By using the finite size extrapolation method and combined with a Monte Carlo simulation we have calculated the zero temperature entropy of the q-state Potts Antiferromagnet in two and three dimensions which is identical to the q-color problem in two and three dimensions. The model is laid on a hypercubic lattice. When q=3 (in two dimensions) the result is in good agreement with Lieb’s exact answer. When q>3 (in two and three dimensions) the results are in strong support of Mattis’ recent conjecture for the q-color problem. This method can also treat the d>3 cases without serious difficulties.