Date: 7 December 1980 00:47-EST From: Alan Bawden Subject: Maximally distant states To: McKeeman.PA at PARC-MAXC cc: CUBE-LOVERS at MIT-MC 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".