Discussions on the mathematics of the cube

New optimal solutions for an important group

Submitted by silviu on Wed, 03/15/2006 - 03:32.
I have computed optimal solutions for every element in the group <U,D,L^2,R^2,F^2,B^2>. For this task I made some minor modifications to Reid's solver. I would like to thank him for sharing it. 

   0q                1 
   1q                4 
   2q               10 
   3q               36 
   4q              123 
   5q              368 
   6q            1,320 
   7q            4,800 
   8q           15,495 
   9q           54,016 
  10q          194,334 
  11q          656,752 
  12q        2,222,295 
  13q        7,814,000 
  14q       26,402,962 
  15q       89,183,776
» 13 comments | read more

Analysis of another two symmetry subgroups of order 4

Submitted by Herbert Kociemba on Wed, 03/08/2006 - 12:46.
The symmetry class C4 defines a 1/4-rotational symmetry around a face (I chose the UD-axis). It took about 8 days to show that all 36160 cubes, which exactly have this symmetry (M-reduced) are solvable in at most 20 moves. There are 39 20f*-cubes. 35 of them also have antisymmetry, 4 only have symmetry, so reduced wrt M+inv there are 37 cubes.

The class D2 (face) consists of all cubes which have a 1/2-rotational symmetry around all faces. Up to M-symmetry there are 23356 cubes, which exactly have this symmetry. It took about 4 days to show, that all cubes of this symmetry class can be solved in 20 moves. There are only 4 cubes which are 20f*, all of them also are antisymmetric. Here are the results:

» 5 comments | read more

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

Submitted by xintax on Mon, 03/06/2006 - 10:35.
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...
» 9 comments | read more

Permutations of the corners and edges: FTM

Submitted by Bruce Norskog on Tue, 02/21/2006 - 17:00.

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.

» 12 comments | read more

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

Submitted by Herbert Kociemba on Tue, 02/21/2006 - 14:58.
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)
» 14 comments | read more

Another four interesting cosets

Submitted by rokicki on Fri, 02/03/2006 - 22:38.
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.
» 6 comments | read more

Symmetric Cube Positions with more than 4 symmetries

Submitted by Herbert Kociemba on Fri, 02/03/2006 - 15:31.
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.
» 10 comments | read more

44 million cubes

Submitted by rokicki on Wed, 02/01/2006 - 01:03.
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:
» 20 comments | read more

Some more interesting groups

Submitted by rokicki on Mon, 01/30/2006 - 18:10.
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:

» 9 comments | read more

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{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}.

» 17 comments | read more