Symmetric Patterns in Detail

You can search for cubes of all symmetry types with the Symmetry Editor module of Cube Explorer.

Look here first for the mathematical background of symmetric patterns and an explanation of the pictograms.

An external page with good information about the schoenflies symbols can be found here.

We know God's algorithm for all the 164,604,041,664 symmetric cubes which exist. The following table gives the distribution:

 Distance Number Distance Number 0f 1 11f 9,732,164 1f 18 12f 35,024,904 2f 51 13f 122,054,340 3f 312 14f 436,197,214 4f 1,335 15f 1,763,452,505 5f 4,380 16f 8,035,307,127 6f 17,782 17f 37,542,012,922 7f 70,188 18f 95,387,902,305 8f 229,336 19f 21,267,102,443 9f 851,139 20f 1,091,994 10f 2,989,204 21f 0

Reducing the 1,091,994 symmetric cubes with 20 moves by symmetry and antisymmetry we find exactly 32,625 essentially different symmetric cubes which need 20 moves to be solved. They are included in the file 20moves.zip.

The details for the different symmetry types can be found below.

 Type Schoenflies-Symbol Number of Symmetries Number of cubes having at least this symmetry Shortest generator for exactly this symmetry More Information Oh 48 4 do nothing yes O 24 4 --- yes Td 24 4 --- yes Th 24 24 U2 L2 F2 D2 U2 F2 R2 U2 yes T 12 72 B F L R B' F' D' U' L R D U yes D3d 12 16 U L D U L' D' U' R B2 U2 B2 L' R' U' yes C3v 6 48 U L' R' B2 U' R2 B L2 D' F2 L' R' U' yes D3 6 432 D B D U2 B2 F2 L2 R2 U' F U yes S6 6 7776 B' D' U L' R B' F U yes C3 3 3,779,136 L' R U2 R2 D2 F2 L R D2 yes D4h 16 128 U2 D2 yes D4 8 512 U D yes C4v 8 1024 D2 yes C4h 8 1536 U D' yes C4 4 147456 D yes S4 4 442368 U R2 L2 U2 R2 L2 D yes D2d (edge) 8 3072 U F2 B2 D2 F2 B2 U yes D2d (face) 8 512 U R L F2 B2 R' L' U yes D2h (edge) 8 2048 U R2 L2 D2 F2 B2 U yes D2h(face) 8 12288 B2 D2 U2 F2 yes D2 (edge) 4 98304 U F2 U2 D2 F2 D yes D2 (face) 4 294912 R2 L2 F B yes C2v (a1) 4 65536 U R2 L2 U2 F2 B2 U' yes C2v (a2) 4 1,179,648 R2 L2 U2 yes C2v (b) 4 98304 B2 R2 B2 R2 B2 R2 yes C2h (a) 4 589824 U' D F2 B2 yes C2h (b) 4 98304 U R2 U D R2 D yes C2 (a) 2 15,288,238,080 L R U2 yes C2 (b) 2 2,548,039,680 U R2 D' U' R2 U' yes Cs (a) 2 18,345,885,696 F2 R2 yes Cs (b) 2 424,673,280 U B2 U D B2 D' yes Ci 2 45,864,714,240 U D' R L' yes C1 1 43,252,003,274,489,856,000 U R yes