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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.