Discussions on the mathematics of the cube

Inner Automorphisms and Outer Automorphisms

An automorphism (specifically, a group automorphism) is simply an isomorphism that is from a group to itself. Most authors define inner automorphisms and outer automorphisms roughly as follows.

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

Hello! (I am new to the group.)

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

I have several questions:

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

Hello everyone!

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

Previously, the number of positions out to 12 moves from Start had been calculated for the quarter turn metric. Patterns are the number of positions unique up to symmetry. This is not a new program or a new algorithm. It's just the old program on a faster machine with more memory.

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

In this posting we introduce the concept of 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]

Modeling subsets of the cube that involve ignoring certain cubies

I'm in the process of developing a C++ class library for modeling various Rubik's cube problems, including some old problems that have already been run on a computer and some new problems that haven't been run before. One of the capabilities I want to include in my class library is the ability to ignore certain cubies. In a certain sense, we already do so when we solve "corners only" or "edges only" problems or some such. But I want a more general facility where the cubies to be ignored could be some of the corner cubies, some of the edge cubies, or both.

I'm having a little difficulty with some of the group theory underpinnings. For example, consider the corners only group and suppose I want to consider the positions of only six of the eight corner cubies. I would like for what I'm modeling to be a group because if it is, a lot of useful group theory concepts come into play such as conjugates, symmetry, Sims tables, and the like. Essentially, the way to model six of the eight cubies is to consider two positions equivalent if they are the same except for the possible transposition and/or rotation of the two particular corner cubies to be ignored. The set of transpositions and rotations of two particular corner cubies can be thought of as a subgroup the corners group, call it H. I'm thinking that what I need to consider is the factor group G/H, where G is the corners group. Trouble is, G/H is only a group if H is normal in G. And I'm not convinced that all possible subgroups H derived in the manner described (ignoring one or more corner cubies) are normal in G.

Order of the Additive List

Conjecture: The order of the additive list always evenly divides the order of the generated group.

Time for some definitions.

First from the possible moves of the cube using Singmaster notation (U, D, F, B, L, R) pick any number of operators, this is the basis for the generated group.

The group generated by < U, F, D > is an example of the "generated group".

From these operators we can generate the "additive list" so the elements are

{ U, UF, UFD, UFDU, UFDUF, UFDUFD ... }

Now for some rules...

Has God's Algorithm been discovered yet?

I apologize if this question is too elementary for your group. My sister is a math teacher & is trying to find the answer for her class. Has an algorithm been discovered that will solve any configuration of the cube in smallest possible number of moves? What is the smallest number of moves that will solve any configuration? Thanx much for any help.

Old cube programs wanted (new ones also welcome)

Recently I was asked about some of the older cube programs written about here. Some of the programs are missing in action, including a couple of my own. It seems worthwhile to try to preserve the older software so if anyone can locate them please send me an email.

If you have written a cube program yourself or know of some forgotten old program feel free to tell people about it in this forum.