From:

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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

XXXX

OOOO

XXXX

XXXX

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.

Jim Saxe