If replacing each number by its square or cube in a Magic Square produces another Magic Square, the square is said to be a trimagic square. Trimagic squares of order 32, 64, 81, and 128 are known. Tarry gave a method for constructing a trimagic square of order 128, Cazalas a method for trimagic squares of orders 64 and 81, and R. V. Heath a method for constructing an order 64 trimagic square which is different from Cazalas's (Kraitchik 1942).

Trimagic squares are also called Trebly Magic Squares, and are 3-Multimagic Squares.

**References**

Ball, W. W. R. and Coxeter, H. S. M. *Mathematical Recreations and Essays, 13th ed.* New York: Dover, pp. 212-213, 1987.

Kraitchik, M. ``Multimagic Squares.'' §7.10 in *Mathematical Recreations.* New York: W. W. Norton, pp. 176-178, 1942.

© 1996-9

1999-05-26