The Group C₂ (b)

Silviu Radu managed to do a complete analysis of the 2,548,039,680 different cubes of this subgroup. Here the the table with the distances, the average distance is 17.82:

Up to M-symmetry there are 636,937,008 cubes which exactly have this symmetry and 3297 of these cubes need 20 moves. All other cubes can be solved in 19 moves or less. 2153 out of these 3297cubes with 20 moves have antisymmetry. Up to M-symmetry and M-antisymmetry it are 2725 20f*-cubes, which are included in the file 20moves.zip. We display only some nicer examples of this symmetry class here.

The next table gives the number of cubes up to M-symmetry which exactly have C₂ (b)-symmetry. To compute this number we used the following identity, where C₂ (b)Total is the table above and the right side hold the tables for the number of cubes mod M with exactly the symmetry of the subgroups of C₂ (b), which are all known except for C₂ (b) itself.

C₂ (b)Total = 4 C₂ (b) + 4 D₂ (edge) + 2 C_2h (b) + 2C_2v (b) + 2 D_2d (edge) + 2 D_2h (edge) + 2 D₄ + 4 D₃ + D_4h + 2 D_3d + O_h

The average maneuver length also is 17.82:

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© 2017  Herbert Kociemba