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Subject:

A QTW cannot take you between two positions of equal distance, I believe

-- is there not some parity quantity obeyed by QTW's (net twist of

the six center cubies)? If so, then there cannot be two positions of

equal distance separated by a QTW, for then there would be an odd

length identity transformation, which would violate parity.

(In the example you gave, there are *not* two positions of

equal distance, but only one -- the halfway point.)