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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

identities

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

16.