Constructing and Enumerating of Magic Squares

Abstract

A magic square of order n, where n is a positive integer, is an n × n square table , say A, filled with distinct positive numbers 1, 2, . . . , n2 such that all cells of A are distinct and the sum of the numbers in each row, column and diagonal of A is equal. Let M(n, s) be the set of all n × n matrices with entries 0 or 1, say T, such that the number of 1 in every row and every column of T is equal to s. In this paper we introduce a new method for constructing magic squares of order 4k, where k is a positive integer. We show that the number of magic squares of order 4k is at least |M(2k, k)|. In particular, we prove that the number of magic squares of order 4k is at least 1 2 (2k k )2.