# An interesting conjugate class

Submitted by mdlazreg on Fri, 01/01/2010 - 11:51.

As you know the number of conjugate classes of the cube is 81120.

Unfortunately as far as I know there is no fast way to calculate the optimal distances distribution for a chosen conjugacy class. The only way is to search for the optimal distance of each position one by one. That is what I did for the following conjugacy class:

CE x CC : 1_1_1_1_1_1_1_1_4 x 1_1_1_1_4

which has 495*6*2^3*70*6*3^3 = 269438400 positions and which include the 12 cube generators.

I did this search however only for one orientation which reduced the number to 495*6*70*6 = 1247400 positions. Here is the optimal distribution:

d pos

1 4

3 0

5 16

7 32

9 584

11 3024

13 31444

15 229429

17 771589

19 211030

21 248

23 0

25 0

27 0

29 0

1247400

The four positions at depth 1 are:

UL UF UR UB DF DR DB DL FR FL BR BL ULF UFR URB UBL DRF DFL DLB DBR

UR UB UL UF DF DR DB DL FR FL BR BL URB UBL ULF UFR DRF DFL DLB DBR

UF UR UB UL DR DB DL DF FR FL BR BL UFR URB UBL ULF DBR DRF DFL DLB

UF UR UB UL DL DF DR DB FR FL BR BL UFR URB UBL ULF DFL DLB DBR DRF

which correspond to U,U',D and D' moves.

The most difficult position is:

UF FL UB UR DF DR DB DL UL FR BR BL UFR URB UBL ULF DLB DRF DBR DFL

which took 35 minutes to calculate its 21 optimal solution.

The solver I used is Kociemba optimal solver.

Unfortunately as far as I know there is no fast way to calculate the optimal distances distribution for a chosen conjugacy class. The only way is to search for the optimal distance of each position one by one. That is what I did for the following conjugacy class:

CE x CC : 1_1_1_1_1_1_1_1_4 x 1_1_1_1_4

which has 495*6*2^3*70*6*3^3 = 269438400 positions and which include the 12 cube generators.

I did this search however only for one orientation which reduced the number to 495*6*70*6 = 1247400 positions. Here is the optimal distribution:

d pos

1 4

3 0

5 16

7 32

9 584

11 3024

13 31444

15 229429

17 771589

19 211030

21 248

23 0

25 0

27 0

29 0

1247400

The four positions at depth 1 are:

UL UF UR UB DF DR DB DL FR FL BR BL ULF UFR URB UBL DRF DFL DLB DBR

UR UB UL UF DF DR DB DL FR FL BR BL URB UBL ULF UFR DRF DFL DLB DBR

UF UR UB UL DR DB DL DF FR FL BR BL UFR URB UBL ULF DBR DRF DFL DLB

UF UR UB UL DL DF DR DB FR FL BR BL UFR URB UBL ULF DFL DLB DBR DRF

which correspond to U,U',D and D' moves.

The most difficult position is:

UF FL UB UR DF DR DB DL UL FR BR BL UFR URB UBL ULF DLB DRF DBR DFL

which took 35 minutes to calculate its 21 optimal solution.

The solver I used is Kociemba optimal solver.