A construction of regular magic squares of odd order

A magic square is an n×nn×n array of numbers whose rows, columns, and the two diagonals sum to μ  . A regular magic square satisfies the condition that the entries symmetrically placed with respect to the center sum to 2μn. Using circulant matrices we describe a construction of regular classical magic squares that are nonsingular for all odd orders. A similar construction is given that produces regular classical magic squares that are singular for odd composite orders. This paper is an extension of [3].