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At last, 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. Note that if we have any legally scrambled Rubik's Cube position as the starting point, then applying the sequence would result in the cube being solved at some point within the sequence.

(Note that the following discussion describing how the solution was found is somewhat technical and assumes some basic knowledge of the branch of mathematics known as group theory and common conventions used in mathematical discussions of Rubik's Cube.)

The Hamiltonian circuit was constructed in a hierarchical manner using nested subgroups. These subgroups are: < UR > < U, R > < U, R, D > < U, R, D, L > < U, R, D, L, F > The last of these subgroups, of course, is actually the whole cube group. The entire Hamiltonian circuit consists of turns of only five face layers. No turns of back layer are used!

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