The Resolution of Keller's Conjecture

We consider three graphs, G7,3, G7,4, and G7,6, related to Keller's conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size 27 = 128. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of ℝ7 contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of ℝ8 exists (which we also verify), this completely resolves Keller's conjecture.