Before we begin to understand a possible way to solve Keller's conjecture we should lay some ground work so the method is understood. First we need to define a graph. A graph is a collection of nodes and arcs. The arcs determine whether two nodes are adjacent, or connected. A map is a simple graph; the cities are the nodes, and the roads between cities are the arcs. A clique is a graph wherein every node in the graph is connected by an arc to every other node in the graph. The size of the clique is the number of nodes in this graph. Any graph can have a clique in it, and the largest clique in a graph is called the max-clique.
Corrádi and Szabó showed that there exists a counterexample of
Keller's conjecture in dimension n if and only if there is a clique
of size 
in certain graphs [CS90]. These graphs, called
Keller graphs, were explicitly defined to represent all possible
tilings of an n-dimensional space by n-dimensional cubes. The
search that Cipra suggested is the search for the max-clique in the
Keller graph for dimension n. If the max-clique in the Keller
graphs is smaller than then Keller's conjecture is true for that
n and false otherwise. The problem no longer is how to find a proof
of Keller's conjecture but what are the max-cliques of the Keller
graphs. It has been shown that finding max-cliques is a NP complete
problem, meaning that any real computer with any known algorithm will
take a long time to completely solve the problem. Cipra stated it will
take 
units of time to solve for n=7 [Cip93].