We know that some positions are "global maxima". We don't
know how many such positions there are. We would dearly love
to know how far away they are.
Suppose that God's Number is N. (For some obscure theological
reason I have the irrational belief that N=28, but we'll
leave such hunches out of the discussion.) Let's say there
are K global maxima at that distance from solved.
What if we could show that there are at least 12K states at
distance N-1? This is a little bit reasonable. All it
means is that global maxima are all separated from each other by
more than 2q.
If that were true, mightn't we be able to increment our lower
bound on N?
Can anybody prove that it's true?
I would also like to hear from ZILCH where he gets those identities,
and whether any of those impressive lists are exhaustive.
Alan, do the other two order 12 nulls enable you to stretch your
I am busy wading through Wielandt's "Finite Permutation Groups" and
will report if I learn anything applicable.