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Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n ‐queens solutions
Author(s) -
Bell Jordan,
Stevens Brett
Publication year - 2007
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.20143
Subject(s) - latin square , magic square , mathematics , combinatorics , orthogonal array , modular design , diagonal , construct (python library) , least squares function approximation , statistics , computer science , geometry , chemistry , rumen , food science , taguchi methods , fermentation , programming language , operating system , estimator
In this article, we show how to construct pairs of orthogonal pandiagonal Latin squares and panmagic squares from certain types of modular n ‐queens solutions. We prove that when these modular n ‐queens solutions are symmetric, the panmagic squares thus constructed will be associative, where for an n × n associative magic square A = ( a ij ), for all i and j it holds that a ij + a n − i −1, n − j −1 = c for a fixed c . We further show how to construct orthogonal Latin squares whose modular difference diagonals are Latin from any modular n ‐queens solution. As well, we analyze constructing orthogonal pandiagonal Latin squares from particular classes of non‐linear modular n ‐queens solutions. These pandiagonal Latin squares are not row cyclic, giving a partial solution to a problem of Hedayat. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 221–234, 2007