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I recently bought a new computer and wanted to put it through its paces. I dusted off my RUF three face coset solver and spruced it up a bit. Since I now have three iMacs in my household connected on an airport network, I rewrote the program using a server–client model. With this I can have all three computers working on a problem in parallel with as many as 14 cores. With these tools I have extended the enumeration of the three face group out to twenty q–turns:

Three Face Enumerator Client

Fixed cubies in subgroup: UF, UR, UB, UL, DF, DR, FR, FL, BR.
92,897,280 cosets of size 1,837,080

Server Status:
Three Face Group Enumerator
Sequential coset iteration
Enumeration to depth: 20

Snapshot: Friday, February 22, 2013 9:28:02 PM Central Standard Time

 Depth             Reduced             Elements
   0                     1                    1 
   1                     1                    6 
   2                     4                   27 
   3                    12                  120 
   4                    51                  534 
   5                   213                2,376 
   6                   914               10,560 
   7                 4,038               46,920 
   8                17,639              208,296 
   9                78,234              923,586 
  10               344,175            4,091,739 
  11             1,524,115           18,115,506 
  12             6,722,358           80,156,049 
  13            29,739,437          354,422,371 
  14           131,158,304        1,565,753,405 
  15           578,971,538        6,908,670,589 
  16         2,546,820,524       30,422,422,304 
  17        11,174,670,698      133,437,351,006 
  18        48,528,827,222      579,929,251,620 
  19       205,901,170,504    2,459,821,160,421 
  20       814,027,054,726    9,731,195,124,049 

 Sum     1,082,927,104,708   12,943,737,711,485 

92,897,280 of 92,897,280 cosets solved

Having repeatedly shot myself in the foot by mishandling the symmetry reduction of coset spaces, I finally sat down, laid out the math and put together a set of notes on the matter. These notes follow.

Coset Spaces

Solving Rubik's cube either manually or by computer usually involves dealing with coset spaces. A group may be partitioned into cosets of a subgroup of the group:

     g * SUB          where g is an element of the parent group and 
	              SUB is a subgroup of the parent group