Keller's Conjecture states that in any tiling of n-dimensional cubes
there will be two cubes sharing a complete (n-1)-dimensional face.
The Conjecture has been proven when 
and disproved when
, leaving the status of dimensions 7, 8, and 9 unknown. The
Keller Conjecture can be described in terms of the existence of large
cliques in certain graphs. A search of these graphs for the cliques
could prove the conjecture one way or the other. This search is known
to take a long time to perform, but a new algorithms exists that may
be able to do the search quickly.