The first answer is that there are exactly 878,880 cube
positions at a distance of 6 quarter-twists from solved, and so
983,926 positions at 6qtw or less. These figures reflect a decrease
of 744 from the previously known upper bounds.
It turns out that the twelve-qtw identities reported by
Chris C. Worrell <ZILCH at MIT-MC> are complete, in a sense. The
only reservation here is that a fifth rule must be added to his
list of the ways in which ``a generator generates other
identities.'' This rule is substitution with shorter identities,
and it's not too surprising that it was left out, since the only
shorter identities are the ``trivial'' ones like XXXX=XYX'Y'=I,
where X and Y are opposite faces. In the case of the twelve-qtw
identities, this means that identities of the form aXXb and aX'X'b
generate each other.
The structure of the 12-qtw identities is clearer if we
write them in a transformed way:
I12-1 FR' F'R UF' U'F RU' R'U I12-2 FR' F'R UF' F'L FL' U'F I12-3 FR' F'R UF' UL' U'L FU'
The fifth rule is necessary so that I12-2 may generate the
I12-2a FR' F'R UF FL FL' U'F and I12-2b F'R' F'R UF FL FL' U'F'.
To see that this rule is necessary, it need only be observed that
inversion, rotation, reflection, and shifting all preserve the
number of clockwise/counterclockwise sign changes between
cyclically adjacent elements.
In what sense are the ``trivial'' identities trivial? I
have come to believe that they are trivial only because they are
short and simple enough that they are well-understood. The only
identities for which I can find any theoretical reasons for calling
trivial are the identities of the form XX'=I. In spite of the
simplicity of the ``trivial'' identities, their occurrence is one
of the major reasons that Alan Bawden and I were unable to show
earlier that I12-1-3 generated all identities of length 12. I fear
that the combination of 4-qtw and 12-qtw identities may turn out to
be a major headache when dealing with identities of length 14 and