The modular \(n\)-queens problem in higher dimensions
type: Item
instance of: scholarly article
title: The modular \(n\)-queens problem in higher dimensions
zbMATH Open document ID: 0837.05009
DOI: 10.1016/0012-365X(94)00161-5
author: Scott P. Nudelman
published in: Discrete Mathematics
publication date: 1996-05-13T00:00:00Z
review text: A modular chessboard is a square board with opposite edges identified like a torus. A queen moves on such a chessboard like in an ordinary chessboard but it may continue along the extended diagonals. The previous results on placements of non-attacking queens are summarized in Theorem 1: (a) \(M(n,2) = n\) if \(\text{gcd} (n,6) = 1\), (b) \(M(n,2) = n - 1\) if \(\text{gcd} (n,12) = 2\), (c) \(M(n,2) = n - 2\) otherwise, where \(M(n,d)\) is the maximum number of non-attacking queens on a \(d\)-dimensional modular chessboard. The author studied the higher dimensional cases and proved Theorem 2: (a) \(M(n,d) = n\) if \(\text{gcd} (n, (2^d - 1)!) = 1\), (b) \(M(n,d) < n\) if \(\text{gcd} (n, (2^d - 1)!) > 1\), (c) \(M(n,d) = 1\) if \(n \leq 2^d - 1\), (d) if \(\text{gcd} (n, (2^d - 1)!) > 1\), there are no \(n\) non- attacking queens on a straight line.
reviewed by: Kequan Ding
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 825350
zbMATH Keywords: chessboard
zbMATH Keywords: queen
zbMATH Keywords: higher dimensional
Wikidata QID: Q56047937
MaRDI profile type: Publication
cites work: On the modular \(n\)-queen problem
cites work: The modular n-queen problem
cites work: The modular n-queen problem. II
full work available at URL: 0012-365x(94)00161-5
OpenAlex ID: W2065696590
DBLP publication ID: journals/dm/Nudelman95
description: scientific article; zbMATH DE number 825350