Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n ‐queens solutions

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