M-symmetric positions

These are the positions that have all the symmetries of the cube.

[Start] Start
[Superflip] Superflip
R' U2 B L' F U' B D F U D' L D2 F' R B' D F' U' B' U D' (24q*, 22f)
U R2 F' R D' L B' R U' R U' D F' U F' U' D' B L' F' B' D' L' (24q*, 23f)
U D' R F U' D' L D' F R U' R U' D' F U' F L B' U F' B' L B' (24q*, 24f)
U D' R F' D L' B L' U' R' D' B' U' D L' F D' R B' R U L D B (24q*, 24f)
U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2 (20f*, 28q)
U R2 F B R B2 R U2 L B2 R U' D' R2 F D2 B2 U2 R' L (20f*, 28q)

These are all minimal maneuvers, up to cyclic shifting, inversion, and conjugation by symmetries of the cube. Dik Winter [8] was the first to find a 20f maneuver. Minimality of the 20f maneuver was first shown in [6]. The first known 24q maneuver (the (24q, 22f) maneuver) was found in [5]. Mark Longridge [4] notes that it has an interesting type of symmetry, namely that it is equivalent to iself under a cyclic shift by 12q and a cube symmetry. The second (24q, 24f) maneuver has a similar kind of symmetry. It was first shown in [7] that superflip requires at least 22 quarter turns. Jerry Bryan [1] was the first to show minimality of a 24q maneuver.

[Pons Asinorum] Pons Asinorum
U2 D2 F2 B2 R2 L2 (12q*, 6f*)
F B' U2 D2 R2 L2 F B' (12q*, 8f)
F B' U' D F B' U' D F B' U' D (12q*, 12f)

These are all minimal maneuvers, up to symmetries of the cube. (Each maneuver is equivalent to its inverse under symmetry.) Dan Hoey [3] originally found the interesting (12q, 12f) maneuver.

[Pons Asinorum composed with Superflip] Pons Asinorum composed with Superflip
B' D' L' F' D' F' B U F' B R2 L U D' F L U R D (20q*, 19f*)
F' U' B' R' F R L' D' R L' U D' L' U D' F R B U F (20q*, 20f)
B' R' F' U' F R L' D' R L' U D' L' U D' F U F R B (20q*, 20f)
B' R' B' D' F U' D L' U' D R L' U' R L' F D B R B (20q*, 20f)
R U R B R' U' D F U' D F B' D F B' R' B' R' U' R' (20q*, 20f)
U D F R L' F B' L D2 R L F' B' U' L2 F B' U2 L' (19f*, 22q)
U D F' B' L' U2 F' B L2 U' R' L' F' U' D F' B D' L2 (19f*, 22q)
U2 R F U F B' L' D' F B' L B R L' U D2 B' R' U2 (19f*, 22q)
U2 R F U2 D' R' L F' L' F B' U L F B' D' B' R' U2 (19f*, 22q)
U2 R U2 D2 R U' L' U B R F2 U' D B' R' F' D B' L2 (19f*, 24q)

These are all minimal maneuvers, up to inversion and symmetries of the cube. Jerry Bryan [2] was the first to find the 20q maneuvers.

References

[1] Jerry Bryan, Qturn Lengths of M-Symmetric Positions, cube-lovers e-mail, February 19, 1995.
[2] Jerry Bryan, Pons Asinorum Superflipped Halfway Positions (corrected), cube-lovers e-mail, February 20, 1995.
[3] Dan Hoey, Pons Asinorum -- Part 2: Pons in the Supergroup, cube-lovers e-mail, January 7, 1981.
[4] Mark Longridge, Superflip 24q, cube-lovers e-mail, January 19, 1995.
[5] michael reid, superflip, cube-lovers e-mail, January 10, 1995.
[6] michael reid, superflip requires 20 face turns, cube-lovers e-mail, January 18, 1995.
[7] michael reid, superflip in quarter turn metric, cube-lovers e-mail, January 20, 1995.
[8] Dik T. Winter, Kociemba's algorithm, cube-lovers e-mail, May 18, 1992.

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Updated May 24, 2005.