From:

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Date: 6 Dec 1980 16:42 PST From: McKeeman.PA at PARC-MAXCI 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.

I am not sure I understand what you are trying to say here. But I do

know that a single quarter twist can never leave you the same distance

from anything. This is because a single quarter twist is a odd

permutation of the "stickers". Thus if you are N quarter twists away

from something, a single quarter twist will leave you N-1 or N+1

quarter twists away. (And hence the proof that any quarter twist will

bring you closer from a maximally distant state.)

I'm not sure how to apply this to your statement that perhaps a "QTW"

can take you "between two solutions that are at the same distance".