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
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.