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.


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