Keller's cube-tiling conjecture is false in high dimensions

O. H. Keller conjectured in 1930 that in any tiling of ${\mathbb{R}^n}$ by unit n-cubes there exist two of them having a complete facet in common. O. Perron proved this conjecture for $n \leq 6$. We show that for all $n \geq 10$ there exists a tiling of ${\mathbb{R}^n}$ by unit n-cubes such that no two n-cubes have a complete facet in common.