Discussions on the mathematics of the cube

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.

Router causing Intermittent problems

The router for the server was acting up and I had to reflash it. Please let me know if there are any other problems. admin mail

Interdimensional Cubes

As a thought experiment consider the case of the familar 4x4x4 cube with a 2x2x2 cube embedded inside it, instead of the usual mechanism. I'll call this the "Interdimensional 4x4x4 cube" for lack of a better name. Now clearly if we turn the slices of the 4x4x4 cube it would have an effect on the internal 2x2x2 cube. Now moving the slice adjacent to the U face and moving the slice adjacent to the R face this would be the equivalent of turning the internal 2x2x2's U face and R face.

My question is: Is it possible to reach all the positions of the internal 2x2x2 without having any constraints on the 4x4x4 cube? How many positions are there?

A question about the commutator subgroup

We all know that commutators can be used to generate half of the Cube group G. My first question is: Can all elements of the commutator subgroup themselves be written as commutators? i.e., the problem is to determine whether the set of commutators is closed under multiplication; it need not be in general, but is it true here?

If it is closed in this case, then a natural question to ask is How do we write a given element of the commutator subgroup as a single commutator?

On the other hand, if the set of commutators is NOT closed under multiplication, then how many elements of G can be written in commutator form?