Optimal supercube cross

[cuBerBruce]

I've done an analysis of the optimal number of moves (face turns) to solve each case of the supercube cross. By supercube cross, I mean solving the 4 cross edges and orienting the 5 centers adjacent to those edge positions. This is for solving a particular color cross. (It's not for color neutral solving.) The distribution is:

  moves   positions
  -----   ---------
    0             1
    1            15
    2           158
    3          1682
    4         17469
    5        166685
    6       1425198
    7      10144474
    8      49800450
    9     104027538
   10      28994240
   11         64004
   12             6
          ---------
 total    194641920

The 6 antipodes include two equivalence classes. Scrambles to generate cases of these two equivalence classes are:

L2 F' R B2 U F2 R B' D B' F' D'
F R B2 U2 L' D' F' D2 L F2 R' D'

(EDIT: The above are for generating antipode cases for the D cross.)

The average number of moves for solving a particular color cross is approximately 8.7636.

 

[qqwref]

Ooh, cool. I'm kind of surprised there are positions that take 12 moves, because the maximum for normal cross is only 8 moves. Is it possible to do the same calculations for color-neutrality? How about for building a (fixed) 2x2x2 block?

 

[Stefan]

L2 F' R B2 U F2 R B' D B' F' D'
F R B2 U2 L' D' F' D2 L F2 R' D'

Are these for the D cross?

On non-super cubes, they take 7 and 8 moves (for D cross).

 

[cuBerBruce]

Ooh, cool. I'm kind of surprised there are positions that take 12 moves, because the maximum for normal cross is only 8 moves.

Well, I'll note that the percentage of 11-move supercube cases is less than the percentage of 8-move regular cube cases.

Is it possible to do the same calculations for color-neutrality?

There are a little over 2 quadrillion positions to consider for an exact color neutral calculation. A distributed effort would seem to be required. Of course, you only need to build a table for the fixed color case, and use symmetry to do 6 table lookups to get the best case cross for each position.

How about for building a (fixed) 2x2x2 block?

A fixed 2x2x2 block should be very doable.

Are these for the D cross?

On non-super cubes, they take 7 and 8 moves (for D cross).

Thanks for pointing out that omission, Stefan. Yes, the antipode scrambles are for the D cross. I'll add that to the post.

 

[cuBerBruce]

I've now done an analysis of the supercube cross in QTM. The distance distributiont table is given below.

Supercube cross (QTM)

  moves   positions
  -----   ---------
    0             1
    1            10
    2            73
    3           536
    4          3922
    5         27620
    6        184728
    7       1151210
    8       6400627
    9      28690546
   10      79587153
   11      72238639
   12       6353219
   13          3632
   14             4
          ---------
 total    194641920

The 4 antipodes are in 2 equivalence classes. The essentially distinct antipodes for the D layer cross can be generated by:

R F' R R F' L B R B R B F D D
R F L D F R B D L B F D' F' D

I've also done the 2x2x2 block for the supercube in both FTM and QTM. This means the three centers that are part of the 2x2x2 block must be correctly oriented.

supercube 2x2x2 block: FTM  supercube 2x2x2 block: QTM

  moves   positions           moves   positions
  -----   ---------           -----   ---------
    0             1             0             1
    1             9             1             6
    2            90             2            39
    3           942             3           288
    4          9606             4          2121
    5         89330             5         14861
    6        713910             6         97460
    7       3949020             7        577222
    8       8924097             8       2718634
    9       2528145             9       7312432
   10          5010            10       5245711
           --------            11        251349
 total     16220160            12            36
                                       --------
                             total     16220160

 

[Stefan]

The 4 antipodes are in 2 equivalence classes. The essentially distinct antipodes for the D layer cross can be generated by:

R F' R R F' L B R B R B F D D
R F L D F R B D L B F D' F' D

Ha, "solved" and "off by 1". I suddenly wish alg.cubing.net had a supercube mode...

The effects on the centers are:
R F' L B'
R2 F2 L2 B2