As Cipra states it, ``Keller's conjecture is concerned with the problem of filling n-dimensional space with `cubes' of equal size, leaving no gaps and no overlaps [Cip93].'' The filling of the space is called a tiling. The ``cubes'' are generalizations of the common three-dimensional cube everyone is familiar with. A cube in dimension two would be a square and a tiling of squares can be seen in any kitchen floor. The three dimensional cube tiling is something like a pile of bricks. In greater than dimension three it is very difficult to visualize what a tiling would look like. The requirements for a tiling, in any dimension, are as follows: there is no empty space left between the cubes, no holes in the brick wall or gaps in the floor tiles; and no cubes cover each other, no floor tile is on top of another or no brick is inside another. Keller conjectured that in any tiling of n-dimensional cubes there exists two cubes having a complete (n-1)-dimensional side in common [LS92]. For our floor tiles this means that one pair of tiles in the entire floor must share at least one full edge, and with the bricks a brick in the pile must have one whole face touching the whole face of another brick.
It has been shown for dimension n when 
Keller's conjecture
is true [CS90] but when 
it has been shown to be false
[LS92]. This leaves dimensions 7, 8, and 9 unknown with respect
to the conjecture. To solve Keller's conjecture it is necessary to
find the status of those last three dimensions. Cipra suggests a
computer could find the solution, ``A computer search through a finite
number of possible ways of filling each space could settle the problem
[Cip93].'' As I will describe later, this computer search is
known to take a prohibitively long amount of time.