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Analysis of another two symmetry subgroups of order 4
Submitted by Herbert Kociemba on Wed, 03/08/2006 - 12:46.
The symmetry class C4 defines a 1/4-rotational symmetry around a face (I chose
the UD-axis). It took about 8 days to show that all 36160 cubes, which exactly
have this symmetry (M-reduced) are solvable in at most 20 moves. There are 39
20f*-cubes. 35 of them also have antisymmetry, 4 only have symmetry, so reduced
wrt M+inv there are 37 cubes.
The class D2 (face) consists of all cubes which have a 1/2-rotational symmetry around all faces. Up to M-symmetry there are 23356 cubes, which exactly have this symmetry. It took about 4 days to show, that all cubes of this symmetry class can be solved in 20 moves. There are only 4 cubes which are 20f*, all of them also are antisymmetric. Here are the results: D B2 D' L2 U L' U F L F U' R' B2 D B U B F' D R' (20f*) //C4 B2 U R2 B D U2 R2 B R' D R2 D2 F' D' L' R' B' F' R' U' (20f*) //D2 (face) D' F2 L2 U B2 F' R D L D' B D' F2 D' B F2 R2 D' B R' (20f*) D U F2 U' F2 L' D2 F' R D2 R F L' D' R B' R' F D U' (20f*) L2 B R2 D' B2 L2 R' F2 D B F' D L' B U L U F2 R' U' (20f*) To complete the analysis of all symmetric subgroups of order 4, there are 6 groups left. The largest has 290880 elements, so this is at the edge what I can compute with one PC within a reasonable time. I strongly believe that all will be solvable in 20 moves too. |
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