Date: 14 August 1981 0111-EDT (Friday) From: Dan Hoey at CMU-10A To: Cube-Lovers at MIT-MC Subject: Results of an exhaustive search to six quarter-twists Message-Id: <14Aug81 011137 DH51@CMU-10A> 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 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.