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The \(n\)-queens problem in higher dimensions. (English) Zbl 1130.05002

The paper considers the \(d\)-dimensional chessboard, where the fields are \((q_1,q_2,\dots,q_d)\) with \(q_i\in\{0,1,\dots,d-1\}\). The attack lines of a queen placed at \((q_1,q_2,\dots,q_d)\) are described as solutions to the system of equations \(\pm(x_1-q_1)=\pm(x_2-q_2)=\dots=\pm(x_d-q_d)\) with a any fixed choices of signes, i.e. there are \(2^{d-1}\) different sets of equations. Later Lemma 2.1 gives a different description to the attack lines of a queen, according to which lines defined by a vector whose every coordinate is \(0\) or \(\pm 1\) are the attack lines. Some elementary facts are observed.

MSC:

05A15 Exact enumeration problems, generating functions
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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