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Consider a finite toroidal lattice of fixed dimension with the vertices colored by a finite number of colors. The lattice has an "energy" defined as the sum of energies of the edges. A challenging problem is to find the infimum of the average energies of these colorings over all finite lattices. In particular, can we find a finite lattice coloring which minimizes the average energy? This leads to some interesting connections in statistical mechanics, combinatorics, and discrete math. We discuss some simple models for which there exists a finite coloring achieving the minimum average energy. Finally, if the question above is true in general, this implies limits on how these models can be used to describe some statistical mechanics and combinatorics problems.