Date: 22 January 1981 0010-EST (Thursday) From: Dan Hoey at CMU-10A, James Saxe at CMU-10A To: Cube-Lovers at MIT-MC Subject: Correction to "Symmetry and Local Maxima" Reply-To: Dan Hoey at CMU-10A Message-Id: <22Jan81 001000 DH51@CMU-10A> In our message "Symmetry and Local Maxima" (14 December 1980 1916-EST) we examined local maxima both in the Rubik group and in the Supergroup. David C. Plummer has discovered a flaw in our argument for the Supergroup, which we now correct. Plummer has previously noted (30 DEC 1980 0109-EST) that the T-symmetric position GIRDLE CUBIES EXCHANGED, depicted near the end of section 4, is an odd distance from SOLVED. This is also true of the composition of GIRDLE CUBIES EXCHANGED with GIRDLE EDGES FLIPPED, ALL EDGES FLIPPED, PONS ASINORUM, or any combination of the three, for a total of eight positions. In addition, there are four different T groups, each corresponding to a choice of opposite corners of the cube. Thus 32 of the 72 positions with Q-transitive symmetry groups are an odd distance from SOLVED. The discussion of the Supergroup in S&LM noted that the only face-center orientations which yield Q-transitive symmetry groups are the home orientation and all face centers twisted 180o (called NOON in Hoey's message of 7 January 1981 1615-EST). Any position with either of these face center orientations must be an even distance from SOLVED, so that any reachable position which is T-symmetric in the Supergroup must be an even distance from SOLVED. In our earlier note, we erroneously calculated the number of Supergroup positions with Q-transitive symmetry groups by simply doubling the number of such positions in the Rubik group to allow for the two allowable face-center orientations. What we failed to notice--until Plummer pointed it out--is that neither of the allowable face-center orientations can occur in conjunction with an odd position. The corrected count of known Supergroup local maxima is determined by counting the 40 *even* symmetric positions, multiplying by two, and subtracting 1 for the identity, yielding 79. As Plummer notes, this is surprisingly close to the number of known local maxima in the Rubik group, which stands at 71. The number of known local maxima modulo M-conjugacy is 25 for the Rubik group and 35 ( = 2*(26-8) - 1 ) for the Supergroup.