Let d≥ 1 be an integer. A unit cube in R^d is a translate of [0,1)^d, and a cube tiling is a family of pairwise disjoint unit cubes whose union is R^d. A tiling is faceshare-free if no two distinct cubes share a complete (d-1)-dimensional face. Keller's conjecture. For all integers d≥ 1, there does not exist a faceshare-free tiling of R^d. Equivalently, every tiling of d-dimensional space by translates of the unit cube contains a pair of cubes that share a complete (d-1)-dimensional face. The conjecture is false in dimension 8, while the paper establishes that it is true in dimension 7; hence the conjecture is resolved.

Keller's conjecture on face-sharing pairs in cube tilings

Let $d\geq 1$ be an integer. A unit cube in $\mathbb{R}^d$ is a translate of $[0,1)^d$ , and a cube tiling is a family of pairwise disjoint unit cubes whose union is $\mathbb{R}^d$ . A tiling is faceshare-free if no two distinct cubes share a complete $(d-1)$ -dimensional face.

Keller's conjecture. For all integers $d\geq 1$ , there does not exist a faceshare-free tiling of $\mathbb{R}^d$ .

Equivalently, every tiling of $d$ -dimensional space by translates of the unit cube contains a pair of cubes that share a complete $(d-1)$ -dimensional face. The conjecture is false in dimension $8$ , while the paper establishes that it is true in dimension $7$ ; hence the conjecture is resolved.

Keller posed the conjecture in 1930, asking whether every tiling by equal cubes contains two cubes sharing a whole face. The final unresolved case was dimension seven.

Known results

Perron, 1940: proved the conjecture for dimensions $d\leq 6$ .
Lagarias and Shor, 1992: constructed counterexamples in dimensions $d\geq 10$ .
Mackey, 2002: constructed a counterexample in dimension $8$ , implying one in dimension $9$ .
Kisielewicz and Łysakowska, 2017: reduced the remaining seven-dimensional analysis to the case $r^+(T)=3$ .

2019–2020 final resolution

Brakensiek, Heule, Mackey, and Narváez ruled out the required clique of size $2^7=128$ in the relevant Keller graphs. Their unsatisfiability certificates were checked by a formally verified checker, proving the seven-dimensional case; the same work verifies a faceshare-free tiling in dimension $8$ . Later literature records the resulting status without reporting a gap or objection.

Current status (as of August 2026): Keller's conjecture is settled, true for $d\leq 7$ and false for $d\geq 8$ .

Keller's conjecture on face-sharing pairs in cube tilings

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Progress summary

Solutions 0