I've been beating my head against the order 4 Rubik's cube for long
enough, and I want a hint. (Not a solution--I have a solution book if
I wanted to use it).
My problem is I cannot flip a pair of adjacent edges (this is
equivalent to not being able to exchange a pair of knights-move
separated edges). All my other transformations have no side effects,
so I can solve the edges first. But I can't see how to just affect
two of them.
I tend to solve using commutators, but I don't see a way here. The
move I use on the top moves the marked pieces clockwise (this pattern
. . 0 . . . . . . . . 0 . 0 . .
rotates and reflects, of course). There is no way to combine these
into a pair exchange (after doing the move, you still have two pieces
out of place--nothing changed from the original).
I tried to find a move that would exchange three pieces, the third
being the correctly placed piece next to one of the incorrectly placed
pieces (ie., treat a right edge cubie as if it should be a left edge
cubie) but this can easily be shown as impossible:
Define the parity of a piece as being left if it is a left
edge cubie when the red facelet is up, right if it is a right
edge cubie when the red facelet is up. The parity is
undefined if there is no red facelet. There are only three
moves available that affect an edge cubie--none of them alter
the parity. QED
So, what am I missing? As I said before, I really just want a hint