Enumeration and construction of pandiagonal Latin squares of prime order

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

“…We have also given additional partial solutions to Hedayat's problem of finding non-cyclic pandiagonal Latin squares to those already given by Atkin, Hay and Larson in [1]. However, the columns of the pandiagonal Latin squares we make are still cyclic permutations of the first column.…”

“…Moreover, Hedayat in [6] notes that the pandiagonal Latin squares constructed by his method are cyclic, and that finding a method for constructing non-cyclic pandiagonal Latin squares is an unsolved problem. However, Atkin, Hay and Larson in [1] do in fact construct pandiagonal Latin squares which are semi-cyclic. Atkin, Hay and Larson's article appears to be the only one in the literature which gives semi-cyclic pandiagonal Latin squares (there are none which give non-cyclic pandiagonal Latin squares).…”

“…An n × n associative magic square is a magic square A = (a ij ) such that for all i and j it holds that a ij + a n−i−1,n−j−1 = c for a fixed c. A modular n-queens solution is a placement of n queens on the n × n modular chessboard such that no two queens share a row, column, modular sum diagonal, or modular difference diagonal, that is, a placement of n non-attacking queens on the modular chessboard for which opposite sides are identified like a torus. An n-queens solution can be represented as a permutation g of Z n , of the columns into the rows; we can further express g by g = (g(0), g (1), . .…”

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 [6] Hedayat proposed a method for construction of cyclic pandiagonal Latin squares, and since that finding a systematic method for constructing non-cyclic pandiagonal Latin squares remains an unsolved problem. 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].…”