God's number for 3-gen 3x3x3 (<U,F,R>) is 21

[Ben Whitmore]

I used my free Claude Fable 5.2 tokens this week to make a coset solver for G = <U,F,R> using the subgroup H = <U,F2,R2>. There are 15676416 cosets of size 10886400. Using reflection symmetry can cut that almost in half, then using 3-fold rotational symmetry, we can find a small subset of cosets such that every remaining coset is equivalent to one of the original ones by some rotation. This left 3072645 cosets which were then solved in around 8 hours (around 100 cosets per second) on my 6-core desktop machine. The cosets are searched up to depth 18 followed by 3 prepasses to get almost all positions in 21 moves. This left 16887175 positions unresolved. Their symmetries and inverse are then checked to see if any of those were already resolved in another coset, which eliminates all but 613 positions. These 613 were then solved with a two-phase solver, all within 21 moves.

Results of solving the cosets (up to depth 18 is the exact count of positions in <U,F,R>, depth 19-21 are not the exact count of positions):

  depth           positions           cumulative   % of all  (log scale)
      0                   1                    1    0.0000%                               
      1                   9                   10    0.0000%  ██                           
      2                  54                   64    0.0000%  ███▋                         
      3                 324                  388    0.0000%  █████▍                       
      4               1,944                2,332    0.0000%  ███████                       
      5              11,664               13,996    0.0000%  ████████▊                     
      6              69,969               83,965    0.0000%  ██████████▍                   
      7             419,526              503,491    0.0000%  ████████████▏                 
      8           2,514,261            3,017,752    0.0000%  █████████████▊               
      9          15,058,551           18,076,303    0.0000%  ███████████████▌             
     10          90,138,505          108,214,808    0.0001%  █████████████████▏           
     11         539,293,253          647,508,061    0.0004%  ██████████████████▉           
     12       3,224,453,612        3,871,961,673    0.0023%  ████████████████████▌         
     13      19,260,064,402       23,132,026,075    0.0136%  ██████████████████████▎       
     14     114,800,043,247      137,932,069,322    0.0808%  ███████████████████████▉     
     15     679,979,206,606      817,911,275,928    0.4793%  █████████████████████████▌   
     16   3,933,938,371,696    4,751,849,647,624    2.7844%  ███████████████████████████▎ 
     17  20,638,076,101,811   25,389,925,749,435   14.8775%  ████████████████████████████▊
     18  72,749,983,009,545   98,139,908,758,980   57.5062%  ██████████████████████████████
     19* 61,614,544,947,293  159,754,453,706,273   93.6099%  █████████████████████████████▊
     20* 10,734,447,281,959  170,488,900,988,232   99.8999%  ████████████████████████████▏
     21*    170,817,266,993  170,659,718,255,225  100.0000%  ████████████████████████▎     
   left          16,887,175  not reached by depth 21 (resolved through rotations, inverses, two-phase solver)
  depths ≤ 18 are exact; * = first reached at that depth by this method (true distance may be lower)