The search for multi-magic squares led to further results during the last years. Meanwhile tetramagic squares of orders 256 and 243 are known. The smallest orders of pentamagic squares are 1024 and 729.
Meanwhile, the list of n-multimagic squares has been dramatically extended. Naturally, the magnitude of these squares is very large. Harm Derksen, Christian Eggermont and Arno van den Essen published an article in 2005, in which they proved that it is possible to construct an n-multi-magic square for each n. This is, what Gaston Tarry suspected in 1906.
An overview of the current state of higher multi-magic squares can be found on the pages of multimagic Christian Boyer.