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Refining Bruce Norskog's 4x4x4 five stage analysis
Submitted by Clement Gallet on Fri, 12/11/2015 - 08:03.Bruce Norskog designed a five stage procedure to solve 4x4x4 positions, where the last stage is solving inside the squares subgroup. When I rewrote the code in Java to be used for the official WCA scrambler program, I found a few improvements on coordinate representation and symmetry reduction that I think is worth sharing.
Except for the last stage where centers need to be solved, every other stage needs to place 8 centers from a single axis (RL, FB or UD centers) in their correct faces (RL, FB or UD faces), in one of the 12 positions that can then be solved using only half turns. However, as pointed by Shuang Chen in his analysis, we don't need to store the exact colours of centers but only if two centers have the same or different colours. This reduces the number of center coordinates by 2.
Also, as opposed to the 3x3x3, 4x4x4 does not have fixed centers, which allows us to do more symmetry reduction. I'm representing a sym-coordinate as:
s1 * g * < H > * s1' * s2'
where g * < H > is a coset, s1 and s2 are symmetries from subgroups S1 and S2 of the group M of symmetries of the cube. s1 is the usual conjugated symmetry, and s2 correspond to a rotation of the cube, which is possible on the 4x4x4. Using carefully chosen subgroups S1 and S2 for each stage, more symmetry reduction is achieved. I will be using the Schoenflies symbols for subgroups of M in the following.
Except for the last stage where centers need to be solved, every other stage needs to place 8 centers from a single axis (RL, FB or UD centers) in their correct faces (RL, FB or UD faces), in one of the 12 positions that can then be solved using only half turns. However, as pointed by Shuang Chen in his analysis, we don't need to store the exact colours of centers but only if two centers have the same or different colours. This reduces the number of center coordinates by 2.
Also, as opposed to the 3x3x3, 4x4x4 does not have fixed centers, which allows us to do more symmetry reduction. I'm representing a sym-coordinate as:
s1 * g * < H > * s1' * s2'
where g * < H > is a coset, s1 and s2 are symmetries from subgroups S1 and S2 of the group M of symmetries of the cube. s1 is the usual conjugated symmetry, and s2 correspond to a rotation of the cube, which is possible on the 4x4x4. Using carefully chosen subgroups S1 and S2 for each stage, more symmetry reduction is achieved. I will be using the Schoenflies symbols for subgroups of M in the following.
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