Message #564

From: "thibaut.kirchner" <thibaut.kirchner@yahoo.fr>
Subject: Re: Upper bound for the number of moves to solve a 3^4
Date: Wed, 17 Sep 2008 00:36:49 -0000

— In 4D_Cubing@yahoogroups.com, David Smith <djs314djs314@…> wrote:
>By the way, I may
> also try to determine an upper bound for the number of moves
required to solve
> MagicCube4D from any position.  Although I think exploring this
question would have a
> more direct impact on and be more of an interest to the group, I am
less confident I can
> 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 = the whole group of the transformations of the 3^4 Hypercube,
G_3 = 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 = 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 = 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 = 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.