A Hamiltonian circuit for Rubik's Cube!

I have found a Hamiltonian circuit for the quarter-turn metric Cayley graph of Rubik's Cube! In fact, it only uses turns of five of the six outer layers of the cube.

In more basic terms, this is a sequence of quarter 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.

Additional information is found at my web site: http://bruce.cubing.net/index.html

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Late to the party on this I k

Late to the party on this I know...

This is a question regarding the 2x2x2 pocket cube Hamiltonian circuit that you have provided as the recursive string z (ultimately boiling down to a long sequence of U, F and R's and their respective inverses V, G and S - if my understanding is correct). How much was done to "optimise" this? Could there be a series with much fewer intermediate variables and of smaller length?

The Hamiltonian path for whole cube on five of the Singmaster generators is a different kettle of fish to fathom - great achievement in producing this. I thought it would be beyond the capability of current computing power.

Nice result!

Nice result!