Keller's Conjecture

Keller conjectured that tiling an $n$-D space with $n$-D Hypercubes of equal size yields an arrangement in which at least two hypercubes have an entire $(n-1)$-D ``side'' in common. The Conjecture has been proven true for $n=1$ to 6, but disproven for $n\geq 10$.

References

Cipra, B. ``If You Can't See It, Don't Believe It.'' Science 259, 26-27, 1993.

Cipra, B. What's Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 24, 1993.