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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.
See also Bimagic Square, Magic Square, Multimagic Square
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.