I have to admit to not quite yet fully understanding all of the
parity issues involved, however I have managed to apply some VERY
SIMPLE logic to the mysterious transform and since it worked I'd
like to share the insight with you-all.
As I mentioned in an earlier note I have been trying to do some
analysis to try to sneak up on the missing transform. Mostly
I've been unsuccessful at fully identifying the parity issues
(my understanding is basically at the same not-so-useful level as
wbe's is: I kind of know that some moves seem to come in pairs and
there are two `classes' of edge cubes and if you interchange one from each
class they both flip... but that mostly didn't help me see what had to be done
to make the `missing transform' happen, nor would it let me look at a scrambled
cube (or even a nearly-done cube) and guess which parity class it was in).
However, I did make one observation: all of my normal transforms contained an
EVEN number of quarter-twists. I fumbled around a bit to try to find a
limited-change transform that used an ODD number of twists and mostly I
couldn't. So I decided to simply take the `simplest' odd-twist transform I
could: a single twist! IT WORKS!! As far as I can tell it must be a slice
twist (but ANY one), a face twist doesn't do it. If I get the cube edges all
solved except for a pair that must be flipped, I simply make a SINGLE quarter
slice and then re-solve the cube from there. Since all of my normal solving
maneuvers are EVEN, when I get the thing solved again I will have preserved the
ODD parity of the configuration, and poof! the edges are solved.
Now to start on the search for pretty patterns and elegant (and short)