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Date: Fri, 21 Aug 1998 23:07:24 -0400
From: michael reid <reid@math.brown.edu>
Message-Id: <199808220307.XAA10899@cauchy.math.brown.edu>
To: cube-lovers@ai.mit.edu
Subject: minimal maneuvers for E symmetric positions

E  is the subgroup of cube symmetries consisting of rotations
(no reflections) that preserve the tetrad of corners  UFR,
UBL, DFL  and  DBR.  of course it preserves the other tetrad as
well.  there are 72 positions that have E symmetry:

     each corner must remain in place, but can be twisted.
     corners in the same tetrad must be twisted in the same
     direction; therefore, by conservation of twist, adjacent
     corners are twisted in opposite directions.

     the UR edge can go in any location in any orientation.
     this determines the location and orientation of all edges.

this gives 3 * 24 = 72 positions.  if the UR edge remains in the
F-B slice, then the position has more symmetry, namely H symmetry
(at least).  this accounts for 24 of the 72 positions; 20 of which
are H symmetric, and 4 of which are M symmetric.

E is a normal subgroup of M; in fact, it's the commutator subgroup.
therefore, any M conjugate of an E symmetric position is also E
symmetric.  the 48 remaining positions form 12 equivalence classes
under M conjugacy, of 4 positions each.  minimal maneuvers for these
are

1.  F' B' U  R' U' D  R  L' U  R' D' L  U' D  R  L' D' L  F' B'  (20q*, 20f)
    F  R  L  F  U' D' F  R2 L2 U2 D2 B  R' L' B  U  D  B   (18f*, 22q)

2.  F' R2 U  D  R' L' U' F  B  R2 F' U' D' F' B  U' D' B'  (20q*, 18f*)

3.  F' B' R' L' F  B  U  D  R' L' U' D'  (12q*, 12f*)

4.  F  U2 R  F  B  R' U2 B  R  L  B  U' D' R' L  U' L2  (20q*, 17f*)

5.  F  R' F  L  U  D' L' U  R  U' D  F' D' B  U  D' B' R' F  R'  (20q*, 20f)
    F2 U  D2 L2 F  U  D  F' L2 U  F  B  U  R' L' F' B  R   (18f*, 22q)

6.  F  B  R' L' F  B  U' D' R  L  U' D'  (12q*, 12f*)

7.  F  R  U2 F  L  B  U  D' F  L  D  F' B  D  R  B  D   (18q*, 17f*)

8.  F  U2 F' B  D  B' L  B  L  D' B2 R' D  F  D  F' R  D2 F'  (22q*, 19f)
    F' R2 D2 F  U' D  F2 R  B2 L  F2 R  D2 F  B' U' F' B'  (18f*, 24q)

9.  F' R  F  U  R2 U  F' L' D  F  U2 R' F' B2 R' D' R' B' U'  (22q*, 19f*)

10.  F  R2 B' L' U' L' F' B  D  B  R  B  R  F  U' R  U2 D2 F'  (22q*, 19f*)

11.  F  B  R2 U' F  L  F  U' F  B' R  F' U' B' R  L' B' R  L   (20q*, 19f)
     F  R2 U  D' F2 L' U  D  L2 B  U  D' F  B' U' D2 F  B   (18f*, 22q)

12.  F  R' D  B  U  D  F' L  B  D  R' F  R' L  B' U' F' B' L  F' D  L
                                                                   (22q*, 22f)
     F  B  U  F  L2 D2 F2 B  R  F  B  R  F' L2 D2 R2 U2 B  U   (19f*, 26q)

as usual, i give a maneuver that is simultaneously minimal in both
metrics, unless one does not exist.  some of these are local maxima.

local maxima in the quarter turn metric: #1, 4, 7, 8, 9, 10, 11, 12.

(strong) local maxima in the face turn metric: #10, 11, 12.

mike