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Subject:

David,

Suppose one could prove local maxima had configurations that were invariant

under the rotation group of the whole cube. (I am not at all sure it is even

true.)

There are a small number of such symmetric configurations, and they could

probably be easily tabulated. One of them would have to be maximally distant

from home. Thus if we had a QTW solution for each of them, the maximum

over that set would bound God's Algorithm.

I see no reason to believe that a QTW cannot take you between two solutions

that are at the same distance. As DPC pointed out, there are a lot of even

identity paths. E.g., (RUR'U')^6. The two furthest points on the path are (by

symmetry) necessarily equally distant, yet connected by a QTW.

Bill