# Optimal supercube cross

#### cuBerBruce

##### Member
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:

Code:
  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.

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#### qqwref

##### Member
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

##### Member
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

##### Member
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

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

Code:
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.
Code:
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

##### Member
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

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