From:

~~~ Subject:

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

here.

Ronnie