The Resolution of Keller’s Conjecture

We consider three graphs, [Formula: see text], [Formula: see text], and [Formula: see text], 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 [Formula: see text]. 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 [Formula: see text] contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of [Formula: see text] exists (which we also verify), this completely resolves Keller’s conjecture.