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