  The transfer matrix is a powerful technique that can be applied to
statistical mechanics systems as for example in the calculus of the entropy of
the ice model. One interesting way to study such systems is to map it onto a
3-color problem. In this paper, we explicitly build the transfer matrix for the
3-color problem in order to calculate the number of possible configurations for
finite systems with free, periodic in one direction and toroidal boundary
conditions (periodic in both directions)
