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

Submitted by Mike G on Wed, 06/15/2005 - 15:29.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]

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## Modeling subsets of the cube that involve ignoring certain cubies

Submitted by Jerry Bryan on Sun, 04/24/2005 - 21:41.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.

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.

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## Order of the Additive List

Submitted by cubex on Thu, 03/10/2005 - 02:39.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...

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...

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## Has God's Algorithm been discovered yet?

Submitted by TonyVarden on Mon, 02/28/2005 - 10:42.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)

Submitted by cubex on Wed, 11/24/2004 - 13:07.Recently I was asked about some of the older cube programs written about

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.

**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.

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## Router causing Intermittent problems

Submitted by cubex on Mon, 09/27/2004 - 13:39.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

Submitted by cubex on Fri, 09/10/2004 - 08:04.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?

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?

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## A question about the commutator subgroup

Submitted by Mike G on Thu, 09/02/2004 - 08:43.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?

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?

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## Two directional serch for cube solutions...

Submitted by crepeau on Wed, 07/14/2004 - 09:06.I was wondering if anyone has ever build a bi-directional tree search for solving specific cube positions.

Let me explain.

Suppose one starts with a solved cube (3x3x3) and find all the configurations after one rotation, and so on say until 10 or 11 rotations. The resulting tree will contain a large number of nodes, but not completely unreasonable.

Now suppose you wish to find the shortest solution for a specific configuration. You may start building another tree similar to the above and look for collisions between nodes of the two trees. After exploring say 10 or 11 levels of the tree it is very likely that the two trees will connect and the shortest path can be obtained.

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## Back from the dead

Submitted by cubex on Wed, 07/07/2004 - 13:04.Service is back up, fixed ip address is 216.138.229.145, enjoy :)