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

Abstract

Abstract: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.

Constructing non-cyclic pandiagonal Latin squares of prime orders

“…According to Theorem 2.7, there exist p − 1 = 12 pairs of such unordered (a, b), which are (1,4), (1, 10), (2,7), (2,8), (3,4), (3,12), (5,7), (5,11), (6,8), (6,11), (9, 10), (9,12) .…”

“…In [1] Atkin, Hay and Larson described a method for constructing pandiagonal Latin squares which are semi-cyclic. Recently another method for constructing semi-cyclic pandiagonal Latin squares is given by Bell and Stevens [2]. In [4] the authors of the current manuscript described a recursive algorithm for constructing non-cyclic pandiagonal Latin squares of any given order n, where n is a positive composite integer not divisible by 2 or 3.…”

Constructing Pandiagonal Latin Squares from Linear Cellular Automaton on Elementary Abelian Groups

“…A Latin square is called pandiagonal if each entry appears only once in the main diagonal, the anti diagonal and all broken diagonals in both directions. These squares have several applications such as construction of pandiagonal magic squares [3], solutions of n-queens problem [4], and construction of statistical designs [12], where in this context they are known as Knut Vik designs, due to Vik [16]. A pandiagonal Latin square is cyclic if each row is a cyclic permutation of the first row and each column is a cyclic permutation of the first column.…”

“…In [1] Atkin, Hay, and Larson described a method for constructing pandiagonal Latin squares that are semi-cyclic. Recently, another method for constructing semi-cyclic pandiagonal Latin squares is proposed by Bell and Stevens [3]. Up to now, finding a direct method for constructing noncyclic pandiagonal Latin squares was an unsolved problem.…”

Polynomial modular n-queens solutions

“…Modular n-queens solutions are related to certain combinatorial structures, in particular Latin squares (cf. [1]). This paper gives three constructions of modular n-queens solutions using permutation polynomials of Z/n.…”