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Thanks to all who responded. I haven't yet got what I consider a

solution for my problem (shift a slice and resolve is my current

method which is slow and ugly) but at least I understand my problem

slightly better.

A few questions:

1. What is the definition of parity by which commutators are even,

but slice turns are odd? I haven't been able to come up with a

cube-wide parity. (I know no group theory).

2. How many orbits does the order 4 cube have? I can only think of

three (twirling a corner cubie). Then again, I haven't painted the

facelets yet, so there could be orbits I haven't begun to see

involving them.

3. Would an order 6 cube have any challenge beyond the order 4? I

think the answer is no--if you are able to solve the 3-cube and the

4-cube you can solve any cube.