Let me try to restate the problem slightly to make sure we are talking
about the same problem.
Basically, you can solve the entire cube, except that the pieces "1" and
"2" in the diagram are flipped/exchanged:
Assuming this is the problem, the hint is as follows:
The problem here is that some of the face pieces are not really in their
right places. In short, one of the center slices is 1/4 turn out of phase.
The simplest way to proceed (at least for me) is to move to the following
XX2X (i.e. rotate one of the central slices 1/4 turn) XXOX XXOX X1OX ^ | Now, you can solve for pieces "1" and "2" and --using these pieces as a landmark-- proceed from there.
Hint #2: You can help avoid this problem by solving the face pieces last.
Extra Credit: Actually, the state as shown in the first diagram above is
pretty interesting in that the analogous position on a 3X3X3 cube (i.e. a
single flipped edge cube) is of course impossible.
>From this state it is relatively easy to get to another, very interesting
state: namely the 4X4X4 appears to be completely solved except that two
opposite corners are exchanged. (Again, this is obviously impossible on
Left as an exercise for the reader. ;)
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