## Another four interesting cosets

Submitted by rokicki on Fri, 02/03/2006 - 22:38.## Symmetric Cube Positions with more than 4 symmetries

Submitted by Herbert Kociemba on Fri, 02/03/2006 - 15:31.For the labeling of cube symmetries there does not seem to exist a really consistent procedure. Michael Reid uses a different labeling than Jaap Scherhuis which also differs from my notation. My notation uses the Schoenflies symbols, I used for example this site for a deeper understanding.

## 44 million cubes

Submitted by rokicki on Wed, 02/01/2006 - 01:03.## Some more interesting groups

Submitted by rokicki on Mon, 01/30/2006 - 18:10.## Calculating Symmetry using Representative Elements

Submitted by Jerry Bryan on Wed, 01/25/2006 - 14:19.I have long been curious how other people who write cube programs and who incorporate symmetry into their programs actually do the symmetry calculations. The general way I do it has been outlined in Cube-Lovers.

For a given position x, I calculate a representative element of its m-conjugates, where M is the standard Cube-Lovers terminology for the 48 symmetries of the cube. I then store and manipulate only the representatives.

We denote this calculation as
y = Rep(x) = min{x^{M}} = min{x^{m} | m in M}.
The minimum element of x^{M} is taken to be the one that
is first in lexicographic order. And as usual,
x^{m} means m^{-1}xm, and x^{M} means
{x^{m} | m in M}.

## Solving Rubik's cube in 36 quarter turns

Submitted by silviu on Sat, 01/14/2006 - 12:44.We call the group of corner edge permutations of the cube for CEP and cube group for C. Let N be the normal subgroup that fixes cubies. Then we have a homomorphism hom:C->C/N=CEP such that hom(g)="permtutation of cubies done by g". Let

**a**be the unique antipode in CEP. Every element in CEP can be written as x*

**a**. Where x is an element requiring at most 18 quarter turns according to Norskog's analysis. There are 220 elements at distance 17 and 1 element at distance 18. So we could say that all elements of CEP except 220+1 elements can be written as x'*

**a**where x' is an element requiring at most 16 quarter turns.

## Partial Permutations

Submitted by Jerry Bryan on Mon, 12/26/2005 - 21:41.
A **permutation** is generally defined as a bijection on a
non-empty set.

Given a permutation on a set W, a **partial permutation** is a bijection
from one subset W_{X} of W to another subset W_{Y} of W.
W_{X} and W_{Y} are the domain and range, respectively,
of the partial permutation. Because a partial permutation
is a bijection, W_{X} and W_{Y} must contain the
same number of elements (or must be of the same cardinality, if they
are infinite).
Note that a partial permutation is
defined only in the context of a specific and previously defined
permutation.
Generally speaking, a partial permutation
is not a permutation, and indeed a partial permutation is a permutation
only if its domain and range are the same set.

## Solving Rubik's cube in 28 face turns

Submitted by silviu on Thu, 12/22/2005 - 20:53.The above method can also be formulated in following way:

Given an arbitrary element g in the cube group we multiply it by an element B^-1 from the right such that gB^-1 is in the group H and the length of B^-1 is at most 12 face turns. Then we multiply gB^-1 by an element A^-1 from the right such that gB^-1A^-1=id --> g=AB. And the length of A is at most 18 face turns.

## Optimal solutions for two important subgroups

Submitted by silviu on Thu, 12/22/2005 - 20:17.Group that fixes cubies Distance Nr of pos Unique wrt M Unique wrt M + inv 0q 1 1 1 2q 0 0 0 4q 0 0 0 6q 0 0 0 8q 0 0 0 10q 0 0 0 12q 441 11 8 14q 3944 87 52 16q 32110 708 396

## Square One God's Algorithm Computed

Submitted by masonjones on Tue, 12/06/2005 - 09:04.The main stumbling block to reporting results at the moment is that I discovered a unexpected position-counting problem, where the shape degeneracy interferes with the 16-fold starting position degeneracy, and which requires me to do a recount on the tables. I will post the results back here in a few days when the recount is finished.