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