A brief question about the "Eight Queens" Puzzle

Question

The classical Eight Queens puzzle asks whether it is possible to arrange $ 8 $ queens on an $ 8 \times 8 $ chess board, so that no two queens attack each other.

It is well-known that such configurations exist (for an $8 \times 8$ board there are $92$ of them), which can be computationally obtained - the discussion proceeds, e.g. even for boards of size $n \times n$, higher dimensional boards, combinations of other chess pieces instead of queens only, etc.

I am not an expert, but it seems to me that solutions are obtained, more or less through a "clever brute-force".

I was wondering whether there are similar related questions or generalizations of the game, where one doesn't really need a computer. For example, the Wikipedia article on the "Eight Queens" puzzle claims that Pólya studied the $n$ queens problem on a toroidal ("donut-shaped") board and showed that there is a solution on an $n \times n$ board if and only if $n$ is not divisible by $2$ or $3$.

In this direction, a reference to a survey, for example, would be great. Thanks!

Answer 1

This might (not certain?) be the most recent summary (OP: "a reference to a survey"):

Bell, Jordan, and Brett Stevens. "A survey of known results and research areas for $n$-queens." Discrete Mathematics. 309.1 (2009): 1--31. (Elsevier link.)

Abstract. In this paper 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 $n$ non-attacking queens can always be placed on an $n \times n$ 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.

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They pose & gather no fewer than 30 conjectures, one of which is:

Conjecture 28 (Klarner). For all $n>6$, there exists a solution to the reflecting queens problem. ("a 'reflecting strip' at the top (i.e., in row $0$), that queens bounce off.").