Discussions on the mathematics of the cube

Regarding God's Algorithm Calculations and lower bounds on face metric

Hi...
Im kinda new here. Just a few intros: Im sheelah from the philippines and im doing an overnight research on the length of God's Algorithm... due tomorrow.

I was kinda ready but then I encountered this page that says that the real size of the cube universe is not 43 quintillion but rather 901 quadrillion.

For one, the repercussions of this on my project are:
1.) Theoretically less runtime and
2.) A different criteria for 'similarity'.

I was trying to read the calculations but I got lost between the numbers. Can someone please clarify this for me?

Secondly, I want to ask about the current lower-bound of the length of God's Algorithm in face turn metric. I keep seeing 20f* but I was wondering if there is a higher lowerbound...

Permutations of the corners and edges: FTM

I have finally completed the face-turn metric version of the analysis of the permutations of the corners and edges of the 3x3x3 Rubik's Cube. That is, I have generated a table of the Cayley graph distances for the positions where the permutation of the corner cubies and the permutation of the edge cubies are considered, but not the orientation of either the edge or corner cubies. This is a set of 8!*12!/2 or 9,656,672,256,000 positions. As with the quarter-turn metric analysis, I used symmetry in the corner permutations to reduce the number of stored distances to 984*12!/2 or 235,668,787,200.

39 cubes with 20f moves in a class with 4 symmetries

After the analysis of cubes with more than 4 symmetries I now try to analyze cubes with 4 symetries. The smallest class has 15552 cubes wrt to M-symmetry. All cubes of this class can be solved in 20f moves. All positions which could not be reduced with the two phase solver to less than 20f moves within a day were solved with my optimal solver within about 4 days. I did not check if the positions are local maxima, so they still are candidates for a 21f maneuver when appending a move.

D L2 U' B2 L2 B2 D B2 U L2 U2 R2 B F L' D U F2 L' R' (20f*) //C2v (a1)

Another four interesting cosets

These are the distributions of the optimal solution lengths for the cosets where the edges start at the given M-symmetric position, using the quarter turn metric. The first run took longer than the other three combined; not sure why. Only a handful of positions at length 24; none at 26 or higher. I'm currently running the coset where the edges begin in the known length-26 position. It's interesting to note that with the edges superflipped and reflected across the center, all solutions took either 18, 20, or 22 moves.

Symmetric Cube Positions with more than 4 symmetries

Symmetric cube positions tend to be deeper positions than positions without symmetry. So in the search for a 21 FTM-positon it seems natural to look for positions with a higher degree of symmetry.

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

Here's another coset distribution. This one is face turn metric, edges in a random initial position (with no symmetry at all), and the distribution of the lengths of the optimal solutions for all corner possibilities:

Some more interesting groups

Here are the results of an exploration of four groups, all of whom leave the edges in some M-symmetric class, for the face turn metric:

Calculating Symmetry using Representative Elements

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{xM} = min{xm | m in M}.  The minimum element of xM is taken to be the one that is first in lexicographic order.  And as usual, xm means m-1xm, and xM means {xm | m in M}.

Solving Rubik's cube in 36 quarter turns

In this paper we present a method of solving Rubik's cube in 36 quarter turns. The proof have many common facts with other methods presented on the forum. In able to prove our claim we have used the corner and edge permutation analysis done for the first time by Bruce Norskog.

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

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 WX of W to another subset WY of W.  WX and WY are the domain and range, respectively, of the partial permutation.  Because a partial permutation is a bijection, WX and WY 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.