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Supercube Squares Group

Submitted by Bruce Norskog on Sat, 09/15/2007 - 12:56.

Before I finish and report the results of a huge analysis that I'm working on, I thought it only fitting that I first perform this much smaller analysis, an analysis of the 3x3x3 supercube squares group (using half-turns only). Since this group has a mere 5,308,416 elements, it wouldn't be surprising to me if others have already done this analysis. However, I did not find any such analysis searching on the internet. I know that Mark had noted the size of this group in a message in the "Cube-lovers" archives: http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Mark_Longridge__Super_Groups.html as well as a God's algorithm calculation of the ordinary 3x3x3 squares group (a group 8 times smaller). Or at least he discussed the antipodes of that group: http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Mark_Longridge__SQUARE'S_GROUP_ANALYS IS.html

The 3x3x3 supercube is the 3x3x3 cube where the orientations of the centers are also considered to be significant. The term squares group refers to the subgroup of positions that are reachable if only half turns (of the face layers) are allowed. Since only half turns are allowed, any given center is either in the correct orientation, or rotated 180 degrees. It would seem that since there are 6 centers with two possible orientations each, there would be 64 times the number of reachable positions than for the ordinary 3x3x3 squares group. Parity effects, however, limit the size of the group to only 8 times larger.

So here is the summary of my God's algorithm calculation for the 3x3x3 supercube squares group (half-turns only).


3x3x3 Supercube Squares Group (half-turns only)

distance positions  unique wrt M  unique wrt M+inv
    0            1             1             1
    1            6             1             1
    2           27             2             2
    3          120             5             4
    4          519            18            13
    5        2,088            59            38
    6        8,368           210           122
    7       31,470           735           403
    8      110,793         2,521         1,345
    9      348,504         7,730         4,094
   10      937,705        20,533        10,724
   11    1,721,148        37,138        19,640
   12    1,532,612        33,250        17,710
   13      541,230        12,166         7,056
   14       62,955         1,833         1,239
   15        9,642           390           252
   16        1,228           112           112
         ---------       -------        ------
         5,308,416       116,704        62,756

I note that the group has quite a few antipodes (1228). It has 112 antipodes that are unique with respect to the cube symmetry group M. The antipodes are only one move deeper than the antipodes (for half-turns only) of the ordinary 3x3x3 squares group. Interestingly, all antipodes are self-inverse positions. One of the antipodes is all corners and edges in the solved position, and all centers rotated. I list move sequences for 12 out of the 112 antipodes (unique wrt M) below. (The first one is the one just mentioned.)

U2 L2 U2 F2 L2 U2 L2 B2 U2 L2 D2 B2 L2 U2 R2 B2
U2 L2 U2 L2 U2 B2 L2 F2 D2 B2 L2 U2 R2 U2 L2 F2
U2 D2 L2 U2 F2 U2 F2 R2 F2 L2 D2 B2 U2 F2 L2 F2
U2 L2 U2 R2 U2 F2 L2 B2 U2 F2 L2 D2 R2 U2 R2 F2
U2 L2 U2 L2 F2 U2 R2 D2 L2 D2 F2 L2 B2 D2 B2 R2
U2 L2 U2 L2 F2 U2 R2 D2 L2 U2 B2 R2 F2 U2 B2 R2
U2 D2 L2 U2 F2 U2 F2 L2 F2 L2 D2 B2 U2 F2 R2 F2
U2 L2 U2 L2 U2 B2 R2 F2 U2 B2 L2 D2 L2 U2 L2 B2
U2 L2 U2 L2 U2 B2 L2 F2 D2 B2 L2 D2 L2 D2 L2 F2
U2 L2 U2 L2 U2 B2 R2 F2 D2 F2 L2 U2 R2 U2 R2 B2
U2 D2 L2 U2 F2 U2 F2 R2 F2 L2 D2 B2 U2 B2 R2 B2
U2 L2 U2 R2 U2 B2 R2 F2 U2 F2 L2 D2 R2 U2 R2 F2
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Did you use only half-turns f

Submitted by Herbert Kociemba on Sun, 09/16/2007 - 05:37.
Did you use only half-turns for the computation of God's algorithm or did you also allow quarter turns? In other words could there for example be a shorter maneuver for an antipode using also quarter turns?
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quarter turns not considered

Submitted by Bruce Norskog on Sun, 09/16/2007 - 19:16.
No, I did not use quarter-turns. (Didn't I say half-turns only a couple of times?)

Since this is a relatively small group, I would guess it's not too large a task to do the analysis in FTM. However, Kunkle/Cooperman indicated it took about a day of CPU time for the smaller ordinary squares group. I have no immediate plans on doing an optimal FTM analysis of this group.

Based upon the analysis results of the ordinary squares group, I would imagine that most and possibly all the antipodes in my analysis would have shorter sequences in FTM.
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bad link

Submitted by Bruce Norskog on Sat, 09/15/2007 - 13:28.
Oops, I had a parenthesis at the start of the URL of the first link in my post, causing it not to work. The displayed text for the link is correct. Can this be corrected?
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Fixed the link

Submitted by cubex on Sat, 09/15/2007 - 14:12.
I'll have to prepare a cube so I can look at those super-antipodes. It will be interesting to compare them to the other antipodes.

Mark
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All 112 antipodes

Submitted by Bruce Norskog on Sat, 09/15/2007 - 19:50.

Since Mark was interested in comparing antipodes (presumably with the antipodes of the ordinary 3x3x3 squares group), I thought I would provide the full list.

As Mark had noted for the ordinary 3x3x3 squares group, parity considerations force all antipodes to be "strict" global maxima. So any move from an antipode gets you one move closer to solved. Since the code I used to generate the sequences always tries U2 before trying the other moves, and U2 (or any of the other five half-turn moves) from an antipode position gets you one move closer to solved, all the sequences start with U2.

  1: U2 L2 U2 F2 L2 U2 L2 B2 U2 L2 D2 B2 L2 U2 R2 B2
  2: U2 L2 U2 L2 U2 B2 L2 F2 D2 B2 L2 U2 R2 U2 L2 F2
  3: U2 D2 L2 U2 F2 U2 F2 R2 F2 L2 D2 B2 U2 F2 L2 F2
  4: U2 L2 U2 R2 U2 F2 L2 B2 U2 F2 L2 D2 R2 U2 R2 F2
  5: U2 L2 U2 L2 F2 U2 R2 D2 L2 D2 F2 L2 B2 D2 B2 R2
  6: U2 L2 U2 L2 F2 U2 R2 D2 L2 U2 B2 R2 F2 U2 B2 R2
  7: U2 D2 L2 U2 F2 U2 F2 L2 F2 L2 D2 B2 U2 F2 R2 F2
  8: U2 L2 U2 L2 U2 B2 R2 F2 U2 B2 L2 D2 L2 U2 L2 B2
  9: U2 L2 U2 L2 U2 B2 L2 F2 D2 B2 L2 D2 L2 D2 L2 F2
 10: U2 L2 U2 L2 U2 B2 R2 F2 D2 F2 L2 U2 R2 U2 R2 B2
 11: U2 D2 L2 U2 F2 U2 F2 R2 F2 L2 D2 B2 U2 B2 R2 B2
 12: U2 L2 U2 R2 U2 B2 R2 F2 U2 F2 L2 D2 R2 U2 R2 F2
 13: U2 L2 U2 L2 F2 U2 R2 D2 L2 U2 F2 R2 B2 U2 B2 L2
 14: U2 L2 U2 L2 F2 U2 R2 U2 R2 U2 B2 L2 F2 D2 B2 L2
 15: U2 D2 L2 U2 F2 D2 F2 L2 F2 R2 U2 B2 U2 F2 R2 F2
 16: U2 L2 U2 L2 U2 F2 R2 B2 U2 F2 L2 D2 L2 U2 R2 F2
 17: U2 L2 U2 L2 U2 F2 L2 B2 D2 F2 L2 U2 R2 U2 R2 B2
 18: U2 D2 L2 U2 F2 D2 F2 R2 F2 R2 U2 B2 U2 F2 L2 F2
 19: U2 L2 U2 R2 U2 B2 L2 F2 U2 B2 L2 D2 R2 U2 L2 B2
 20: U2 D2 L2 F2 L2 U2 L2 U2 F2 R2 B2 L2 D2 R2 U2 F2

 21: U2 D2 L2 F2 L2 U2 L2 U2 B2 L2 B2 R2 U2 L2 D2 B2
 22: U2 D2 L2 F2 L2 D2 L2 U2 B2 R2 F2 L2 U2 L2 U2 F2
 23: U2 D2 L2 F2 L2 D2 L2 U2 B2 R2 F2 L2 D2 R2 D2 F2
 24: U2 D2 L2 F2 L2 D2 L2 U2 F2 L2 F2 R2 U2 L2 U2 B2
 25: U2 D2 L2 F2 L2 U2 L2 U2 B2 L2 B2 R2 D2 R2 U2 B2
 26: U2 D2 L2 F2 L2 D2 L2 U2 F2 R2 F2 R2 D2 L2 D2 B2
 27: U2 D2 L2 F2 L2 U2 L2 U2 B2 R2 B2 R2 D2 L2 U2 B2
 28: U2 D2 L2 F2 L2 D2 L2 U2 B2 L2 F2 L2 D2 L2 D2 F2
 29: U2 D2 L2 U2 F2 D2 F2 R2 F2 R2 U2 F2 U2 F2 L2 B2
 30: U2 D2 L2 U2 F2 U2 F2 R2 F2 L2 U2 B2 D2 B2 R2 F2
 31: U2 D2 L2 U2 F2 D2 F2 R2 F2 R2 D2 B2 D2 F2 L2 B2
 32: U2 D2 L2 U2 F2 U2 F2 L2 F2 L2 D2 F2 U2 B2 L2 F2
 33: U2 D2 L2 U2 F2 D2 F2 L2 F2 R2 D2 B2 D2 F2 R2 B2
 34: U2 D2 L2 U2 F2 U2 F2 R2 F2 L2 D2 F2 U2 B2 R2 F2
 35: U2 L2 U2 L2 U2 F2 R2 B2 U2 F2 L2 U2 R2 U2 R2 B2
 36: U2 L2 U2 L2 F2 U2 R2 U2 L2 U2 F2 R2 B2 U2 F2 R2
 37: U2 L2 U2 R2 U2 B2 R2 F2 U2 F2 L2 D2 R2 D2 R2 B2
 38: U2 L2 U2 R2 U2 B2 L2 F2 U2 B2 L2 U2 L2 U2 L2 F2
 39: U2 L2 U2 L2 U2 F2 L2 B2 D2 F2 L2 D2 L2 U2 R2 F2
 40: U2 L2 U2 L2 U2 F2 R2 B2 U2 F2 L2 D2 L2 D2 R2 B2

 41: U2 L2 U2 L2 U2 F2 L2 B2 U2 B2 L2 U2 R2 U2 L2 F2
 42: U2 L2 U2 L2 F2 U2 R2 U2 L2 U2 B2 R2 F2 U2 F2 L2
 43: U2 L2 U2 R2 U2 F2 R2 B2 U2 B2 L2 U2 L2 U2 L2 F2
 44: U2 L2 U2 R2 U2 F2 L2 B2 U2 F2 L2 U2 L2 U2 R2 B2
 45: U2 L2 U2 L2 U2 B2 L2 F2 D2 B2 L2 D2 L2 U2 L2 B2
 46: U2 L2 U2 L2 U2 B2 R2 F2 U2 B2 L2 U2 R2 U2 L2 F2
 47: U2 L2 U2 L2 F2 U2 R2 D2 R2 U2 F2 L2 B2 D2 F2 L2
 48: U2 D2 L2 U2 F2 D2 F2 R2 F2 L2 U2 B2 U2 B2 R2 F2
 49: U2 L2 U2 R2 U2 F2 L2 B2 U2 B2 L2 D2 R2 U2 L2 F2
 50: U2 D2 L2 U2 F2 U2 F2 L2 F2 R2 D2 B2 U2 F2 R2 B2
 51: U2 L2 U2 L2 U2 B2 L2 F2 U2 B2 L2 D2 L2 U2 L2 F2
 52: U2 D2 L2 F2 L2 U2 L2 U2 F2 R2 F2 R2 D2 L2 U2 F2
 53: U2 D2 L2 F2 L2 U2 L2 U2 B2 L2 F2 L2 D2 L2 U2 B2
 54: U2 D2 L2 F2 L2 D2 L2 U2 B2 R2 B2 R2 D2 L2 D2 F2
 55: U2 L2 U2 L2 F2 U2 L2 U2 R2 D2 F2 R2 B2 D2 B2 L2
 56: U2 L2 U2 L2 U2 F2 R2 B2 D2 F2 L2 D2 L2 U2 R2 B2
 57: U2 L2 U2 F2 L2 U2 L2 B2 U2 L2 U2 F2 R2 U2 L2 F2
 58: U2 L2 U2 L2 F2 U2 L2 F2 R2 D2 R2 D2 B2 R2 D2 B2
 59: U2 L2 U2 R2 B2 U2 L2 F2 L2 U2 L2 D2 F2 R2 U2 F2
 60: U2 L2 U2 L2 F2 U2 L2 F2 R2 U2 L2 D2 F2 L2 U2 B2

 61: U2 L2 U2 L2 F2 U2 L2 F2 R2 U2 L2 U2 B2 R2 D2 B2
 62: U2 L2 U2 L2 F2 U2 F2 L2 B2 U2 R2 U2 L2 D2 B2 L2
 63: U2 L2 U2 L2 F2 L2 B2 U2 R2 F2 B2 D2 R2 B2 L2 B2
 64: U2 L2 U2 R2 F2 U2 L2 B2 R2 D2 L2 U2 B2 L2 U2 B2
 65: U2 L2 U2 L2 F2 L2 F2 U2 R2 F2 B2 D2 L2 B2 R2 F2
 66: U2 L2 U2 D2 F2 R2 F2 D2 F2 D2 L2 B2 R2 F2 U2 B2
 67: U2 L2 U2 D2 F2 L2 F2 D2 F2 D2 L2 B2 L2 F2 U2 B2
 68: U2 L2 U2 D2 F2 R2 F2 U2 F2 U2 L2 F2 R2 B2 U2 F2
 69: U2 L2 U2 D2 F2 L2 F2 D2 F2 U2 R2 F2 R2 B2 D2 B2
 70: U2 D2 L2 F2 D2 F2 U2 L2 F2 R2 F2 U2 B2 U2 L2 B2
 71: U2 L2 U2 L2 F2 U2 D2 L2 B2 D2 L2 D2 R2 F2 R2 F2
 72: U2 D2 L2 F2 D2 F2 U2 R2 F2 L2 B2 U2 F2 U2 L2 F2
 73: U2 L2 U2 L2 F2 U2 D2 L2 F2 U2 R2 U2 L2 F2 L2 B2
 74: U2 D2 L2 F2 U2 F2 U2 R2 F2 L2 F2 U2 B2 D2 L2 B2
 75: U2 L2 U2 L2 F2 L2 U2 D2 F2 D2 R2 D2 R2 B2 L2 F2
 76: U2 D2 L2 F2 D2 F2 U2 R2 F2 L2 F2 D2 F2 D2 R2 B2
 77: U2 L2 U2 D2 F2 R2 F2 D2 F2 U2 L2 B2 R2 B2 D2 B2
 78: U2 D2 L2 F2 U2 F2 U2 R2 F2 L2 B2 D2 B2 U2 R2 F2
 79: U2 D2 L2 F2 D2 F2 U2 R2 F2 L2 F2 U2 B2 U2 R2 B2
 80: U2 D2 L2 F2 D2 F2 U2 L2 F2 R2 B2 U2 F2 U2 R2 F2

 81: U2 L2 U2 D2 F2 L2 F2 D2 F2 D2 R2 F2 R2 F2 U2 B2
 82: U2 D2 L2 F2 U2 F2 U2 L2 F2 R2 F2 U2 B2 D2 R2 B2
 83: U2 L2 U2 L2 F2 L2 F2 D2 L2 F2 B2 D2 R2 B2 L2 F2
 84: U2 L2 U2 L2 B2 U2 L2 F2 R2 U2 L2 U2 B2 R2 U2 B2
 85: U2 L2 U2 L2 B2 U2 F2 R2 B2 U2 L2 U2 R2 U2 B2 R2
 86: U2 L2 U2 L2 F2 L2 F2 U2 R2 F2 B2 U2 R2 B2 L2 F2
 87: U2 L2 U2 L2 F2 L2 B2 D2 L2 F2 B2 D2 L2 B2 R2 B2
 88: U2 L2 U2 L2 F2 U2 L2 B2 L2 U2 R2 D2 B2 R2 D2 F2
 89: U2 L2 U2 L2 F2 U2 L2 B2 L2 D2 L2 D2 F2 L2 U2 F2
 90: U2 L2 U2 L2 F2 L2 F2 U2 F2 B2 R2 U2 R2 F2 L2 B2
 91: U2 L2 U2 D2 F2 R2 F2 D2 F2 U2 L2 B2 R2 F2 U2 F2
 92: U2 L2 U2 D2 F2 L2 F2 U2 F2 U2 R2 B2 R2 F2 D2 B2
 93: U2 L2 U2 D2 F2 R2 F2 U2 F2 U2 R2 B2 L2 F2 D2 B2
 94: U2 L2 U2 D2 F2 L2 F2 D2 F2 D2 R2 F2 R2 B2 D2 F2
 95: U2 L2 U2 D2 F2 R2 F2 U2 F2 D2 L2 F2 R2 B2 U2 B2
 96: U2 D2 L2 F2 U2 F2 U2 L2 F2 L2 B2 U2 F2 D2 R2 B2
 97: U2 L2 U2 L2 F2 U2 D2 L2 F2 U2 R2 U2 R2 B2 R2 B2
 98: U2 L2 U2 L2 F2 U2 D2 L2 F2 D2 L2 D2 R2 B2 R2 B2
 99: U2 D2 L2 F2 D2 F2 U2 R2 F2 R2 B2 U2 F2 U2 R2 B2
100: U2 L2 U2 L2 F2 U2 D2 L2 B2 U2 R2 U2 L2 B2 L2 F2

101: U2 L2 U2 L2 F2 L2 U2 D2 F2 D2 R2 D2 L2 F2 R2 F2
102: U2 L2 U2 L2 F2 L2 U2 D2 B2 D2 R2 D2 R2 F2 L2 B2
103: U2 D2 L2 F2 D2 F2 U2 R2 F2 R2 F2 D2 F2 D2 L2 F2
104: U2 L2 U2 D2 F2 L2 F2 D2 F2 U2 R2 B2 R2 B2 D2 F2
105: U2 D2 L2 F2 U2 F2 U2 R2 F2 R2 B2 U2 F2 D2 L2 B2
106: U2 L2 U2 D2 F2 R2 F2 D2 F2 U2 R2 B2 L2 B2 D2 F2
107: U2 D2 L2 F2 D2 F2 U2 L2 F2 L2 B2 U2 F2 U2 L2 B2
108: U2 L2 U2 D2 F2 R2 F2 D2 F2 D2 L2 F2 R2 F2 U2 F2
109: U2 L2 U2 L2 B2 U2 L2 B2 R2 U2 L2 U2 F2 R2 U2 F2
110: U2 L2 U2 L2 B2 U2 B2 R2 F2 U2 L2 U2 R2 U2 F2 R2
111: U2 L2 U2 L2 F2 U2 L2 F2 L2 D2 L2 D2 B2 L2 U2 B2
112: U2 L2 U2 D2 F2 R2 F2 D2 F2 D2 L2 F2 R2 B2 D2 B2
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