The Existence of Irrational Diagonally Ordered Magic Squares

Abstract

Let $M=(m_{i,j})$ be a magic square, where $0\le m_{i,j}\le n^2-1$, $0\le i, j\le n-1$. M is called diagonally ordered if both the main diagonal and the back diagonal, when traversed from left to right, have strictly increasing values. Let $M=nA+B$, where $A=(a_{i,j})$, $B=(b_{i,j})$, $0\le a_{i,j}, \ b_{i,j}\le n-1$ for $0\le i, j\le n-1$. M is called rational if both A and B possess the property that the sums of the n numbers in every row and every column are the same; otherwise, M is said to be irrational. In this paper, a pair of weakly diagonally ordered irrational orthogonal matrices (WDOIOM for short) is introduced to construct an irrational diagonally ordered magic square (IDOMS). It is proved that there exists a WDOIOM(n) for each positive integer $n\ge 5$, and there does not exist a WDOIOM(n) for $n\in \{2,3,4\}$. Consequently, it is proved that there exists an IDOMS(n) for each positive integer $n\ge 5$, and there does not exist an IDOMS(n) for $n\in \{2,3,4\}$.