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Solving the 4x4x4 in 57 moves(OBTM)
Submitted by CS on Mon, 09/30/2013 - 13:31.
According to my computation, the 4x4x4 cube can be solved no more than 57 moves.
The solving algorithm is based on tsai's 8-step method which can be found here: link The only modification is that I merged step 3 & step 4 to one step. For some reasons, the algorithm cannot be defined to the conversions between subsets, but can be defined to the conversions between sets: S0: <U R F D L B Uw Rw Fw Dw Lw Bw> step 1 => S1: <U R F D L B> * <U R F D L B Uw2 Rw Fw2 Dw2 Lw Bw2> step 2 => S2: <U R F D L B> * <U R2 F D L2 B Uw2 Rw2 Fw2 Dw2 Lw2 Bw2> step 3 & step4 => S3: <U R F D L B> 3x3x3 solver => S4: Solved State In step 1, we should: Place R&L centers to R&L faces which can be done in 8 moves: depth states 0 1 1 4 2 82 3 1206 4 14116 5 123404 6 422508 7 173254 8 896 In step 2, we should: Separate low edges and high edges (or the orientation of the edges. I have no idea how to deal with it in a simple way, so I modified it to "let low edges to low places and high edges to high places, or low to high & high to low",totally C(24,12)/2 = 2,704,156 different states or 86048 symmetry classes) U&D centers to U&D faces, F&Bcenters to F&B faces ( C(16, 8)/2 ) R&L centers to one of 12 states which can be solved in step 3 (12 solved states in 70, or 6 in 35 if we only care about the separation of R L pieces) The parity of all edges ( 2 ) which can be done in 14 moves: depth states mod M 0 6 1 12 2 48 3 381 4 3643 5 45030 6 606937 7 8154706 8 102867620 9 1114713818 10 8194798024 11 21637154427 12 7652855512 13 49121344 14 92 total 38760321600 = 86048 * 6435 *35 * 2Notice that in depth 14, there're only 92 symmetry classes and no more than 92*16 = 1472 states while the orientation of the edges can be defined in 2048 different ways. After redefining edges, the 14-move states won't occur. And the step 2 can be solved in 13 moves. In merged step 3 & step 4, we should: Pairing Edges. (12!/2 different states or 14999140 symmetry classes) Solving centers. (C(8, 4)/2 * C(8, 4)/2* 12) Fix the parity of centers, edges and corners. (2) which can be solved in 16 moves: depth states mod M 0 1 1 2 2 11 3 85 4 547 5 3757 6 29154 7 249155 8 2287529 9 21385620 10 197829833 11 1788075646 12 14489430468 13 86675095891 14 242701855493 15 94978862660 16 119610148 total 440974716000 = 14999140 * 35 *35 * 12 * 2 After step 3 & step 4, the reduction is finished and the cube can be solved by <U, R, F, D,L, B>. According to the god's number of 3x3x3, the step 5 to step 8 can be solved in 20 moves. Thus, the 4x4x4 cube can be solved in 8 + 13 + 16 + 20 = 57 moves. |
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