The smallest magic square is, of course, 1 by 1. There is no 2 by 2 magic square. There are, however, 3 by 3 magic squares. According to the Guiness Book of Records, the largest known magic square is 3001 by 3001 and was computed by Louis Caya (Sainte-Foy, Canada) in 1994.

Many examples of magic squares can be found online.

The smallest anti-magic square is 3 by 3. Constructions exist for building anti-magic squares of larger sizes, but these do not give all possible squares.

Eliminating symmetry would appear to be very important in these problems, especially when counting solutions or proving none exits.

Results on can be found in [PesantQZ12].