  In this paper, we present the problem of counting magic squares and we focus
on the case of multiplicative magic squares of order 4. We give the exact
number of normal multiplicative magic squares of order 4 with an original and
complete proof, pointing out the role of the action of the symmetric group.
Moreover, we provide a new representation for magic squares of order 4. Such
representation allows the construction of magic squares in a very simple way,
using essentially only five particular 4X4 matrices.
