These are the positions that have all the symmetries of the cube.
Start |
Superflip |
R' U^{2} B L' F U' B D F U D' L D^{2} F' R B' D F' U' B' U D' | (24q*, 22f) |
U R^{2} 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 R^{2} F B R B^{2} R U^{2} L B^{2} R U' D' R^{2} F R' L B^{2} U^{2} F^{2} | (20f*, 28q) |
U R^{2} F B R B^{2} R U^{2} L B^{2} R U' D' R^{2} F D^{2} B^{2} U^{2} 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 |
U^{2} D^{2} F^{2} B^{2} R^{2} L^{2} | (12q*, 6f*) |
F B' U^{2} D^{2} R^{2} L^{2} 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 |
B' D' L' F' D' F' B U F' B R^{2} 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 D^{2} R L F' B' U' L^{2} F B' U^{2} L' | (19f*, 22q) |
U D F' B' L' U^{2} F' B L^{2} U' R' L' F' U' D F' B D' L^{2} | (19f*, 22q) |
U^{2} R F U F B' L' D' F B' L B R L' U D^{2} B' R' U^{2} | (19f*, 22q) |
U^{2} R F U^{2} D' R' L F' L' F B' U L F B' D' B' R' U^{2} | (19f*, 22q) |
U^{2} R U^{2} D^{2} R U' L' U B R F^{2} U' D B' R' F' D B' L^{2} | (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.