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mark writes

This brings up the idea of a "Rubik's Tour". Such a tour would

touch on a set of interesting patterns within a given subgroup,

or potentially the entire cube group. Of course, "God's Tour"

would not only touch on all the interesting patterns, it would

also sequence all the patterns AND orient them in space such that

the number of q turns would be minimal for the tour! I am currently

working on "God's Tour" for some of the lesser subgroups, touching on

say a dozen patterns for the square's group. If humans and computers

ever resolve "God's Algorithm" there is some solace that there are

problems even more intractible.

there's a general graph theory conjecture that cayley graphs are

hamiltonian (i.e. have hamiltonian circuits).

if we take the cayley graph formed by generators

{F, F', L, L' U, U', R, R', B, B', D, D'}, the conjecture asserts

that there is a sequence of N quarter turns that visits every position

exactly once and returns to START. (here N = 43252003274489856000

is the order of the group.)

so the proposed "simplest" solution to the cube is to apply such a

hamiltonian sequence. at some point, in the middle of the sequence,

the cube will be solved! no need to continue with the rest of the

sequence.

i don't think the general conjecture is close to being proved, but

it is known for some special groups and generators. it would be

interesting to know if anyone can verify the conjecture for the cube

group with quarter turn generators. (face turn generators would also

be interesting.)

mike