}, booktitle = {Proceedings AAAI Conference on Artificial Intelligence AAAI-92}, year = {1992}, url = {www.aaai.org/Papers/AAAI/1992/AAAI92-066.pdf} }
Nadel, B.	Representation Selection for Constraint Satisfaction: A Case Study Using $n$-Queens [Abstract] [BibTeX]	1990	IEEE Expert Vol. 5, pp. 16-23	article	DOI
Abstract: Representation selection for a constraint satisfaction problem (CSP) is addressed. The CSP problem class is introduced and the classic $n$-Queens problem is used to show that many different CSP representations may exist for a given real-world problem. The complexities of solving these alternative representations are compared empirically and theoretically. The good agreement found is due to two key features of the analytic results, their generality and their precision (or instance specificity), which are also discussed. The $n$-Queens problem is used only to provide a convenient case study; the approach to CSP representation selection applies to arbitrary problems that can be formulated in terms of CSP and, when the corresponding mathematical results are available, should also be readily applicable when selecting representations in domains other than CSP
BibTeX: @article{Nadel1990, author = {B.A. Nadel}, title = {Representation Selection for Constraint Satisfaction: A Case Study Using $n$-Queens}, journal = {IEEE Expert}, year = {1990}, volume = {5}, pages = {16-23}, doi = {http://dx.doi.org/10.1109/64.54670} }
Nakaguchi, T., Jin'no, K. & Tanaka, M.	Theoretical Analysis of Hysteresis Neural Network solving $n$-Queens Problems [Abstract] [BibTeX]	1999	Proceedings IEEE International Symposium on Circuits and Systems (ISCAS'99), pp. 555-558	inproceedings	DOI
Abstract: We propose a hysteresis neural network system solving NP-hard optimization problems, the $N$-Queens Problem. The continuous system with binary outputs searches a solution of the problem without energy function. The output vector corresponds to a complete solution when the output vector becomes stable. That is, this system does never become stable without satisfying the constraints of the problem. Through it is very hard to remove limit cycles completely from this system, we can propose a new method to reduce the possibility of limit cycle by controlling time constants.
BibTeX: @inproceedings{Nakaguchi1999, author = {Nakaguchi, T. and Jin'no, K. and Tanaka, M.}, title = {Theoretical Analysis of Hysteresis Neural Network solving $n$-Queens Problems}, booktitle = {Proceedings IEEE International Symposium on Circuits and Systems (ISCAS'99)}, year = {1999}, pages = {555-558}, doi = {http://dx.doi.org/10.1109/ISCAS.1999.777632} }
Nauck, F.	Brief Wechseln mit Allen für Alle [BibTeX]	1850	Leipziger Illustrierte Zeitung Vol. 377, pp. 182	article
BibTeX: @article{Nauck1850, author = {F. Nauck}, title = {Brief Wechseln mit Allen für Alle}, journal = {Leipziger Illustrierte Zeitung}, year = {1850}, volume = {377}, pages = {182} }
Naur, P.	An experiment on Program Development [Abstract] [BibTeX]	1972	BIT Vol. 12, pp. 347-365	article	DOI
Abstract: As a contribution to programming methodology, the paper contains a detailed, step-by-step account of the considerations leading to a program for solving the 8-queens problem. The experience is related to the method of stepwise refinement and to general problem solving techniques.
BibTeX: @article{Naur1972, author = {P. Naur}, title = {An experiment on Program Development}, journal = {BIT}, year = {1972}, volume = {12}, pages = {347-365}, doi = {http://dx.doi.org/10.1007/BF01932307} }
Netto, E.	Lehrbuch der Combinatorik [BibTeX]	1901		book
BibTeX: @book{Netto1901, author = {E. Netto}, title = {Lehrbuch der Combinatorik}, publisher = {B.G. Teubner, Leipzig}, year = {1901} }
Nivasch, G. & Lev, E.	Non-Attacking Queens on a Triangle [BibTeX]	2005	Mathematics Magazine Vol. 78, pp. 399-403	article	URL
BibTeX: @article{Nivasch2005, author = {G. Nivasch and E. Lev}, title = {Non-Attacking Queens on a Triangle}, journal = {Mathematics Magazine}, year = {2005}, volume = {78}, pages = {399-403}, url = {http://www.jstor.org/stable/30044202} }
Noguchi, W. & Pham, C.-K.	A Proposal to Solve $n$-Queens Problems Using Maximum Neuron Model with A Modified Hill-Climbing Term [Abstract] [BibTeX]	2006	Proceedings International Joint Conference on Neural Networks (IJCNN'06), pp. 2679-2682	inproceedings	DOI
Abstract: An effective solving method with a modified hill-climbing term which is applied to a maximum neuron model for the $N$-Queens problems is proposed. In which, a first model using a gradient ascent learning for determining A and B coefficients, a second model using fixed A and B coefficients which are determined by an upper bound of an input value to a neuron, and a third model using modified initial values which applied to the second model, have been adopted. As a result, calculation times are reduced when compared with the previous methods.
BibTeX: @inproceedings{Noguchi2006, author = {W. Noguchi and C.-K. Pham}, title = {A Proposal to Solve $n$-Queens Problems Using Maximum Neuron Model with A Modified Hill-Climbing Term}, booktitle = {Proceedings International Joint Conference on Neural Networks (IJCNN'06)}, year = {2006}, pages = {2679-2682}, doi = {http://dx.doi.org/10.1109/IJCNN.2006.247149} }
Noon, H.	Surreal Numbers and the $n$-Queens Game [BibTeX]	2002	School: Bennington College, Bennington, Vermont, US	mastersthesis	URL
BibTeX: @mastersthesis{Noon2002, author = {H. Noon}, title = {Surreal Numbers and the $n$-Queens Game}, school = {Bennington College, Bennington, Vermont, US}, year = {2002}, url = {http://www.liacs.nl/home/kosters/nqueens/papers/noon2002.pdf} }
Noon, H. & Van Brummelen, G.	The Non-Attacking Queens Game [Abstract] [BibTeX]	2006	College Mathematics Journal Vol. 37, pp. 223-227	article	URL
Abstract: Gauss found a solution to the problem (first posed by Max Bezzel in 1848) of placing $n$ queens on an $nn$ chessboard so that no queen is attacked by another. The $n$alfaro-queens game considered here is this: Two players alternately place queens on a board so that no two attack one another, and the winner is the player who places a queen so that all squares are attacked.
BibTeX: @article{Noon2006, author = {H. Noon and G. Van Brummelen}, title = {The Non-Attacking Queens Game}, journal = {College Mathematics Journal}, year = {2006}, volume = {37}, pages = {223-227}, url = {http://www.jstor.org/stable/27646335} }
Nudelman, S.	The Modular $n$-Queens Problem in Higher Dimensions [Abstract] [BibTeX]	1995	Discrete Mathematics Vol. 146, pp. 159-167	article	DOI
Abstract: Let $M(n, d)$ denote the maximum number of queens on a $d$-dimensional modular chessboard such that no two attack each other. We show that if $n, (2d - 1)!) = 1$ then $M (n, d) = n$. We also prove that if $n, (2d - 1)!) > 1$ then there are no complete linear solutions, and if $n, (2d - 1)!) > 1$ then $M (n, d) < n$. Moreover, if $n leq 2^d - 1$ we show $M (n, d) = 1$.
BibTeX: @article{Nudelman1995, author = {S.P. Nudelman}, title = {The Modular $n$-Queens Problem in Higher Dimensions}, journal = {Discrete Mathematics}, year = {1995}, volume = {146}, pages = {159-167}, doi = {http://dx.doi.org/10.1016/0012-365X(94)00161-5} }
Oestergård, P. & Weakley, W.	Values of Domination Numbers of the Queen's Graph [BibTeX]	2001	The Electronic Journal of Combinatorics Vol. 8(1)(R29), pp. 1-19	article	URL
BibTeX: @article{Oestergard2001, author = {P.R.J. Oestergård and W.D. Weakley}, title = {Values of Domination Numbers of the Queen's Graph}, journal = {The Electronic Journal of Combinatorics}, year = {2001}, volume = {8(1)}, number = {R29}, pages = {1-19}, url = {http://www.combinatorics.org/Volume_8/PDF/v8i1r29.pdf} }
Oh, S.	An Analytical Evidence for Kalé's Heuristic for the $N$ Queens Problem [BibTeX]	1993	Information Processing Letters Vol. 46, pp. 51-54	article	DOI
BibTeX: @article{Oh1993, author = {S.B. Oh}, title = {An Analytical Evidence for Kalé's Heuristic for the $N$ Queens Problem}, journal = {Information Processing Letters}, year = {1993}, volume = {46}, pages = {51-54}, doi = {http://dx.doi.org/10.1016/0020-0190(93)90196-G} }
Okunev, L.	Kombinatornye Zadachi na Shakhmatnoi Doske [BibTeX]	1935		book
BibTeX: @book{Okunev1935, author = {L.Y. Okunev}, title = {Kombinatornye Zadachi na Shakhmatnoi Doske}, publisher = {ONTI, Moscow, Leningrad}, year = {1935} }
Olson, A.	The Eight Queens Problem [BibTeX]	1993	Journal of Computers in Mathematics and Science Teaching Vol. 12, pp. 93	article
BibTeX: @article{Olson1993, author = {A.T. Olson}, title = {The Eight Queens Problem}, journal = {Journal of Computers in Mathematics and Science Teaching}, year = {1993}, volume = {12}, pages = {93} }
Panayotopoulos, A.	Generating Stable Permutations [BibTeX]	1986	Discrete Mathematics Vol. 62, pp. 219-221	article	DOI
BibTeX: @article{Panayotopoulos1986, author = {A. Panayotopoulos}, title = {Generating Stable Permutations}, journal = {Discrete Mathematics}, year = {1986}, volume = {62}, pages = {219-221}, doi = {http://dx.doi.org/10.1016/0012-365X(86)90121-4} }
Parmentier, T.	Problème des $n$-reines [BibTeX]	1883	Comptes Rendus de l'Association Française pour l'Avancement des Sciences, pp. 197-213	misc
BibTeX: @misc{Parmentier1883, author = {T. Parmentier}, title = {Problème des $n$-reines}, journal = {Comptes Rendus de l'Association Française pour l'Avancement des Sciences}, year = {1883}, pages = {197-213} }
Pauls	Das Maximalproblem der Damen auf dem Schachbrete [BibTeX]	1874	Deutsche Schachzeitung, Organ für das Gesammte Schachleben Vol. 29, pp. 129-134, 257-267	article
BibTeX: @article{Pauls1874, author = {Pauls}, title = {Das Maximalproblem der Damen auf dem Schachbrete}, journal = {Deutsche Schachzeitung, Organ für das Gesammte Schachleben}, year = {1874}, volume = {29}, pages = {129-134, 257-267} }
Pearson, C. & Pearson, M.	Analysis of the n-Queens Puzzle in 2 and 3 Dimensions [BibTeX]	2009		misc	URL
BibTeX: @misc{Pearson, author = {C.S. Pearson and M.S. Pearson}, title = {Analysis of the n-Queens Puzzle in 2 and 3 Dimensions}, year = {2009}, url = {http://queens.cspea.co.uk/} }
Pegg Jr., E.	Math Games: Chessboard Tasks [BibTeX]	2005		misc	URL
BibTeX: @misc{Pegg2005, author = {Pegg Jr., E.}, title = {Math Games: Chessboard Tasks}, year = {2005}, url = {http://www.maa.org/editorial/mathgames/mathgames_04_11_05.html} }
Petković, M.	Mathematics and Chess (110 Entertaining Problems and Solutions) [BibTeX]	1997		book
BibTeX: @book{Petkovic1997, author = {M. Petković}, title = {Mathematics and Chess (110 Entertaining Problems and Solutions)}, publisher = {Dover Publications Inc.}, year = {1997} }
Pickover, C.	The Zen of Magic Squares, Circles, and Stars (An Exhibition of Surprising Structures Across Dimensions) [BibTeX]	2002		book
BibTeX: @book{Pickover2002, author = {C.A. Pickover}, title = {The Zen of Magic Squares, Circles, and Stars (An Exhibition of Surprising Structures Across Dimensions)}, publisher = {Princeton University Press, Princeton, NJ}, year = {2002} }
Planck, C.	The $n$-Queens Problem [BibTeX]	1900	British Chess Mag Vol. 20(4), pp. 94-97	article
BibTeX: @article{Planck1900, author = {C. Planck}, title = {The $n$-Queens Problem}, journal = {British Chess Mag}, year = {1900}, volume = {20(4)}, pages = {94-97} }
Polster, B.	A Geometrical Picture Book [BibTeX]	1998		book
BibTeX: @book{Polster1998, author = {B. Polster}, title = {A Geometrical Picture Book}, publisher = {Springer}, year = {1998} }
Poulet, P.	Suites de Nombres [BibTeX]	1922	L'Intermediaire des mathématiciens Vol. 21, pp. 92-93	article
BibTeX: @article{Poulet1922, author = {P. Poulet}, title = {Suites de Nombres}, journal = {L'Intermediaire des mathématiciens}, year = {1922}, volume = {21}, pages = {92-93} }
Pólya, G.	Mathematische Unterhaltungen und Spiele [BibTeX]	1918		inbook
BibTeX: @inbook{Polya1918, author = {G. Pólya}, title = {Mathematische Unterhaltungen und Spiele}, publisher = {B.G. Teubner}, year = {1918} }
Qiu, W.	The $n$-Queens Problem [BibTeX]	1986	Journal of Mathematics (Wuhan) Vol. 6(2), pp. 117-130	article
BibTeX: @article{Qiu1986, author = {W.S. Qiu}, title = {The $n$-Queens Problem}, journal = {Journal of Mathematics (Wuhan)}, year = {1986}, volume = {6(2)}, pages = {117-130} }
Qiu, Z.	Bit-Vector Encoding of $n$-Queen Problem [Abstract] [BibTeX]	2002	ACM SIGPLAN Notices Vol. 37, pp. 68-70	article	DOI
Abstract: 8-queen problem and its generalization, n-queen problem are well-known examples in the textbooks on elementary programming, data structures, and algorithms. Different methods are proposed to solve these problems, for example, in teWirth1976. In this paper, we present a purely bit-vector encoding of the $n$-queen problem. It is very natural, simple to understand, and efficient. It involves only bit-wise operations.
BibTeX: @article{Qiu2002, author = {Z. Qiu}, title = {Bit-Vector Encoding of $n$-Queen Problem}, journal = {ACM SIGPLAN Notices}, year = {2002}, volume = {37}, pages = {68-70}, doi = {http://dx.doi.org/10.1145/568600.568613} }
Raghavan, V. & Venkatesan, S.	On Bounds for a Board Covering Problem [BibTeX]	1987	Information Processing Letters Vol. 25, pp. 281-284	article	DOI
BibTeX: @article{Raghavan1987, author = {V. Raghavan and S.M. Venkatesan}, title = {On Bounds for a Board Covering Problem}, journal = {Information Processing Letters}, year = {1987}, volume = {25}, pages = {281-284}, doi = {http://dx.doi.org/10.1016/0020-0190(87)90201-8} }
Reichling, M.	A Simplified Solution of the $N$ Queens' Problem [BibTeX]	1987	Information Processing Letters Vol. 25, pp. 253-255	article	DOI
BibTeX: @article{Reichling1987, author = {M. Reichling}, title = {A Simplified Solution of the $N$ Queens' Problem}, journal = {Information Processing Letters}, year = {1987}, volume = {25}, pages = {253-255}, doi = {http://dx.doi.org/10.1016/0020-0190(87)90171-2} }
Rivin, I., Vardi, I. & Zimmerman, P.	The $n$-Queens Problem [BibTeX]	1994	The American Mathematical Monthly Vol. 101(7), pp. 629-639	article	DOI
BibTeX: @article{Rivin1994, author = {I. Rivin and I. Vardi and P. Zimmerman}, title = {The $n$-Queens Problem}, journal = {The American Mathematical Monthly}, year = {1994}, volume = {101(7)}, pages = {629-639}, doi = {http://dx.doi.org/10.2307/2974691} }
Rivin, I. & Zabih, R.	A Dynamic Programming Solution to the $n$-Queens Problem [Abstract] [BibTeX]	1992	Information Processing Letters Vol. 41, pp. 253-256	article	DOI
Abstract: The $n$-queens problem is to determine in how many ways $n$ queens may be placed on an $n$-by-$n$ chessboard so that no two queens attack each other under the rules of chess. We describe a simple $O(f(n)8^n)$ solution to this problem that is based on dynamic programming, where $f(n)$ is a low-order polynomial. This appears to be the first nontrivial upper bound for the problem.
BibTeX: @article{Rivin1992, author = {I. Rivin and R. Zabih}, title = {A Dynamic Programming Solution to the $n$-Queens Problem}, journal = {Information Processing Letters}, year = {1992}, volume = {41}, pages = {253-256}, doi = {http://dx.doi.org/10.1016/0020-0190(92)90168-U} }
Rivin, I. & Zabih, R.	An Algebraic Approach to Constraint Satisfaction Problems [Abstract] [BibTeX]	1989	Proceedings Eleventh International Joint Conference on Artificial Intelligence (IJCAI), pp. 284-289	inproceedings	URL
Abstract: A constraint satisfaction problem, or CSP, can be reformulated as an integer linear programming problem. The reformulated problem can be solved via polynomial multiplication. If the CSP has $n$ variables whose domain size is $m$, and if the equivalent programming problem involves $M$ equations, then the number of solutions can be determined in time $0(nm2^M-n)$. This surprising link between search problems and algebraic techniques allows us to show improved bounds for several constraint satisfaction problems, including new simply exponential bounds for determining the number of solutions to the $n$-queens problem. We also address the problem of minimizing $M$ for a particular CSP.
BibTeX: @inproceedings{Rivin1989, author = {I. Rivin and R. Zabih}, title = {An Algebraic Approach to Constraint Satisfaction Problems}, booktitle = {Proceedings Eleventh International Joint Conference on Artificial Intelligence (IJCAI)}, year = {1989}, pages = {284-289}, url = {http://dli.iiit.ac.in/ijcai/IJCAI-89-VOL1/PDF/045.pdf} }
Rohl, J.	A Faster Lexicographical $N$ Queens Algorithm [BibTeX]	1983	Information Processing Letters Vol. 17, pp. 231-233	article	DOI
BibTeX: @article{Rohl1983, author = {J.S. Rohl}, title = {A Faster Lexicographical $N$ Queens Algorithm}, journal = {Information Processing Letters}, year = {1983}, volume = {17}, pages = {231-233}, doi = {http://dx.doi.org/10.1016/0020-0190(83)90104-7} }
Rolfe, T.	Las Vegas does $n$-Queens [Abstract] [BibTeX]	2006	ACM SIGCSE Bulletin Vol. 38, pp. 37-38	article	DOI
Abstract: This paper presents two Las Vegas algorithms to generate single solutions to the $n$-queens problem. One algorithm generates and improves on random permutation vectors until it achieves one that is a successful solution, while the other algorithm randomly positions queens within each row in positions not under attack from above.
BibTeX: @article{Rolfe2006, author = {T.J. Rolfe}, title = {Las Vegas does $n$-Queens}, journal = {ACM SIGCSE Bulletin}, year = {2006}, volume = {38}, pages = {37-38}, doi = {http://dx.doi.org/10.1145/1138403.1138429} }
Rolfe, T.	Queens on a Chessboard: Making the Best of a Bad Situation [Abstract] [BibTeX]	1995	SCCS: Proceedings of the 28th Annual Small College Computing Symposium Vol. 28, pp. 201-210	article	URL
Abstract: Placing Queens on a chessboard is a classic use of backtracking to speed up a worse than exponential-time algorithm. After the discussion of the basic problem and its solution, two algorithm optimizations are presented (both optimizations together increase the processing speed by an order of magnitude for sufficiently large boards), along with a symmetry constraint on acceptable board configurations. The fully optimized algorithm is then used to show three separate approaches to using parallel processing to further speed the solution: (1) using fork() on a UNIX multiprocessor, (2) using a shared-memory multiprocessor (Silicon Graphics 4D/380), and (3) programming in a message-passing distributed-memory environment (PVM on RS/6000 computers).
BibTeX: @article{Rolfe1995, author = {T.J. Rolfe}, title = {Queens on a Chessboard: Making the Best of a Bad Situation}, journal = {SCCS: Proceedings of the 28th Annual Small College Computing Symposium}, year = {1995}, volume = {28}, pages = {201-210}, url = {http://penguin.ewu.edu/~trolfe/SCCS-95/SCCS-95.html} }
Ruskey, F.	Information on the $n$-Queens Problem [BibTeX]			misc	URL
BibTeX: @misc{Ruskey, author = {F. Ruskey}, title = {Information on the $n$-Queens Problem}, url = {http://www.theory.csc.uvic.ca/~cos/inf/misc/Queen.html} }
Sagols, F. & Colbourn, C.	NS1D0 Sequences and Anti-Pasch Steiner Triple Systems [BibTeX]	2002	Ars Combinatoria Vol. 62, pp. 17-31	article
BibTeX: @article{Sagols2002, author = {F. Sagols and C.J. Colbourn}, title = {NS1D0 Sequences and Anti-Pasch Steiner Triple Systems}, journal = {Ars Combinatoria}, year = {2002}, volume = {62}, pages = {17-31} }
Sainte-Lague, A.	Mémorial des Sciences Mathématiques [BibTeX]	1926	Vol. 18	inbook
BibTeX: @inbook{Sainte-Lagu:e1926, author = {A. Sainte-Lague}, title = {Mémorial des Sciences Mathématiques}, publisher = {Gauthier-Villars, Paris}, year = {1926}, volume = {18} }
Scheid, F.	Some Packing Problems [BibTeX]	1960	The American Mathematical Monthly Vol. 67(3), pp. 231-235	article	DOI
BibTeX: @article{Scheid1960, author = {F. Scheid}, title = {Some Packing Problems}, journal = {The American Mathematical Monthly}, year = {1960}, volume = {67(3)}, pages = {231-235}, doi = {http://dx.doi.org/10.2307/2309682} }
Schlude, K. & Specker, E.	Zum Problem der Damen auf dem Torus [BibTeX]	2003	(412)	techreport
BibTeX: @techreport{Schlude2003, author = {K. Schlude and E. Specker}, title = {Zum Problem der Damen auf dem Torus}, year = {2003}, number = {412} }
Schrage, G.	The Eight Queens Problem as a Strategy Game [Abstract] [BibTeX]	1989	International Journal of Mathematical Education in Science and Technology Vol. 17, pp. 143-148	article	DOI
Abstract: A strategy game is presented that is strongly connected to the classical `eight queens problem' for checkerboards. Two different versions of the game are analysed with computer assistance. The algorithm for this analysis is developed in terms of a general game model. Thus it can be used, at least in principal, for any two-person strategy game.
BibTeX: @article{Schrage1989, author = {G. Schrage}, title = {The Eight Queens Problem as a Strategy Game}, journal = {International Journal of Mathematical Education in Science and Technology}, year = {1989}, volume = {17}, pages = {143-148}, doi = {http://dx.doi.org/10.1080/0020739860170203} }
Schroeder, M.	Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise [BibTeX]	1991		book
BibTeX: @book{Schroeder1991, author = {M. Schroeder}, title = {Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise}, publisher = {W.H. Freeman and Company, New York}, year = {1991} }
Schwartz, J., Dewar, R., Dubinsky, E. & Schonberg, E.	An Introduction to SETL [BibTeX]	1986		book
BibTeX: @book{Schwartz1986, author = {J.T. Schwartz and R.B.K. Dewar and E. Dubinsky and E. Schonberg}, title = {An Introduction to SETL}, publisher = {Springer-Verlag}, year = {1986} }
Sebastian, J.	Some Computer Solutions to the Reflecting Queens Problem [BibTeX]	1969	The American Mathematical Monthly Vol. 76(4), pp. 399-400	article	DOI
BibTeX: @article{Sebastian1969, author = {J.D. Sebastian}, title = {Some Computer Solutions to the Reflecting Queens Problem}, journal = {The American Mathematical Monthly}, year = {1969}, volume = {76(4)}, pages = {399-400}, doi = {http://dx.doi.org/10.2307/2316435} }
Selfridge, J.	Abstract 63T-80 [BibTeX]	1963	Notices of the American Mathematical Society Vol. 19, pp. 195	article
BibTeX: @article{Selfridge1963, author = {J.L. Selfridge}, title = {Abstract 63T-80}, journal = {Notices of the American Mathematical Society}, year = {1963}, volume = {19}, pages = {195} }
Sforza, G.	Una Regola pel Gioco della $n$ Regine Quando $n$ é Primo [BibTeX]	1925	Periodicodi Matematiche. Organo della Mathesis, Societá Italiana di Scienze Mathematichee Fisiche Vol. 5(4), pp. 107-109	article
BibTeX: @article{Sforza1925, author = {G. Sforza}, title = {Una Regola pel Gioco della $n$ Regine Quando $n$ é Primo}, journal = {Periodicodi Matematiche. Organo della Mathesis, Societá Italiana di Scienze Mathematichee Fisiche}, year = {1925}, volume = {5(4)}, pages = {107-109} }
Shagrir, O.	A Neural Net with Self-inhibiting Units for the $n$-Queens Problem [Abstract] [BibTeX]	1992	International Journal of Neural Systems Vol. 3, pp. 249-252	article	DOI
Abstract: Suggested here is a neural net algorithm for the $n$-Queens problem. The net is basically a Hopfield net but with one major difference: every unit is allowed to inhibit itself. This distinctive characteristic enables the net to escape efficiently from all local minima. The net’s dynamics then can be described as a travel in paths of low-level energy spaces until it finds a solution (global minimum). The paper explains why standard Hopfield nets have failed to solve the Queens problem and proofs that the self-inhibiting net (NQ2 algorithm in the text) never stabilizes in local minima and relaxes when it falls into a global minimum are provided. The experimental results supported by theoretical explanation indicate that the net never continually oscillates but relaxes into a solution in polynomial time. In addition, it appears that the net solves the Queens problem regardless of the dimension n or the initialized values. The net uses only few parameters to fix the weights; all globally determined as a function of $n$.
BibTeX: @article{Shagrir1992, author = {O. Shagrir}, title = {A Neural Net with Self-inhibiting Units for the $n$-Queens Problem}, journal = {International Journal of Neural Systems}, year = {1992}, volume = {3}, pages = {249-252}, doi = {http://dx.doi.org/10.1142/S0129065792000206} }
Shapiro, H.	Generalized Latin Squares on the Torus [BibTeX]	1978	Discrete Mathematics Vol. 24, pp. 63-77	article	DOI
BibTeX: @article{Shapiro1978, author = {H.D. Shapiro}, title = {Generalized Latin Squares on the Torus}, journal = {Discrete Mathematics}, year = {1978}, volume = {24}, pages = {63-77}, doi = {http://dx.doi.org/10.1016/0012-365X(78)90173-5} }
Shapiro, H.	Theoretical Limitations on the Efficient Use of Parallel Memories [Abstract] [BibTeX]	1978	IEEE Transactions on Computers Vol. C-27, pp. 421-428	article	DOI
Abstract: The effective utilization of single-instruction-multiple-data stream machines depends heavily on being able to arrange the data elements of arrays in parallel memory modules so that memory conflicts are avoided when the data are fetched. Several classes of storage algorithms are presented. Necessary and sufficient conditions are derived which can be used to determine if all conflict can be avoided. For the matrix subparts most often demanded in numerical analysis computations, whenever the class of storage algorithms called periodic skewing schemes provides conflict-free access, the subclass called linear skewing schemes also provides such access.
BibTeX: @article{Shapiro1978a, author = {H.D. Shapiro}, title = {Theoretical Limitations on the Efficient Use of Parallel Memories}, journal = {IEEE Transactions on Computers}, year = {1978}, volume = {C-27}, pages = {421-428}, doi = {http://dx.doi.org/10.1109/TC.1978.1675122} }
Shen, M.-K. & Shen, T.-P.	Research Problem 39 [BibTeX]	1962	Bulletin of the American Mathematical Society Vol. 68, pp. 557	article	DOI
BibTeX: @article{Shen1962, author = {M.-K. Shen and T.-P. Shen}, title = {Research Problem 39}, journal = {Bulletin of the American Mathematical Society}, year = {1962}, volume = {68}, pages = {557}, doi = {http://dx.doi.org/10.1090/S0002-9904-1962-10842-8} }
da Silva, I., de Souza, A. & Bordon, M.	A Modified Hopfield Model for Solving the $N$-Queens Problem [Abstract] [BibTeX]	2000	Neural Networks, Proceedings of the IEEE-INNS-ENNS International Joint Conference on, pp. $509 - 514$	inproceedings	DOI
Abstract: A neural network model for solving the $N$-Queens problem is presented in this paper. More specifically, a modified Hopfield network is developed and its internal parameters are computed using the valid-subspace technique. These parameters guarantee the convergence of the network to the equilibrium points. The network is shown to be completely stable and globally convergent to the solutions of the $N$-Queens problem. Simulation results are presented to validate the proposed approach.
BibTeX: @inproceedings{Silva2000, author = {I.N. da Silva and A.N. de Souza and M.E. Bordon}, title = {A Modified Hopfield Model for Solving the $N$-Queens Problem}, booktitle = {Neural Networks, Proceedings of the IEEE-INNS-ENNS International Joint Conference on}, year = {2000}, pages = {$509 - 514$}, doi = {http://dx.doi.org/10.1109/IJCNN.2000.859446} }
Slater, M.	Research Problem 1 [BibTeX]	1963	Bulletin of the American Mathematical Society Vol. 69, pp. 333	article	DOI
BibTeX: @article{Slater1963, author = {M. Slater}, title = {Research Problem 1}, journal = {Bulletin of the American Mathematical Society}, year = {1963}, volume = {69}, pages = {333}, doi = {http://dx.doi.org/10.1090/S0002-9904-1963-10907-6} }
Sloane, N.	Sequence A000170: Number of Ways of Placing $n$ Nonattacking Queens on $nn$ Board [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 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, ldots
BibTeX: @misc{Sloane000170, author = {N.J.A. Sloane}, title = {Sequence A000170: Number of Ways of Placing $n$ Nonattacking Queens on $nn$ Board}, url = {http://www.research.att.com/~njas/sequences/?q=A000170} }
Sloane, N.	Sequence A001366: Maximal Number of Unattacked Squares with $n$-Queens on $nn$ Board (Answers for $n geq 17$ only Probable) [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 0, 0, 0, 1, 3, 5, 7, 11, 18, 22, 30, 36, 47, 56, 72, 82, 97, 111, 132, 145, 170, 186, 216, 240, 260, 290, 324, 360, 381, 420, ldots
BibTeX: @misc{Sloane001366, author = {N.J.A. Sloane}, title = {Sequence A001366: Maximal Number of Unattacked Squares with $n$-Queens on $nn$ Board (Answers for $n geq 17$ only Probable)}, url = {http://www.research.att.com/~njas/sequences/?q=A001366} }
Sloane, N.	Sequence A002562: Number of Ways of Placing $n$ Nonattacking Queens on $nn$ Board (Symmetric Solutions Count only Once) [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 1, 0, 0, 1, 2, 1, 6, 12, 46, 92, 341, 1787, 9233, 45752, 285053, 1846955, 11977939, 83263591, 621012754, 4878666808, 39333324973, 336376244042, 3029242658210, 28439272956934, 275986683743434, ldots
BibTeX: @misc{Sloane002562, author = {N.J.A. Sloane}, title = {Sequence A002562: Number of Ways of Placing $n$ Nonattacking Queens on $nn$ Board (Symmetric Solutions Count only Once)}, url = {http://www.research.att.com/~njas/sequences/?q=A002562} }
Sloane, N.	Sequence A006717: Toroidal Semi-Queens on a $(2n+1) (2n+1)$ Board [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 1, 3, 15, 133, 2025, 37851, 1030367, 36362925, 1606008513, 87656896891, 5778121715415, 452794797220965, 41609568918940625, ldots
BibTeX: @misc{Sloane006717, author = {N.J.A. Sloane}, title = {Sequence A006717: Toroidal Semi-Queens on a $(2n+1) (2n+1)$ Board}, url = {http://www.research.att.com/~njas/sequences/?q=A006717} }
Sloane, N.	Sequence A007705: Number of Ways of Arranging $ 2n+1 $ Nonattacking Queens on a $(2n+1)(2n+1)$ Toroidal Board [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 1, 0, 10, 28, 0, 88, 4524, 0, 140692, 820496, 0, 128850048, 1957725000, 0, 605917055356, 13404947681712, 0, ldots
BibTeX: @misc{Sloane007705, author = {N.J.A. Sloane}, title = {Sequence A007705: Number of Ways of Arranging $ 2n+1 $ Nonattacking Queens on a $(2n+1)(2n+1)$ Toroidal Board}, url = {http://www.research.att.com/~njas/sequences/?q=A007705} }
Sloane, N.	Sequence A019317: Place $n$ Queens on an $nn$ Board so as to Leave the Maximal Number of Unattacked Squares; Sequence Gives Number of Different Solutions [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 1, 2, 16, 25, 1, 3, 38, 7, 1, 1, 2, 7, 1, 4, 3, 1, ldots
BibTeX: @misc{Sloane019317, author = {N.J.A. Sloane}, title = {Sequence A019317: Place $n$ Queens on an $nn$ Board so as to Leave the Maximal Number of Unattacked Squares; Sequence Gives Number of Different Solutions}, url = {http://www.research.att.com/~njas/sequences/?q=A019317} }
Sloane, N.	Sequence A051906: Number of Ways of Placing $n$ Nonattacking Toroidal Queens on an $n b$ Chess Board [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 1, 0, 0, 0, 10, 0, 28, 0, 0, 0, 88, 0, 4524, 0, 0, 0, 140692, 0, 820496, 0, 0, 0, 128850048, 0, 1957725000, 0, 0, 0, 605917055356, ldots
BibTeX: @misc{Sloane051906, author = {N.J.A. Sloane}, title = {Sequence A051906: Number of Ways of Placing $n$ Nonattacking Toroidal Queens on an $n b$ Chess Board}, url = {http://www.research.att.com/~njas/sequences/?q=A051906} }
Sloane, N.	Sequence A053994: Nonattacking Queens on a $(2n+1)(2n+1)$ Toroidal Board, Solutions which Differ only by Rotation, Reflection or Torus Shift Count only Once [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 1, 0, 1, 1, 0, 2, 11, 0, 97, 354, 0, 31381, 395551, 0, 90120677, ldots
BibTeX: @misc{Sloane053994, author = {N.J.A. Sloane}, title = {Sequence A053994: Nonattacking Queens on a $(2n+1)(2n+1)$ Toroidal Board, Solutions which Differ only by Rotation, Reflection or Torus Shift Count only Once}, url = {http://www.research.att.com/~njas/sequences/?q=A053994} }
Sloane, N.	Sequence A054500: Indicator Sequence for Classification of Nonattacking Queens on $nn$ Toroidal Board [Abstract] [BibTeX]		The On-Line Encyclopedia of Integer Sequences	misc	URL
Abstract: 1, 5, 7, 11, 13, 13, 13, 13, 17, 17, 17, 17, 17, 19, 19, 19, 23, 23, 23, 25, 25, 25, 25, 25, 25, 25, 25, 29, 29, 29, 29, 29, ldots
BibTeX: @misc{Sloane054500, author = {N.J.A. Sloane}, title = {Sequence A054500: Indicator Sequence for Classification of Nonattacking Queens on $nn$ Toroidal Board}, url = {http://www.research.att.com/~njas/sequences/?q=A054500} }
Sloane, N. & Plouffe, S.	Figure M0180 in The Encyclopedia of Integer Sequences. [BibTeX]	1995	San Diego: Academic Press	misc
BibTeX: @misc{Sloane1995, author = {N.J.A. Sloane and S. Plouffe}, title = {Figure M0180 in The Encyclopedia of Integer Sequences.}, year = {1995} }
Sosic, R.	A Parallel Search Algoritm for the $n$-Queens Problem [BibTeX]	1994	Parallel Computing and Transputer Conference, Wollongong, pp. 162-172	inproceedings
BibTeX: @inproceedings{Sosic1994, author = {Sosic, R.}, title = {A Parallel Search Algoritm for the $n$-Queens Problem}, booktitle = {Parallel Computing and Transputer Conference, Wollongong}, publisher = {IOS Press}, year = {1994}, pages = {162-172} }
Sosic, R. & Gu, J.	Efficient Local Search with Conflict Minimization: A Case Study of the $n$-Queens Problem [Abstract] [BibTeX]	1994	IEEE Transactions on Knowledge and Data Engineering Vol. 6(5), pp. 661-668	article	DOI
Abstract: Backtracking search is frequently applied to solve a constraint-based search problem, but it often suffers from exponential growth of computing time. We present an alternative to backtracking search: local search with conflict minimization. We have applied this general search framework to study a benchmark constraint-based search problem, the $n$-Queens problem. An efficient local search algorithm for the $n$-Queens problem was implemented. This algorithm, running in linear time, does not backtrack. It is capable of finding a solution for extremely large size $n$-Queens problems. For example, on a workstation it can find a solution for 3000000 Queens in less than 55 s.
BibTeX: @article{Sosic1994a, author = {R. Sosic and J. Gu}, title = {Efficient Local Search with Conflict Minimization: A Case Study of the $n$-Queens Problem}, journal = {IEEE Transactions on Knowledge and Data Engineering}, year = {1994}, volume = {6(5)}, pages = {661-668}, doi = {http://dx.doi.org/10.1109/69.317698} }
Sosic, R. & Gu, J.	$3,000,000$ Queens in Less than One Minute [Abstract] [BibTeX]	1991	ACM SIGART Bulletin Vol. 2, pp. 22-24	article	DOI
Abstract: The $n$-queens problem is a classical combinatorial search problem. In this paper we give a linear time algorithm for this problem. The algorithm is an extension of one of our previous local search algorithms [3, 4, 6]. On an IBM RS 6000 computer, this algorithm is capable of solving problems with 3,000,000 queens in approximately 55 seconds.
BibTeX: @article{Sosic1991, author = {R. Sosic and J. Gu}, title = {$3,000,000$ Queens in Less than One Minute}, journal = {ACM SIGART Bulletin}, year = {1991}, volume = {2}, pages = {22-24}, doi = {http://dx.doi.org/10.1145/122319.122325} }
Sosic, R. & Gu, J.	Fast Search Algorithms for the Queens Problem [Abstract] [BibTeX]	1991	IEEE Transactions on Systems, Man and Cybernetics Vol. 21(6), pp. 1572-1576	article	DOI
Abstract: The $n$-queens problem is to place $n$ queens on an $nn$ chessboard so that no two queens attack each other. The authors present two new algorithms, called queen search 2 (QS2) and queen search 3 (QS3). QS2 and QS3 are probabilistic local search algorithms, based on a gradient-based heuristic. These algorithms, running in almost linear time, are capable of finding a solution for extremely large $n$-queens problems. For example, QS3 can find a solution for 500000 queens in approximately 1.5 min.
BibTeX: @article{Sosic1991a, author = {R. Sosic and J. Gu}, title = {Fast Search Algorithms for the Queens Problem}, journal = {IEEE Transactions on Systems, Man and Cybernetics}, year = {1991}, volume = {21}, number = {6}, pages = {1572-1576}, doi = {http://dx.doi.org/10.1109/21.135698} }
Sosic, R. & Gu, J.	A Polynomial Time Algorithm for the $n$-Queens Problem [Abstract] [BibTeX]	1990	ACM SIGART Bulletin Vol. 1, pp. 7-11	article	DOI
Abstract: The $n$-Queens problem is a classical combinatorial problem in the artificial intelligence (AI) area. Since the problem has a simple and regular structure, it has been widely used as a testbed to develop and benchmark new AI search problem-solving strategies. Recently, this problem has found practical applications in VLSI testing and traffic control. Due to its inherent complexity, currently even very efficient AI search algorithms developed so far can only find a solution for the $n$-Queens problem with n up to about 100. In this paper we present a new, probabilistic local search algorithm which is based on a gradient-based heuristic. This efficient algorithm is capable of finding a solution for extremely large size $n$-Queens problems. We give the execution statistics for this algorithm with $n$ up to 500,000.
BibTeX: @article{Sosic1990, author = {R. Sosic and J. Gu}, title = {A Polynomial Time Algorithm for the $n$-Queens Problem}, journal = {ACM SIGART Bulletin}, year = {1990}, volume = {1}, pages = {7-11}, doi = {http://dx.doi.org/10.1145/101340.101343} }
Sosic, R. & Gu, J.	How to Search For Million Queens [BibTeX]	1988	(UUCS-TR-88-008)	techreport
BibTeX: @techreport{Sosic1988a, author = {R. Sosic and J. Gu}, title = {How to Search For Million Queens}, year = {1988}, number = {UUCS-TR-88-008} }
Sosic, R. & Gu, J.	$n$-Queen Search on VAX and Bobcat Machines [BibTeX]	1988	CS 547 AI Class Student Project Report	misc
BibTeX: @misc{Sosic1988b, author = {R. Sosic and J. Gu}, title = {$n$-Queen Search on VAX and Bobcat Machines}, journal = {CS 547 AI Class Student Project Report}, year = {1988} }
Sprague, T.	On the Eight Queens Problem [Abstract] [BibTeX]	1898	Proceedings of the Edinburgh Mathematical Society Vol. 17, pp. 43-68	article	DOI
Abstract: This is the problem discussed in my paper bearing the not very happy title ``On the different non-linear arrangements of eight men on a chess-board”, which was read to the Edinburgh Mathematical Society on 14th March 1890, and is printed in its Transactions, Vol. VIII, p. 30. At that time I was not aware that the problem had been discussed by any previous writer, and I treated it as an entirely new one. I have since learnt that a good deal has been written about it, and I propose on the present occasion to give briefly the history of the problem, and the results which have been arrived at; also to communicate some new results which I have obtained.
BibTeX: @article{Sprague1898, author = {T.B. Sprague}, title = {On the Eight Queens Problem}, journal = {Proceedings of the Edinburgh Mathematical Society}, year = {1898}, volume = {17}, pages = {43-68}, doi = {http://dx.doi.org/10.1017/S0013091500029096} }
Sprague, T.	On the Different Non-Linear Arrangements of Eight Men on a Chess-board [Abstract] [BibTeX]	1889	Proceedings of the Edinburgh Mathematical Society Vol. 8, pp. 30-43	article	DOI
Abstract: The question having been proposed to me as a puzzle: To arrange eight men on a chess-board, so that no two of them shall be in the same line,—--that is to say, that no two are to be in the same column, nor in the same row, nor in the same diagonal line,—--I succeeded before very long in solving it by finding the annexed arrangement.
BibTeX: @article{Sprague1889, author = {T.B. Sprague}, title = {On the Different Non-Linear Arrangements of Eight Men on a Chess-board}, journal = {Proceedings of the Edinburgh Mathematical Society}, year = {1889}, volume = {8}, pages = {30-43}, doi = {http://dx.doi.org/10.1017/S0013091500030522} }
Stanley, R.	Enumerative Combinatorics [BibTeX]	1986	Vol. I	book
BibTeX: @book{Stanley1986, author = {R.P. Stanley}, title = {Enumerative Combinatorics}, publisher = {Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, California}, year = {1986}, volume = {I} }
Steinhaus, H.	Mathematical Snapshots [BibTeX]	1938		book
BibTeX: @book{Steinhaus1938, author = {H. Steinhaus}, title = {Mathematical Snapshots}, publisher = {Oxford University Press}, year = {1938} }
Stern, E.	General Formulas for the Number of Magic Squares Belonging to Certain Classes [BibTeX]	1939	The American Mathematical Monthly Vol. 46(9), pp. 555-581	article	DOI
BibTeX: @article{Stern1939, author = {E. Stern}, title = {General Formulas for the Number of Magic Squares Belonging to Certain Classes}, journal = {The American Mathematical Monthly}, year = {1939}, volume = {46(9)}, pages = {555-581}, doi = {http://dx.doi.org/10.2307/2302760} }
Stern, E.	Über irregulare Pan Diagonale Lateinische Quadrate mit Primzahlseitenlange und ihre Bedeutung für das $n$-Königinnenproblem sowie für die Bildung magischer Quadrate [BibTeX]	1938	Nieuw Archief voor Wiskunde Vol. 19, pp. 257-270	article
BibTeX: @article{Stern1938, author = {E. Stern}, title = {Über irregulare Pan Diagonale Lateinische Quadrate mit Primzahlseitenlange und ihre Bedeutung für das $n$-Königinnenproblem sowie für die Bildung magischer Quadrate}, journal = {Nieuw Archief voor Wiskunde}, year = {1938}, volume = {19}, pages = {257-270} }
Stoffel, A.	Totally Diagonal Latin Squares [BibTeX]	1976	Stud. Cerc. Mat. Vol. 28(1), pp. 113-119	article
BibTeX: @article{Stoffel1976, author = {A. Stoffel}, title = {Totally Diagonal Latin Squares}, journal = {Stud. Cerc. Mat.}, year = {1976}, volume = {28(1)}, pages = {113-119} }
Stone, H. & Stone, J.	Efficient Search Techniques --- An empirical Study of the $n$-Queens Problem [BibTeX]	1987	IBM Journal of Research and Development Vol. 31, pp. 464-474	article
BibTeX: @article{Stone1987, author = {H.S. Stone and J.M. Stone}, title = {Efficient Search Techniques --- An empirical Study of the $n$-Queens Problem}, journal = {IBM Journal of Research and Development}, year = {1987}, volume = {31}, pages = {464-474} }
Sumitaka, A.	Explicit Solutions of the $n$-Queens Problem [BibTeX]	2001	(060-002)	techreport
BibTeX: @techreport{Sumitaka2001, author = {A. Sumitaka}, title = {Explicit Solutions of the $n$-Queens Problem}, year = {2001}, number = {060-002} }
Tambouratzis, T.	A Simulated Annealing Artificial Neural Network Implementation of the $n$-Queens Problem [Abstract] [BibTeX]	1997	International Journal of Intelligent Systems Vol. 12, pp. 739-752	article	DOI
Abstract: A Harmony Theory artificial neural network implementation of the $n$-Queens problem is presented in this piece of research. The problem is encoded in the two layers of the artificial neural network in such a manner that the inherent constraints of the problem are made directly available. Subsequently, during the simulated annealing procedure of Harmony Theory, maximal constraint satisfaction is accomplished in parallel and an optimal solution of the $n$-Queens problem is produced. This solution indicates the appropriate locations of the greatest possible number of Queens that can be placed on the $nn$ chessboard in a valid configuration, i.e., so that no Queen threatens or is threatened by another Queen. The proposed parallel implementation of the $n$-Queens problem, combined with the application of the simulated annealing procedure, offers an interesting alternative to existing techniques (e.g., search, constraint propagation) in terms of optimality as well as computational and time efficiency.
BibTeX: @article{Tambouratzis1997, author = {T. Tambouratzis}, title = {A Simulated Annealing Artificial Neural Network Implementation of the $n$-Queens Problem}, journal = {International Journal of Intelligent Systems}, year = {1997}, volume = {12}, pages = {739-752}, doi = {http://dx.doi.org/10.1002/(SICI)1098-111X(199710)12:10<739::AID-INT3>3.0.CO;2-Z} }
Tanaka, I., Nishio, Y. & Hasegawa, M.	An Approach to Finding All Solutions of $n$-Queens Problem Using Chaos Neural Network [BibTeX]	2002		techreport
BibTeX: @techreport{Tanaka2002, author = {I. Tanaka and Y. Nishio and M. Hasegawa}, title = {An Approach to Finding All Solutions of $n$-Queens Problem Using Chaos Neural Network}, year = {2002} }
Tanik, M.	A Graph Model for Deadlock Prevention [BibTeX]	1978	School: Texas A&M University	phdthesis
BibTeX: @phdthesis{Tanik1978, author = {M.M. Tanik}, title = {A Graph Model for Deadlock Prevention}, school = {Texas A&M University}, year = {1978} }
Tarry, H.	Problème des $n$ Reines sur Léchiquier de $n^2$ Cases [BibTeX]	1897	Compte rendu de l'Association Française pour l'Avancement des Sciences 26, Congrès de Saint Etienne, pp. 176	inproceedings
BibTeX: @inproceedings{Tarry1897a, author = {H. Tarry}, title = {Problème des $n$ Reines sur Léchiquier de $n^2$ Cases}, booktitle = {Compte rendu de l'Association Française pour l'Avancement des Sciences 26, Congrès de Saint Etienne}, year = {1897}, pages = {176} }
Tarry, H.	Problème des Reines (Problème 605) [BibTeX]	1895	L'Intermédiaire des Mathématiciens Ser Vol. 12, pp. 205	article
BibTeX: @article{Tarry1895, author = {H. Tarry}, title = {Problème des Reines (Problème 605)}, journal = {L'Intermédiaire des Mathématiciens Ser}, year = {1895}, volume = {12}, pages = {205} }
Taylor, H.	Mathematical Properties of Sequences and Other Combinatorial Structures [BibTeX]	2003		inbook
BibTeX: @inbook{Taylor2003, author = {H. Taylor}, title = {Mathematical Properties of Sequences and Other Combinatorial Structures}, publisher = {Kluwer Acad. Publ., Boston, MA}, year = {2003} }
Taylor, H.	Florentine Rows or Left-Right Shifted Permutation Matrices with Cross-correlation Values $leq 1$ [Abstract] [BibTeX]	1991	Discrete Mathematics Vol. 93, pp. 247-260	article	DOI
Abstract: (1) Find $nn$ permutation matrices---as many as possible---whose aperiodic horizontal shifting cross-correlation function takes only the values 0 or 1. (2) Find values of $F(n)$ = the maximum number of Florentine rows on $n$ symbols. (3) It turns out that problem (1) is isomorphic to problem (2), so that optimum constructions are available for (1) whenever $n + 1$ is prime. Also on exhibit is S. Alquaddoomi's recent discovery that $F(8) = 7$.
BibTeX: @article{Taylor1991, author = {H. Taylor}, title = {Florentine Rows or Left-Right Shifted Permutation Matrices with Cross-correlation Values $leq 1$}, journal = {Discrete Mathematics}, year = {1991}, volume = {93}, pages = {247-260}, doi = {http://dx.doi.org/10.1016/0012-365X(91)90259-5} }
Thangavel, P. & Gladisa, D.	Hysteretic Hopfield Network with Dynamic Tunneling for Crossbar Switch and $n$-Queens Problem [Abstract] [BibTeX]	2007	Neurocomputing Vol. 70, pp. 2544-2551	article	DOI
Abstract: An efficient hysteretic Hopfield network with dynamic tunneling is proposed. The hysteretic activation function is used for training. The dynamic tunneling approach is employed to detrap the network from local minima. The network gives better convergence results for the selected problems namely crossbar switch problem with exclusive switching and concurrent switching, and $n$-Queens problem.
BibTeX: @article{Thangavel2007, author = {P. Thangavel and D. Gladisa}, title = {Hysteretic Hopfield Network with Dynamic Tunneling for Crossbar Switch and $n$-Queens Problem}, journal = {Neurocomputing}, year = {2007}, volume = {70}, pages = {2544-2551}, doi = {http://dx.doi.org/10.1016/j.neucom.2006.06.006} }
Theron, W. & Burger, A.	Queen Domination of Hexagonal Hives [BibTeX]	2000	Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 32, pp. 161-172	article
BibTeX: @article{Theron2000, author = {W.F.D. Theron and A.P. Burger}, title = {Queen Domination of Hexagonal Hives}, journal = {Journal of Combinatorial Mathematics and Combinatorial Computing}, year = {2000}, volume = {32}, pages = {161-172} }
Theron, W. & Geldenhuys, G.	Domination by Queens on a Square Beehive [Abstract] [BibTeX]	1998	Discrete Mathematics Vol. 178, pp. 213-220	article	DOI
Abstract: A chessboard-like game board consisting of hexagonal cells and a board piece called a queen are defined. We determine bounds on the upper and lower domination and independence numbers and on the diagonal domination number for queens on square hives of any order.
BibTeX: @article{Theron1998, author = {W.F.D. Theron and G. Geldenhuys}, title = {Domination by Queens on a Square Beehive}, journal = {Discrete Mathematics}, year = {1998}, volume = {178}, pages = {213-220}, doi = {http://dx.doi.org/10.1016/S0012-365X(97)81828-6} }
Tolpygo, A.	Follow-up: Queens on a Cylinder [BibTeX]	1996	Quantum: The Student Magazine of Math and Science Vol. 6, pp. 38-42	article
BibTeX: @article{Tolpygo1996, author = {A. Tolpygo}, title = {Follow-up: Queens on a Cylinder}, journal = {Quantum: The Student Magazine of Math and Science}, year = {1996}, volume = {6}, pages = {38-42} }
Topor, R.	Fundamental Solutions of the Eight Queens Problem [Abstract] [BibTeX]	1982	BIT Numerical Mathematics Vol. 22, pp. 42-52	article	DOI
Abstract: Previous algorithms presented to solve the eight queens problem have generated the set of all solutions. Many of these solutions are identical after applying sequences of rotations and reflections. In this paper we present a simple, clear, efficient algorithm to generate a set of fundamental (or distinct) solutions to the problem.
BibTeX: @article{Topor1982, author = {R.W. Topor}, title = {Fundamental Solutions of the Eight Queens Problem}, journal = {BIT Numerical Mathematics}, year = {1982}, volume = {22}, pages = {42-52}, doi = {http://dx.doi.org/10.1007/BF01934394} }
Vaderlind, P., Guy, R. & Larson, L.	The Inquisitive Problem Solver [BibTeX]	2002		book
BibTeX: @book{Vaderlind2002, author = {P. Vaderlind and R.K. Guy and L.C. Larson}, title = {The Inquisitive Problem Solver}, publisher = {Mathematical Association of America, Washington, DC}, year = {2002} }
Valtorta, M.	Correspondence: Response to ``Explicit Solutions to the $N$-Queens Problem for all $N$'' [BibTeX]	1991	ACM SIGART Bulletin Vol. 2, pp. 4-5	article	DOI
BibTeX: @article{Valtorta1991, author = {M. Valtorta}, title = {Correspondence: Response to ``Explicit Solutions to the $N$-Queens Problem for all $N$''}, journal = {ACM SIGART Bulletin}, year = {1991}, volume = {2}, pages = {4-5}, doi = {http://dx.doi.org/10.1145/122344.1063799} }
Van Rees, G.	On Latin Queen Squares [BibTeX]	1981	Vol. IIProceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, pp. 267–273	inproceedings
BibTeX: @inproceedings{Rees1981, author = {G.H.J. Van Rees}, title = {On Latin Queen Squares}, booktitle = {Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing}, year = {1981}, volume = {II}, pages = {267–273} }
Vardi, I.	The $n$-Queens Problem [BibTeX]	1991	Computational Recreations in Mathematica, pp. 107-125	incollection
BibTeX: @incollection{Vardi1991, author = {Vardi, I.}, title = {The $n$-Queens Problem}, booktitle = {Computational Recreations in Mathematica}, publisher = {Redwood City, CA: Addison-Wesley}, year = {1991}, pages = {107-125} }
Vasquez, M.	Coloration des Graphes de Reines [Abstract] [BibTeX]	2006	Comptes Rendus de l'Académie des Sciences Paris, Série I Mathématique Vol. 342, pp. 157-160	article	DOI
Abstract: Until 2003 no chromatic numbers ($i_n$) for the queen graphs were available for $n>9$ except where n is not a multiple of 2 or 3. In this research announcement we present an exact algorithm which provides coloring solutions for $n$=12,14,15,16,18,20,21,22,24,26,28 and 32 such as $i_n=n$. Then we prove that there exists an infinite number of values for $n$ such that $n=2p$ or $n=3p$, and $i_n=n$.
BibTeX: @article{Vasquez2006, author = {M. Vasquez}, title = {Coloration des Graphes de Reines}, journal = {Comptes Rendus de l'Académie des Sciences Paris, Série I Mathématique}, year = {2006}, volume = {342}, pages = {157-160}, doi = {http://dx.doi.org/10.1016/j.crma.2005.11.022} }
Vasquez, M.	New Result on the Queens $n^2$ Graph Coloring Problem, [Abstract] [BibTeX]	2004	Journal of Heuristics Vol. 10, pp. 407-413	article	DOI
Abstract: For the Queens $n^2$ graph coloring problems no chromatic numbers are available for $n > 9$ except where $n$ is not a multiple of 2 or 3. In this paper we propose an exact algorithm that takes advantage of the particular structure of these graphs. The algorithm works on the independent sets of the graph rather than on the vertices to be colored. It combines branch and bound, for independent set assignment, with a clique based filtering procedure. A first experimentation of this approach provided the coloring number values ranging for $n = 10$ to $n = 14$.
BibTeX: @article{Vasquez2004, author = {M. Vasquez}, title = {New Result on the Queens $n^2$ Graph Coloring Problem,}, journal = {Journal of Heuristics}, year = {2004}, volume = {10}, pages = {407-413}, doi = {http://dx.doi.org/10.1023/B:HEUR.0000034713.28244.e1} }
Vasquez, M.	On the Queen Graph Coloring Problem [BibTeX]	2004	Proceedings of the 3rd International Conference on Information (INFO’04), pp. 109–112	inproceedings
BibTeX: @inproceedings{Vasquez2004a, author = {M. Vasquez}, title = {On the Queen Graph Coloring Problem}, booktitle = {Proceedings of the 3rd International Conference on Information (INFO’04)}, year = {2004}, pages = {109–112} }
Vasquez, M. & Habet, D.	Complete and Incomplete Algorithms for the Queen Graph Coloring Problem [BibTeX]	2004	Proceedings of the 16th European Conference on Artiﬁcial Intelligence (ECAI’04), pp. 226–230	inproceedings
BibTeX: @inproceedings{Vasquez2004b, author = {M. Vasquez and D. Habet}, title = {Complete and Incomplete Algorithms for the Queen Graph Coloring Problem}, booktitle = {Proceedings of the 16th European Conference on Artiﬁcial Intelligence (ECAI’04)}, year = {2004}, pages = {226–230} }
Vasquez, M. & Habet, D.	Algorithmes Complet et Incomplet pour la Coloration des Graphes de Reines [BibTeX]	2004	Programmation en Logique avec Contraintes (JFPLC2004)	inproceedings
BibTeX: @inproceedings{Vasquez2004c, author = {M. Vasquez and D. Habet}, title = {Algorithmes Complet et Incomplet pour la Coloration des Graphes de Reines}, booktitle = {Programmation en Logique avec Contraintes (JFPLC2004)}, year = {2004} }
Velucchi, M.	For Me, This Is the Best Chess-Puzzle! Non-Dominating Queens Problem [BibTeX]	1998		misc	URL
BibTeX: @misc{Velucchi1998, author = {M. Velucchi}, title = {For Me, This Is the Best Chess-Puzzle! Non-Dominating Queens Problem}, year = {1998}, url = {http://anduin.eldar.org/~problemi/papers.html} }
Velucchi, M.	Different Dispositions on the Chessboard [BibTeX]	1998		misc	URL
BibTeX: @misc{Velucchi1998a, author = {M. Velucchi}, title = {Different Dispositions on the Chessboard}, year = {1998}, url = {http://anduin.eldar.org/~problemi/papers.html} }
Wagner, R. & Geist, R.	The Crippled Queen Placement Problem [Abstract] [BibTeX]	1984	Science of Computer Programming Vol. 4, pp. 221-248	article	DOI
Abstract: We describe the outcome of various combinations of choices made by individuals in the solution of a non-trivial combinatorial problem on a computer. The programs which result are analyzed with respect to execution speed, design time, and difficulty in debugging. The solutions obtained vary dramatically as a result of choices made in the overall design of the solution. Choices made at lower levels in the top-down tree of design choices seem to have less effect on the parameters analyzed. A tradeoff between mathematical effort in algorithm design, and program speed is evident, since some solutions required solution-time which grows exponentially with the case size, while another solution presented here gives a closed-form expression for the required answers for all large cases.
BibTeX: @article{Wagner1984, author = {R.A. Wagner and R.H. Geist}, title = {The Crippled Queen Placement Problem}, journal = {Science of Computer Programming}, year = {1984}, volume = {4}, pages = {221-248}, doi = {http://dx.doi.org/10.1016/0167-6423(84)90001-7} }
Wang, C.-N., Yang, S.-W., Liu, C.-M. & Chiang, T.	A Hierarchical $N$-Queen Decimation Lattice and Hardware Architecture for Motion Estimation [Abstract] [BibTeX]	2004	IEEE Transactions on Circuits and Systems for Video Technology Vol. 14, pp. 429-440	article	DOI
Abstract: A subsampling structure, an $N$-Queen lattice, for spatially decimating a block of pixels is presented. Despite its use for many applications, we demonstrate that the $N$-Queen lattice can be used to speed up motion estimation with nominal loss of coding efficiency. With a simple construction, the $N$-Queen lattice characterizes the spatial features in the vertical, horizontal, and diagonal directions for both texture and edge areas. Especially in the 4-Queen case, every skipped pixel has the minimal and equal distance of unity to the selected pixel. It can be hierarchically organized for variable nonsquare block-size motion estimation. Despite the randomized lattice, we design compact data storage architecture for efficient memory access and simple hardware implementation. Our simulations show that the $N$-Queen lattice is superior to several existing sampling techniques with improvement in speed by about $N$ times and small loss in peak SNR (PSNR). The loss in PSNR is negligible for slow-motion video sequences and is less than 0.45 dB at worst for high-motion estimation sequences.
BibTeX: @article{Wang2004, author = {C.-N. Wang and S.-W. Yang and C.-M. Liu and T. Chiang}, title = {A Hierarchical $N$-Queen Decimation Lattice and Hardware Architecture for Motion Estimation}, journal = {IEEE Transactions on Circuits and Systems for Video Technology}, year = {2004}, volume = {14}, pages = {429-440}, doi = {http://dx.doi.org/10.1109/TCSVT.2004.825550} }
Wang, C.-N., Yang, S.-W., Liu, C.-M. & Chiang, T.	A Hierarchical Decimation Lattice Based on $N$-Queen with an Application for Motion Estimation [Abstract] [BibTeX]	2003	IEEE Signal Processing Letters Vol. 10, pp. 228-231	article	DOI
Abstract: We present a novel technique, $N$-queen lattice, to spatially subsample a block of pixels. Although this lattice is pertinent to many applications, we present an application to speed up motion estimation with minimal loss of coding efficiency. The $N$-queen lattice is constructed to characterize spatial features in all directions. It can be hierarchically organized for motion estimation with variable nonsquare block size. Despite the randomized lattice structure, we demonstrate that it is possible to achieve compact data storage architecture for efficient memory access and simple hardware implementation. Our simulations show that the $N$-queen lattice is superior to several existing sampling techniques with improvement in speed by about $N$ times and small loss in peak SNR.
BibTeX: @article{Wang2003, author = {C.-N. Wang and S.-W. Yang and C.-M. Liu and T. Chiang}, title = {A Hierarchical Decimation Lattice Based on $N$-Queen with an Application for Motion Estimation}, journal = {IEEE Signal Processing Letters}, year = {2003}, volume = {10}, pages = {228-231}, doi = {http://dx.doi.org/10.1109/LSP.2003.814403} }
Watkins, J.	Across the Board: The Mathematics of Chessboard Problems [BibTeX]	2004		book
BibTeX: @book{Watkins2004, author = {J. Watkins}, title = {Across the Board: The Mathematics of Chessboard Problems}, publisher = {Princeton, NJ: Princeton University Press}, year = {2004} }
Wikipedia	Eight Queens Puzzle [BibTeX]	2009		misc	URL
BibTeX: @misc{Wikipedia, author = {Wikipedia}, title = {Eight Queens Puzzle}, year = {2009}, url = {http://en.wikipedia.org/wiki/Eight_queens_puzzle} }
Wirth, N.	Algorithms + Data Structures = Programs [BibTeX]	1976		book
BibTeX: @book{Wirth1976, author = {N. Wirth}, title = {Algorithms + Data Structures = Programs}, publisher = {Prentice-Hall}, year = {1976} }
Wirth, N.	Program Development by Stepwise Refinement [Abstract] [BibTeX]	1971	Communications of the ACM Vol. 14, pp. 221-227	article	URL
Abstract: The creative art of programming---to be distinguished from coding---is usually taught by examples serving to exhibit certain techniques. It is here considered as a sequence of design decisions concerning the decomposition of tasks into subtasks and of data into data structures. The process of successive refinement of specifications is illustrated by a short but nontrivial example, from which a number of conclusions are drawn regarding the art and the instruction of programming.
BibTeX: @article{Wirth1971, author = {N. Wirth}, title = {Program Development by Stepwise Refinement}, journal = {Communications of the ACM}, year = {1971}, volume = {14}, pages = {221-227}, url = {http://doi.acm.org/10.1145/362575.362577} }
Wu, J.	A Solution to the $n$-Queens Problem [BibTeX]	1994	J. Huazhong Univ. Sci. Tech. Vol. 22, pp. 195-198	article
BibTeX: @article{Wu1994, author = {J.B. Wu}, title = {A Solution to the $n$-Queens Problem}, journal = {J. Huazhong Univ. Sci. Tech.}, year = {1994}, volume = {22}, pages = {195-198} }
Yaglom, A. & Yaglom, I.	Challenging Mathematical Problems with Elementary Solutions; Volume 1: Combinatorial Analysis and Probability Theory [BibTeX]	1964		book	URL
BibTeX: @book{Yaglom1964, author = {A.M. Yaglom and I.M. Yaglom}, title = {Challenging Mathematical Problems with Elementary Solutions; Volume 1: Combinatorial Analysis and Probability Theory}, publisher = {Holden-Day, Inc.}, year = {1964}, url = {http://www.liacs.nl/home/kosters/nqueens/papers/yaglom1964.pdf} }
Yamamoto, K., Kitamura, Y. & Yoshikura, H.	Computation of Statistical Secondary Structure of Nucleic Acids [Abstract] [BibTeX]	1984	Nucleic Acids Research Vol. 12, pp. 335-346	article
Abstract: This paper presents a computer analysis of statistical secondary structure of nucleic acids. For a given single stranded nucleic acid, we generated ``structure map" which included all the annealig structures in the sequence. The map was transformed into ``energy map" by rough approximation; here, the energy level of every pairing structure consisting of more than 2 successive nucleic acid pairs was calculated. By using the ``energy map", the probability of occurrence of each annealed structure was computed, i.e., the structure was computed statistically. The basis of computation was the 8-queen problem in the chess game. The validity of our computer programme was checked by computing tRNA structure which has been well established. Successful application of this programme to small nuclear RNAs of various origins is demonstrated.
BibTeX: @article{Yamamoto1984, author = {K. Yamamoto and Y. Kitamura and H. Yoshikura}, title = {Computation of Statistical Secondary Structure of Nucleic Acids}, journal = {Nucleic Acids Research}, year = {1984}, volume = {12}, pages = {335-346} }
Yang, S.-W., Wang, C.-N., Liu, C.-M. & Chiang, T.	Fast Motion Estimation Using $N$-Queen Pixel Decimation [Abstract] [BibTeX]	2001	Vol. 2195Advances in Multimedia Information Processing (PCM 2001), pp. 126-133	inproceedings	DOI
Abstract: We present a technique to improve the speed of block motion estimation using only a subset of pixels from a block to evaluate the distortion with minimal loss of coding efficiency. To select such a subset we use a special sub-sampling structure, $N$-queen pattern. The $N$-queen pattern can characterize the spatial information in the vertical, horizontal and diagonal directions for both texture and edge features. In the 4-queen case, it has a special property that every skipped pixel has the minimal and equal distance of one to the selected pixel. Despite of the randomized pattern, our technique has compact data storage architecture. Our results show that the pixel decimation of $N$-queen patterns improves the speed by about $N$ times with small loss in PSNR. The loss in PSNR is negligible for slow motion video sequence and has 0.45 dB loss in PSNR at worst for high motion video sequence.
BibTeX: @inproceedings{Yang2001, author = {S.-W. Yang and C.-N. Wang and C.-M. Liu and T. Chiang}, title = {Fast Motion Estimation Using $N$-Queen Pixel Decimation}, booktitle = {Advances in Multimedia Information Processing (PCM 2001)}, publisher = {Springer-Verlag, Berlin}, year = {2001}, volume = {2195}, pages = {126-133}, doi = {http://dx.doi.org/10.1007/3-540-45453-5} }
Yoshio, H., Baba, T., Funabiki, N. & Nishikawa, S.	Proposal of an $N$-Parallel Computation Method for a Neural Network for the $n$-Queens Problem [BibTeX]	1997	Electronics and Communications in Japan Vol. 80, pp. 12-20	article
BibTeX: @article{Yoshio1997, author = {H. Yoshio and T. Baba and N. Funabiki and S. Nishikawa}, title = {Proposal of an $N$-Parallel Computation Method for a Neural Network for the $n$-Queens Problem}, journal = {Electronics and Communications in Japan}, year = {1997}, volume = {80}, pages = {12-20} }
Yuen, C. & Feng, M.	Breadth-First Search in the Eight Queens Problem [Abstract] [BibTeX]	1994	ACM SIGPLAN Notices Vol. 29, pp. 51-55	article	DOI
Abstract: The Eight Queens Problem is used to illustrate some different approaches to recursive programming and parallel processing.
BibTeX: @article{Yuen1994, author = {C.K. Yuen and M.D. Feng}, title = {Breadth-First Search in the Eight Queens Problem}, journal = {ACM SIGPLAN Notices}, year = {1994}, volume = {29}, pages = {51-55}, doi = {http://dx.doi.org/10.1145/185009.185019} }
Zeng, C. & Gu, T.	A Novel Assembly Evolutionary Algorithm for $n$-Queens Problem [Abstract] [BibTeX]	2007	Computational Intelligence and Security Workshops	article	DOI
Abstract: Individuals in nowadays evolutionary algorithms for $n$-Queens problem do not satisfy some basic constraint conditions. Motivated by self-assembly computing, a novel assembly evolutionary algorithm for $n$-Queens problem is presented. Each individual is made up of assembly-parts, assembly-seeds and status information. Some important notions and rules regarding the novel assembly evolutionary algorithm are discussed. Experimental results show that the algorithm finds a solution faster than other latest evolutionary algorithms.
BibTeX: @article{Zeng2007, author = {C. Zeng and T. Gu}, title = {A Novel Assembly Evolutionary Algorithm for $n$-Queens Problem}, journal = {Computational Intelligence and Security Workshops}, year = {2007}, doi = {http://dx.doi.org/10.1109/CISW.2007.4425472} }
Zhang, C. & Ma, J.	Counting Solutions for the $n$-Queens and Latin Square Problems by Efficient Monte Carlo Simulations [Abstract] [BibTeX]	2009	Pysical Review E Vol. 79(016703)	article	DOI
Abstract: We apply Monte Carlo simulations to count the numbers of solutions of two well-known combinatorial problems: the $n$-Queens problem and Latin square problem. The original system is first converted to a general thermodynamic system, from which the number of solutions of the original system is obtained by using the method of computing the partition function. Collective moves are used to further accelerate sampling: swap moves are used in the $n$-Queens problem and a cluster algorithm is developed for the Latin squares. The method can handle systems of $10^4$ degrees of freedom with more than $10^10000$ solutions. We also observe a distinct finite size effect of the Latin square system: its heat capacity gradually develops a second maximum as the size increases.
BibTeX: @article{Zhang2008, author = {C. Zhang and J. Ma}, title = {Counting Solutions for the $n$-Queens and Latin Square Problems by Efficient Monte Carlo Simulations}, journal = {Pysical Review E}, year = {2009}, volume = {79}, number = {016703}, doi = {http://dx.doi.org/10.1103/PhysRevE.79.016703} }
Zhao, K.	The Combinatorics of Chessboards [BibTeX]	1998	School: City University of New York	phdthesis
BibTeX: @phdthesis{Zhao1998, author = {K. Zhao}, title = {The Combinatorics of Chessboards}, school = {City University of New York}, year = {1998} }

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