A survey of known results and research areas for \(n\)-queens. (English) Zbl 1228.05002

Bell, Jordan; Stevens, Brett

Discrete Math. 309, No. 1, 1-31 (2009).

Summary: We survey known results for the \(n\)-queens problem of placing \(n\) nonattacking queens on an \(n\times n\) 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 \(n\times n\) board for \(n>3\) is by E. Pauls [“Das Maximalproblem der Damen auf dem Schachbrette. II,” Deutsche Schachzeitung. Organ für das Gesamte Schachleben 29, No. 9, 257–267 (1874)], rather than by W. Ahrens [Mathematische Unterhaltungen und Spiele, Leipzig: B.G. Teubner (1901; JFM 31.0220.02), Chapter IX] 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.

MSC:

05-02	Research exposition (monographs, survey articles) pertaining to combinatorics
05-03	History of combinatorics
01A55	History of mathematics in the 19th century
05C85	Graph algorithms (graph-theoretic aspects)

Keywords:

\(n\)-queens problem; modular \(n\)-queens problem; queens graph; chessboard graph; chessboard problems

Citations:

JFM 31.0220.02

Software:

OEIS; FreeSquares

References:

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