Diagonally ordered orthogonal Latin squares and their related elementary diagonally ordered magic squares

Abstract

Let N = {0, 1,⋯, n - 1}. A pair of orthogonal Latin squares A, B of order n is called diagonally ordered (DOOLS(n) for short) if the following properties are satisfied: (1) the diagonal and back diagonal of A, B have non-decreasing values when traversed from left to right; (2) the sum of the n numbers in main diagonal and back diagonal of A, B is n(n-1)/2. In this paper, it is proved that a DOOLS(n) exists for each positive integer n with two exceptions of n = 2, 6 and two possible exceptions of n = 22, 26; a strong symmetric DOOLS(n) (SSDOOLS(n) for short) exists if and only if n = 0, 1, 3 (mod 4). As applications, it is proved that an elementary diagonally ordered magic square of order n exists for each positive integer n with two exceptions of n = 2, 6 and two possible exceptions of n = 22, 26; also it is proved that an elementary symmetrical diagonally ordered magic square exists if and only if n = 0, 1, 3 (mod 4).