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Alan, I enjoyed your presentation. I am convinced that counting the RLFBUD

manipulations will not give a metric. I do not, however, see an easy way to

compute twists T(M). I think one gets a metric only if one takes the minimum

over some set of manipulations. That is, take a set, AM, of atomic moves

including their inverses, let AM* be the strings of AM, and |M| be the length

of M in AM*. Then

D(a, b) = min |M| such that a = bM

defines a metric. D(a,b) would sometimes be undefined if AM did not generate

the whole group. The recent discussion on shortest solutions is in fact about the

maximum of such a T(M) for all M in some AM*.

Bill