## An analysis of the corner and edge permutations of the 3x3x3 cube

Submitted by Bruce Norskog on Sat, 10/01/2005 - 11:15.I have generated a distance table using the quarter-turn metric for positions of the Rubik's cube (the standard 3x3x3 cube) where the orientation of the cubies is ignored. That is, the permutation of the corner cubies and the permutation of the edge cubies are considered, but not the orientation of either the edge or corner cubies. This is a set of 8!*12!/2 or 9,656,672,256,000 positions. To the best of my knowledge, this is (now) the largest "subset" of the 3x3x3 cube that this has been done for!

I used symmetry in the corner permutations, so that my programs "only" needed to store values for 984*12!/2 or 235,668,787,200 positions. The "real" size (considering symmetry in both the corner permutations and edge permutations) I found to be 201,181,803,792 positions. So about 17% of the stored positions are redundant with respect to the set of symmetrically distinct positions.

## The 4x4x4 centres can be solved in 22 moves

Submitted by jaap on Wed, 09/21/2005 - 02:47.Finding God's Algorithm for the centres only was too hard a task since there are 24!/4!^6 = 3246670537110000 possible centre arrangements. I therefore split it into two stages.

First of all, solving 2 particular opposite centre colours on the 4x4x4 cube, placing them on any 2 opposite faces.

depth 0, positions 6, total 6

depth 1, positions 36, total 42

depth 2, positions 624, total 666

## Exhaustive search of the cube space

Submitted by Joe Miller on Thu, 08/04/2005 - 09:58.## Antisymmetry and enumeration of LL algorithms

Submitted by Joe Miller on Thu, 08/04/2005 - 07:38.## Inner Automorphisms and Outer Automorphisms

Submitted by Jerry Bryan on Wed, 08/03/2005 - 23:12.An inner automorphism is an automorpism of the form p(g)=G^h=h'gh for all g in G and for a specific, fixed h in G. An outer automorphism is an automorphism that is not inner.

But occasionally the definition takes a slightly different form. The alternate definition says that automorphisms are of the form p(g)=G^h=h'gh for all g in G. If h is a fixed element of G, then p is an inner automorphism. Otherwise, h is not in G but rather is a fixed element of a larger group of which G is a subgroup, and p is an outer automormphism. The latter definition clearly motivates the names "inner" and "outer".

## 3x3x3 cube ignoring edge locations

Submitted by Bruce Norskog on Tue, 07/26/2005 - 16:18.I have done an analysis of the 3x3x3 Rubik's cube ignoring the locations of the edges (but not ignoring the orientations). That is, the locations and orientations of the corners were used, as well as the orientation of the edges. I used symmetry to reduce the number of positions from 180,592,312,320 to 3,772,354,560 (1,841,970 corner sym-coordinate values * 2048).

I have created files giving the distances for each of the positions of this problem space for the face-turn and quarter-turn metrics. I have listed the summary of the results I got below (where the numbers given are not symmetry-reduced). I was wondering if anyone has done this analysis before as I haven't been able to find any other such data on the internet to verify my results against.

## Distance preserving automorphisms

Submitted by Joe Miller on Mon, 07/18/2005 - 20:30.Have the number of distance preserving automorphisms of the Rubik's cube and/or Junior cube been counted/enmerated? Is there a way to count/enumerate them with GAP?

Thank you for your time,

-Joe Miller

## Rubik's GAP file

Submitted by Joe Miller on Sun, 07/17/2005 - 20:30.I am a Senior student of Kent State studying permutation puzzles. (the cube in particular)

Here are the GAP definitions I use for analyzing the cube. I thought that maybe someone else would like to use it.

Notation: U2 is a 180 turn of the top layer, Uc is a 90 counterclockwise turn of the top layer.

U := ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19);

U2 := U*U;

Uc := U2*U;

L := ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35);

L2 := L*L;

Lc := L2*L;

F := (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11);

## New Results, 13 Moves from Start, Quarter Turn Metric

Submitted by Jerry Bryan on Sat, 07/09/2005 - 00:08.Distance Patterns Positions

from

Start

0 1 1

1 1 12

2 5 114

3 25 1068

4 219 10011

5 1978 93840

6 18395 878880

## Rubik's Cube antisymmetry and the shrinking of Cube Space

Submitted by Mike G on Wed, 06/15/2005 - 15:29.*antisymmetry*of Cube positions: antisymmetric positions include self-inverse positions as a special case. We show that the size of Cube Space is reduced by approximately a factor of 2 after taking account of positions that are related to one another by antisymmetry. The "new" real size of Cube Space is found to be 450541810590509978.

[Contribution also posted on sci.math]