Construction and Enumeration of Pandiagonal Magic Squares of Order \(n\) using Step Method

A Pandiagonal magic square (PMS) of order \(n\) is a square matrix which is an arrangement of integers \(0, 1, \ldots, n^2-1\) such that the sums of each row, each column, and each extended diagonal are the same. In this paper, we use the Step method to construct a PMS of order \(n\) for each \(n > 3\) and \(n\) is not singly-even. We discuss how to enumerate the number of PMSs of order \(n\) constructed by the Step method, and we get the number of nonequivalent PMSs of order \(8\) with the top left cell \(0\) is \(4,176,000\) and the number of nonequivalent PMSs of order \(9\) with the top left cell \(0\) is \(1,492,992\).