4x4x4 Supercube Squares Group

At last, I have completed my God's algorithm calculation for the 4x4x4 supercube squares group, in the single-slice half-turns only metric. It was found to have 47,968,768 antipodes, many more elements than are in the entire 3x3x3 supercube squares group. (OK, it's only 11,992,192 antipodes after reducing the group by four cube orientations.) The analysis showed that the elements in the group can always be solved using no more than 20 half-turns, only one more half-turn than the number required if the four centers for each face are considered indistinguishable. (The 4x4x4 supercube differs from the usual 4x4x4 cube in that all 24 centers are considered to be distinguishable from one another.)

The 4x4x4 supercube squares group has 9,393,093,476,352 elements. There are 96 possible configurations of the corners, 96 possible configurations of the centers for each of three sets of opposite faces, and 96 possible configurations of each of three sets of eight edges. There are parity effects reducing the total number of reachable positions by a factor of eight. So the group has (96^7)/8 = 9,393,093,476,352 elements. The group has 64 times the number of configurations of the set of the corresponding set of positions if the centers for each face are considered indistinguishable.

For each position, there are three other positions corresponding to the same configuration of the cube, but with a different orientation of the cube. (Quarter-turn cube rotations need not be considered, and half-turn cube rotations generate a total of four possible orientations.) Considering such sets of four to be equivalent would further reduce the number of elements of the group by a factor of four to 2,348,273,369,088. In my analysis, however, I did not use this reduced size group, so I could take advantage of all 48 symmetries of the cube. I also used antisymmetry to reduce the number of positions that I needed to store data for. Since I considered four equivalent orientations of the solved cube as "solved" positions, the table below starts with four positions of distance 0. Note all numbers in the positions columns are divisible by four because there are always sets of four positions differing only by cube orientation.

In my analysis, I used a sym-coordinate for the center configurations (12,331 with 96 antisymmetry group elements), and "parity-reduced" coordinates for corners (48) and edges (221,148). This gives a total coordinate space of 12,331 * 48 * 221,148 = 130,916,155,392 positions. I used 59 files to store distances for each position, using 5 bits per position, packing 6 values in 4 bytes. In the table below, the "Reduced count" column represents the position counts in this coordinate space.

4x4x4 Supercube Squares Group (single-slice half-turns only metric)

Distance      Positions      Reduced count   Unique wrt M+inv
   0                  4                  2                  2
   1                 48                  6                  6
   2                420                 20                 17
   3              3,648                119                 84
   4             30,540                865                492
   5            244,624              4,971              3,180
   6          1,918,408             36,394             22,199
   7         14,481,096            244,288            157,932
   8        104,265,996          1,691,070          1,111,460
   9        707,532,096         10,865,490          7,453,633
  10      4,413,991,040         65,844,035         46,300,374
  11     24,381,323,856        355,598,016        255,114,903
  12    113,468,246,152      1,620,265,693      1,185,903,197
  13    421,706,542,552      5,919,553,410      4,403,422,070
  14  1,189,403,465,984     16,524,243,027     12,412,484,307
  15  2,359,825,595,824     32,570,946,016     24,624,024,712
  16  2,895,046,677,616     40,066,113,814     30,220,825,890
  17  1,864,204,674,656     26,205,700,816     19,474,596,454
  18    494,060,173,248      7,178,711,192      5,175,559,998
  19     25,706,339,776        395,164,564        271,822,858
  20	     47,968,768          1,171,584            654,096
      -----------------    ---------------    ---------------
Total 9,393,093,476,352    130,916,155,392     98,079,457,864