Gauss and the eight queens problem: A study in miniature of the propagation of historical error

Campbell, Paul J.

doi:10.1016/0315-0860(77)90076-3

Cited by 30 publications

(12 citation statements)

References 3 publications

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“…However Gauss only found 72 of the solutions before becoming aware of Nauck's 92 solutions. According to Campbell [13], in these letters Gauss reformulated the 8-queens problem as an arithmetic one and related it to the representation of complex numbers, though seemingly this got Gauss neither further in enumerating the solutions, nor in proving an upper bound. Again, various different attributions are made as to who first proposed the generalised n-queens variation of the problem.…”

Section: The N-queens Problemmentioning

confidence: 99%

The $n$-queens problem

Bowtell,

Keevash

2021

Preprint

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Chapter 1. Introduction 1.1. The n-queens problem 1.2. Graphical reformulations of the n-queens problem 1.3. Semi-queens 1.4. Strategy overview Chapter 2. Definitions, notation and preliminaries 2.1. Bachmann-Landau Notation 2.2. Inequalities and Bounds Chapter 3. Key structural properties of the toroidal n-queens hypergraph 3.1. Proof Overview 3.2. The lattice of T 3.3. Zero-sum configurations 3.4. Degree-type conditions 3.5. Wrap-around edges 3.6. Facts about T Chapter 4. The absorber 4.1. Finding an integral decomposition for L * 4.2. Building and using the absorber 4.3. The bounded integral decomposition lemma Chapter 5. The random greedy count 5.1. Details of the process 5.2. Proof of the main result 5.3. Reaching H Chapter 6. The iterative matching process 6.1. Overview 6.2. The weight shuffle 6.3. Using the random matching tool 6.4. Reaching L * 6.5. Initial steps Chapter 7. Classical queens and concluding remarks 7.1. The classical n-queens problem 7.2. Concluding remarks Bibliography iii

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Section: The N-queens Problemmentioning

confidence: 99%

The $n$-queens problem

Bowtell,

Keevash

2021

Preprint

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“…The n-Queens problem has an extraordinary history for such an apparently unassuming problem, both generally and inside Artificial Intelligence. Formerly, and incorrectly, attributed to Gauss, the problem's history was clarified by Campbell [1977]. The 8-Queens problem was introduced by Bezzel [1848] and by Nauck [1850] (possibly independently).…”

Section: History Of N-queensmentioning

confidence: 99%

Complexity of n-Queens Completion (Extended Abstract)

Gent

Jefferson

Nightingale

2018

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence

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The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem. We introduce generators of random instances for n-Queens Completion and the closely related Blocked n-Queens and Excluded Diagonals Problem. We describe three solvers for these problems, and empirically analyse the hardness of randomly generated instances. For Blocked n-Queens and the Excluded Diagonals Problem, we show the existence of a phase transition associated with hard instances as has been seen in other NP-Complete problems, but a natural generator for n-Queens Completion did not generate consistently hard instances. The significance of this work is that the n-Queens problem has been very widely used as a benchmark in Artificial Intelligence, but conclusions on it are often disputable because of the simple complexity of the decision problem. Our results give alternative benchmarks which are hard theoretically and empirically, but for which solving techniques designed for n-Queens need minimal or no change.

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“…Carl Friedrich Gauss considered this problem in its original form on an 8 by 8 chess board (Campbell, 1977). Given that a queen can move horizontally in its row, it follows that we can have at most one queen in each row.…”

Section: Local Versus Global Searchmentioning

confidence: 99%