The Existence of Irrational Diagonally Ordered Magic Squares

Abstract

Let $$M=(m_{i,j})$$M=(mi,j) be a magic square, where $$0\le m_{i,j}\le n^2-1$$0≤mi,j≤n2-1, $$0\le i, j\le n-1$$0≤i,j≤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 $$M=nA+B$$M=nA+B, where $$A=(a_{i,j})$$A=(ai,j), $$B=(b_{i,j})$$B=(bi,j), $$0\le a_{i,j}, \ b_{i,j}\le n-1$$0≤ai,j,bi,j≤n-1 for $$0\le i, j\le n-1$$0≤i,j≤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 $$n\ge 5$$nź5, and there does not exist a WDOIOM(n) for $$n\in \{2,3,4\}$$nź{2,3,4}. Consequently, it is proved that there exists an IDOMS(n) for each positive integer $$n\ge 5$$nź5, and there does not exist an IDOMS(n) for $$n\in \{2,3,4\}$$nź{2,3,4}.