Consider all manipulations of Rubik's revenge as consisting of two
sorts of moves, namely (1) shallow moves, which turn an outer layer
with respect to the remaining three layers, and (2) deep moves, which
turn an outer layer and the adjacent inner layer with respect to the
remaining two layers. [For the purposes of this problem, we will
regard a manipulation that turns only an inner layer--resulting, for
example, in faces that look like
when applied to a solved cube--as consisting of two moves, one deep and
one shallow, in opposing directions.] If only shallow moves are
permitted, the 4x4x4 simulates a 3x3x3. If only deep moves are
permitted, the 4x4x4 simulates a 2x2x2. Define a shallow (deep)
hypermove as an arbitrary sequence of shallow (resp. deep) moves.
My question is:
What is the maximum number of hypermoves required
to solve the 4x4x4?
Notice that the answer to this question may depend on whether or not
one considers identically-colored face centers to be distinct (as
Hoey points out, the puzzle is not a group if identically-colored
face centers are not distinguished) and on whether or not one worries
about the positions of the eight hypothetical stomach cubies. Also,
if the minimal number of hypermoves is odd, then it might be
important to start with one class of move. That is, it is plausible
that sequences of the form SDSDS may be sufficient while sequences of
the form DSDSD may not.