New Infinite Classes for Normal Trimagic Squares of Even Orders Using Row–Square Magic Rectangles

As matrix representations of magic labelings of related hypergraphs, magic squares and their various variants have been applied to many domains. Among various subclasses, trimagic squares have been investigated for over a hundred years. The existence problem of trimagic squares with singly even orders and orders 16n has been solved completely. However, very little is known about the existence of trimagic squares with other even orders, except for only three examples and three families. We constructed normal trimagic squares by using product constructions, row–square magic rectangles, and trimagic pairs of orthogonal diagonal Latin squares. We gave a new product construction: for positive integers p, q, and r having the same parity, other than 1, 2, 3, or 6, if normal p × q and r × q row–square magic rectangles exist, then a normal trimagic square with order pqr exists. As its application, we constructed normal trimagic squares of orders 8q3 and 8pqr for all odd integers q not less than 7 and p, r ∈ {7, 11, 13, 17, 19, 23, 29, 31, 37}. Our construction can easily be extended to construct multimagic squares

On the Existence of a Normal Trimagic Square of Order 16n

The study of magic squares has a long history, and magic squares have been applied to many mathematical fields. In this paper, we give a complete solution to the existence of normal trimagic squares of all orders 16n. In particular, we obtain a unified solution for the normal trimagic square of order 16n for n>3 by means of set partitions, semibimagic squares, Latin squares, and new product construction. Since there exist normal trimagic squares of orders 16, 32, and 48, we prove that there exists a normal trimagic square of order 16n for every positive integer n