The Resolution of Keller’s Conjecture

Abstract

We consider three graphs, $$G_{7,3}$$ , $$G_{7,4}$$ , and $$G_{7,6}$$ , 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 $$2^7 = 128$$ . 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 $$\mathbb {R}^7$$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $$\mathbb {R}^8$$ exists (which we also verify), this completely resolves Keller’s conjecture.