# 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