  We give a method for exhaustive generation of a huge number of Kochen-Specker
contextual sets, based on the 600-cell, for possible experiments and quantum
gates. The method is complementary to our previous parity proof generation of
these sets, and it gives all sets while the parity proof method gives only sets
with an odd number of edges in their hypergraph representation. Thus we obtain
35 new kinds of critical KS sets with an even number of edges. Using a random
sample of the sets generated with our method, we give a statistical estimate of
the number of sets that might be obtained in an eventual exhaustive
enumeration.
