# 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 http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/13163), 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!)*(2^{5}) = 3041280 edge configurations, and (8!/6!)*(3^{2}) = 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