Blockbuilding analyses

I've done a few more analyses that may be of some interest to the speedcubing community. I'm guessing the first two may have been done before. Such an analysis has been talked about on speedcubing forums (such as in this thread, but I haven't located any actual results. I'll be happy to give credit for any prior result, if I'm made aware of it.

The goal in these analyses is to build a 2x2x2 sub-block from a scrambled cube state. These analyses do not consider choosing the easiest of eight possible such blocks, but rather one such block is picked, and the distance distribution for all possible scrambles is determined for building that block. Only the three edges and the one corner for that block need to be considered. There are 10560 edge configurations and 24 corner configurations, for a total of 253440 positions. The analysis was carried out in both FTM and QTM.

Building a specific 2x2x2 sub-block

distance  FTM positions   QTM positions
    0                 1               1
    1                 9               6
    2                90              39
    3               852             276
    4             7,169           1,899
    5            44,182          11,245
    6           131,636          49,412
    7            68,940         117,221
    8               561          70,767
    9                 0           2,574
                -------         -------
       total    253,440         253,440

     avg dist  6.033834        6.988060

The other two analyses are similar, but are for building a 3x2x2 block. There are 12 possible 3x2x2 blocks to choose from, but the analyses only considers one specific such block. This has a total of (12!/7!)*(25) = 3041280 edge configurations, and (8!/6!)*(32) = 504 corner configurations, for a total of 1,532,805,120 positions.

Building a specific 3x2x2 sub-block

distance  FTM positions   QTM positions
    0                 1               1
    1                12               8
    2               141              64
    3             1,746             532
    4            20,935           4,533
    5           243,092          38,328
    6         2,698,935         317,688
    7        27,258,179       2,553,916
    8       216,204,042      19,267,822
    9       830,686,751     124,739,618
   10       453,825,501     531,537,338
   11         1,865,784     757,813,925
   12                 1      96,498,849
   13                 0          32,498
          -------------   -------------
   total  1,532,805,120   1,532,805,120

     avg       9.115899       10.507878

A sequence for building the 3x2x2 block for the position at distance 12 in FTM is:
D F B' U2 F2 U L2 D' L' B' R U2
(Do the inverse to create from a solved cube.) It certainly appears to me that there exists no 3x3x3 position can have this configuration for all 12 possible 3x2x2 blocks. Therefore, no 3x3x3 scramble requires 12 face turns to build some 3x2x2 block.

From superflip, a 3x2x2 block can be built using the following sequence:
U2 R2 F' U' L D' R2 F L' D R

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2x2x2 to 3x2x2 block

My analysis of UF cosets in the UFR group may be relevant here. This gives the number of turns necessary to build a 3x2x2 block once a 2x2x2 block has been solved, using just UFR q turns:

2x2x2 to 3x2x2 block vs 2x2x2 to two face group

To clarify, the analysis of the UFR/UF cosets gives the distances for extending a 2x2x2 block to a 3x2x2 block using turns of three faces in a manner such that the final two faces of the cube may be solved using just those two faces. The distances for simply extending the block with turns of three faces are considerably shorter:

    Depth  2x2x3       UF Group
      0        1            1
      1        2            2
      2        5            9
      3       18           40
      4       66          172
      5      236          725
      6      692        2,972
      7    1,513       11,991
      8    2,058       46,117
      9    1,252      157,775
     10      201      436,771
     11        4      799,123
     12        0      688,908
     13               172,118
     14                 5,702
     15                     6
     16                     0

Restricting one's self to turns of the three faces which leave the 2x2x2 block alone, it takes at most 11 q turns to extend the block to 3x2x2 but 15 turns to do so such that the cube may solved using turns of the final two faces.

My analysis is optimal, not extending 2x2x2 to 3x2x2

The <U,F> group also has all the edges oriented, while my analysis ignores the remaining 7 edges. OK, your latest reply has a table separating out just the block and full <U,F> group. My analysis considers turns on all six faces. It is also one-step - does not (necessarily) first build a 2x2x2 block, then extend it to 3x2x2. I note that the link provided by Johannes also has 2x2x2->2x2x3 in FTM. I'm guessing that analysis is equivalent to B MacKenzie's, only FTM instead of QTM.

I've thought about extending the analysis to include orienting the remaining edges. That would bring it up to nearly 100 billion positions (reducible about 4x by symmetry). So it is a fairly big calculation to carry out. Extending the analysis to first two layers minus one corner/edge slot would be 4.6 trillion cases (not including orientation of last 5 edges) with only 2x symmetry reduction to simplify it. So that would be a rather monumental task.

By the way, in my original post, the algs listed are for solving the 3x2x2 block with UFL and URF corners. You could figure that out from the superflip alg, but it would not be so obvious for the distance-12 alg.

Going directly to the two face group

The group generated by the turns of two faces has restrictions on the corner position permutation as well. This ups the number by another factor of six. This gives 588,597,166,080 cosets within the full Rubik group. Too much for me to handle.

Here's one link that has the

Here's one link that has the numbers for 2x2x2 block in FTM: I've also written an optimal solver for it (and thus got the same results), and there are so little positions that I wouldn't be surprised if some others have done it, too.

I haven't seen the numbers for 2x2x3 block before, thanks for posting! It's nice to know the optimal average and worst case.

Johannes Laire

Re: Here's one link that ...

OK, I see that the former 2x2x2 block analysis is on a non-public forum. That's probably why I couldn't find such a post with internet searches.

Thanks for that info, Johannes. I kind of thought you may have known the distribution for the 2x2x2 block analysis. So I give credit to Morley Davidson and Joe Miller for their earlier post (FTM anyway). And secondary credit to Johannes who has also independently done it previously.

My program will also allow you to produce an optimal alg for a position. I do this as a sanity check against possible bugs in the program. It's also how I got algs for interesting cases.