Discussions on the mathematics of the cube

## 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?

## Two directional serch for cube solutions...

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.

Service is back up, fixed ip address is 216.138.229.145, enjoy :)

## Server will be down for a short time

I'm changing to a better ISP with a fixed IP address. The current service will go down sometime on July 5th, 2004 and should be back up by the 6th.

olympic cube 6a