I discuss a model of spinless fermions hopping on the square lattice where the repulsive interactions are tuned to give the model a supersymmetry. As a result, there is an extensive ground-state entropy (i.e. flat directions on the Fermi surface). I present substantial evidence that counting certain types of rhombus tilings yields the exact number of ground states. This behavior is akin to that of frustrated magnets, but the exact degeneracy persists in the quantum system.