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A Hamiltonian Circuit for Rubik's Cube
, the Hamiltonian circuit problem for Rubik's Cube has a solution! To be a little more mathematically precise, a Hamiltonian circuit of the quarter-turn metric Cayley graph for the Rubik's Cube group has been found. Basically it is a sequence of quarter-turn moves that would (in theory) put a Rubik's cube through all of its 43,252,003,274,489,856,000 positions without repeating any of them, and then one more move restores the cube to the starting position.
To be a little more mathematically precise, a Hamiltonian circuit of the quarter-turn metric Cayley graph for the Rubik's Cube group has been found. Of course, the Hamilton circuit for this group is simply the two-move sequence UR repeated 105 times. Since R'D has order 63, a cycle of 63 cosets is easily created using the above Hamiltonian path followed by the move D, and repeating this 63 times.
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