# 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.