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).

Yes, it is not really simple.

...
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).

Still you are halfway there if you are willing to forgo the pattern of
the centers (which can always be done later). Turn the front face (at
the bottom in the frawing), the right face and the back face one quarter
turn before your turn, and back after. Observe that that constitutes a
cycle of three edge cubies in a single middle slice. Combine with a
quarter turn of that middle slice.

So, what am I missing? As I said before, I really just want a hint
here.

I hope that is enough of a hint and not enough of a giveaway.

(I thought there was a shorter sequence, but I disremember at the moment.)

My solution for the 4x4x4 always was: first corners, next edges and finally
centers. Because there are many identical pieces for the centers those are
reasonably simple. It would be much more difficult if each center had its
own place.

dik
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland
home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl