--- In 4D_Cubing@yahoogroups.com, David Smith
>By the way,=A0I may=20
> also try to=A0determine an upper bound for the number of moves
required to solve=20
> MagicCube4D from any position.=A0 Although I think=A0exploring this
question=A0would have a=20
> more direct impact on=A0and be more of an interest to the group,=A0I am
less confident I can=20
> provide any answers here.
I suggest you to explore the following groups, and their successive
quotients (each group is a subgroup of the previous one, therefore
each group G_k has a natural action on the quotient G_k / G_k-1, and
that's what we have to study).
G_4 =3D the whole group of the transformations of the 3^4 Hypercube,
G_3 =3D the sub-group of G4 generated by the rotations of 3 pairs of
opposite cells and the even rotations of the last pair of opposite
cells (even rotations of a cube are the identity, the half-turns
around the faces, and the third-turns around corners, whereas the odd
rotations of a cube are the quarter-turns arounds the centers, and
half-turns around the edges)
G_2 =3D the sub-group of G4 generated by the rotations of 2 pairs of
opposite cells and the even rotations of the other 2 pairs of opposite
cells
G_1 =3D the sub-group of G4 generated by the rotations of one pair of
opposite cells and the even rotations of the other 3 pairs of opposite
cells.
G_0 =3D the sub-group of G4 generated by the even rotations of the 8 cells.
The similar study for the 3^3 has been useful to compute upper bounds
to solve the 3^3. I wonder if this decomposition of G_4 is as
relevant, and if a single computer can get the least upper bound of
moves for each quotient in a reasonable time.