The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

Let N = {0, 1, · · ·, n − 1}. A strongly idempotent self-orthogonal row Latin magic array of order n (SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties: (1) each row of M is a permutation of N, and at least one column is not a permutation of N; (2) the sums of the n numbers in every row and every column are the same; (3) M is orthogonal to its transpose; (4) the main diagonal and the back diagonal of M are 0, 1, · · ·, n − 1 from left to right. In this paper, it is proved that an SISORLMA(n) exists if and only if n ∉ {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ∉ {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠ 2.