# 4x4x4 Parity Algorithms

Parity (also known as Orientation Parity and Permutation Parity) on the 4x4x4 is situation (occurring in 3/4 of all solves) commonly identified when only two or four edge pieces need to be cycled in order to complete solving the 4x4x4 or at least successfully bring the 4x4x4 into a pseudo 3x3x3 state. However, as is shown on this page, parity cases can take many other forms.

This page attempts to list all efficient algorithms for every common form of parity as well as those only common in specific solving methods.

Solutions listed which are not as efficient as others in their categories are at least relatively efficient for their specific effect on the cube or for the move set they are confined to.

Introduction

• The shortest odd parity fix (PLL Parity algorithms are even parity fixes for wing edges) algorithm which preserves the colors of the centers is simply:
 (2R U2)4 2R (13,9) [X]

For those who are familiar with commutators and conjugates, this quick parity fix can be represented as [2R: [U2, 2R] [2R2 U2: 2R] ].

In fact, we can do the same 4-cycle of wing edges with just one conjugate [2R2 2D' 2R2 u2 S': 2R'].

The phrase "there is more than one way to solve any given problem" holds true with tackling 4x4x4 parity situations. In fact, there are different categories of parity algorithms, and algorithms can consist of different move patterns (the move set of one algorithm might be entirely different than the algorithm above and/or below it).

This page not only contains commonly practiced speedsolving algorithms, but it also contains algorithms which illustrate the veracity of the 4x4x4 cube parity algorithm domain.

NOTES

• All algorithms are in SiGN Notation
• Some algorithms have been named, and their names are in the first column.
Names such as "Alg(v1)", "Alg(v2)" are not actual names: they are just a notification that consecutive ordered version algorithms are different versions of the same algorithm.
• The algorithms below each case image solve the permutation in the case image.
• All algorithms' lengths are written next to them (slice quarter turn, slice half turn).
Algorithms with fewer slice half turns (STM) are listed first in each category.
Algorithms which have fewer slice quarter turn moves (SQTM) are listed before other algorithms which have the same number of STM as them.
• Most of these algorithms affect centers on the 4x4x4 supercube: not all algorithms affect the supercube centers in the same manner.
• All algorithms can be applied to the 6x6x6 if instead of turning the outer 2 layers, turn the outer 3 layers; instead of turning 1 inner layer slice, turn 2 inner layer slices.

# PLL Parity

## Dedges

• Algorithms marked as "Safe" are supercube safe.

### Two Dedges (Oriented)

#### Opposite

 2R2 U2 2R2 u2 2R2 2U2 (12,6) Chris Hardwick [X]
 SP01 (u2 r2 U2) 2R2 (U2 r2 u2) (14,7) Stefan Pochmann [X]
 (r2 F2 U2) 2R2 (U2 F2 r2) (14,7) Stefan Pochmann [X]
 (r2 B2 U2) 2R2 (U2 B2 r2) (14,7) Stefan Pochmann [X]
 2R2 U2 B2 2L 2R U2 M' U2 2R2 B2 U2//Safe (19,11) [X]
 Alg(v1) y r2 U2 r U2 r2 U2 r2 U2 r U2 r2 y' (20,11) [X]
 Alg(v2) y r2 U2 r' U2 r2 U2 r2 U2 r' U2 r2 y' (20,11) [X]
 2R2 U2 2R U2 2R2 U2 2R2 U2 2R U2 2R2 U2//Safe (22,12) [X]
 2R' F U' R F' U 2L 2R U' F R' U F' 2L'//Safe (14,14) [X]

 (R2 D' x) 2R2 U2 2R2 u2 2R2 2U2 (x' D R2) (18,10) [X]
 SP02 (R2 D' x u2 r2 U2) 2R2 (U2 r2 u2 x' D R2) (20,11) Stefan Pochmann [X]
 FB02 (R2 D' r2 U2 F2) 2R2 (F2 U2 r2 D R2) (20,11) Frédérick Badie [X]
 (F2 U r2 U2 F2) 2R2 (F2 U2 r2 U' F2) (20,11) [X]
 (R2 D' x r2 F2 U2) 2R2 (U2 F2 r2 x' D R2) (20,11) Stefan Pochmann [X]
 (R2 D' x r2 B2 U2) 2R2 (U2 B2 r2 x' D R2) (20,11) Stefan Pochmann [X]
 y' R' F 2L E F2 E' 2L' 2R' E F2 E' 2R F' R y//Safe (16,14) Christopher Mowla [X]
 y' R' F 2L E' F2 E 2L' 2R' E' F2 E 2R F' R y//Safe (16,14) Christopher Mowla [X]
 (R U R' U') 2R2 U2 2R2 u2 2R2 u2 (U' R U' R') (20,14) [X]
 (R2 D' x) 2R2 U2 B2 2L 2R U2 M' U2 2R2 B2 U2 (x' D R2)//Safe (24,15) [X]
 Alg(v1) (r' U R U l' U2 r' U2) 2R2 (U2 r U2 l U' R' U' r) (22,17) Christopher Mowla [X]
 Alg(v2) y' (r U' R' U' r B2) (r B2 2R2 B2 r') (B2 r' U R U r') y (22,17) Christopher Mowla [X]
 y2 r U r' R U' r' U' r U r U' r' U' r' R U r U R' U' R' U y2 (20,20) cubizh [X]
 y2 R' U' R' U r U R r' U' r' U' r U r U' r' U' R r' U r U y2 (20,20) cubizh [X]
 y R' U r U R r' U' r' U' r U r U' r' U' R r' U r U R' U' y' (20,20) cubizh [X]
 y r' U2 r U2 r' U2 r' U' r U' r U r' U' r U r2 U r U' r U r' U r U y' (30,26) Ben Whitmore [X]

### Two Dedges (Unoriented)

#### Opposite

 F2 2L E F2 E' 2L' 2R' E F2 E' 2R F2//Safe (16,12) Christopher Mowla [X]
 F2 2L E' F2 E 2L' 2R' E' F2 E 2R F2//Safe (16,12) Christopher Mowla [X]
 (y R' U F') 2R2 U2 2R2 u2 2R2 2U2 (F U' R y') (18,12) [X]
 2R U2 2L D2 2L' U2 M U2 2R D2 2R' U2 2L'//Safe (19,13) Kenneth Gustavsson [X]
 (y' R' F U' r2 U2 F2) 2R2 (F2 U2 r2 U F' R y) (20,13) [X]
 (y R' U F' r2 F2 U2) 2R2 (U2 F2 r2 F U' R y') (20,13) [X]
 2L U2 M 2L U2 M' 2L' U2 M' U2 r M' U2 M 2R' U2 r'//Safe (23,17) Kenneth Gustavsson [X]
 (F 2R U' R U' l U2 r U2) 2R2 (U2 r' U2 l' U R' U 2R' F') (24,19) Christopher Mowla [X]
 (r U2 r' U 2L' U' l' U2 r' U' 2L2 U') 2R2 (U 2L2 U r U2 l U 2L U' r U2 r') (32,25) Christopher Mowla [X]

 (R B) 2R2 U2 2R2 u2 2R2 2U2 (B' R') (16,10) [X]
 (R B r2 F2 U2) 2R2 (U2 F2 r2 B' R') (18,11) [X]
 (3l U r2 U2 F2) 2R2 (F2 U2 r2 U' 3l') (18,11) [X]
 R B U2 2R2 U2 B2 2L 2R U2 M' U2 2R2 B R'//Safe (21,14) [X]

### Four Dedges (Oriented)

#### Circular 4-cycle of Dedges (O Perm)

 CG03 2R2 u2 2R2 2B2 U' 2R2 2B2 U 2B2 u2 2R2 (20,11) Clément Gallet [X]
 PKF03 u2 2R2 u2 2R2 U2 2L2 U M2 U' M' E2 M' D2 y2 (22,13) Per Kristen Fredlund [X]
 M2 U' 2R2 U2 F2 2R2 F2 U2 2L 2R U2 M' U' M2 (23,14) [X]
 2L 2R 3d' L R 2U' L' R' 3d M2 3d L' R' 2U' L R 3d' 2L 2R//Safe (20,19) Christopher Mowla [X]
 2L' 2R' 3d' L' R' 2U L R 3d M2 3d L R 2U L' R' 3d' 2L' 2R'//Safe (20,19) Christopher Mowla [X]
 2L 2R 3d L R 2U L' R' 3d' M2 3d' L' R' 2U L R 3d 2L 2R//Safe (20,19) Christopher Mowla [X]
 2L' 2R' U y L' R' 2U' L R 3d' M2 3d' L R 2U' L' R' U y 2L' 2R'//Safe (20,19) Christopher Mowla [X]
 (F R U R' U' F') (2R2 U2 2R2 u2 2R2 u2) (R' U' F R' F' R U R) U2 (28,21) Antoine Cantin [X]
 (F2 U2 M U f2 2R2 2U S' r2 2U2) M' (2U2 r2 S 2U' 2R2 f2 U' M' U2 F2) (33,21) Christopher Mowla [X]
 U r2 U2 R U' R' U r U2 r2 U2 r2 U2 r R' U' R U R' U' R2 U' r2 (31,22) cubizh [X]
 U r2 U' R2 U' R' U R U' r R' U2 r2 U2 r2 U2 r U R' U' R U2 r2 (31,22) cubizh [X]
 r' U r U R' r U' r' U R' U' R' U' R2 U r' U' R' r U R r U R' r' (24,23) cubizh [X]
 U r2 U2 r U' r2 R U2 r2 R' U2 r2 R U' r' U' R' U R' U' R2 U' r2 (29,23) cubizh [X]
 U r2 U' R2 U' R' U R' U' r' U' r2 R U2 r2 R' U2 r2 R U' r U2 r2 (29,23) cubizh [X]
 U r2 R2 U' R' U R' U2 r' U2 r2 U2 r2 U2 r' U R' U' R' U R U r2 (30,23) cubizh [X]
 U r2 R2 U' R' U R' U2 r U2 r2 U2 r2 U2 r U R' U' R' U R U r2 (30,23) cubizh [X]
 U r2 U R U R' U' R' U r' U2 r2 U2 r2 U2 r' U2 R' U R' U' r2 R2 (30,23) cubizh [X]
 U r2 U R U R' U' R' U r U2 r2 U2 r2 U2 r U2 R' U R' U' r2 R2 (30,23) cubizh [X]
 R2 U r U' r2 U' R2 U r2 U r' U r2 U r2 U R2 U' r2 R2 U' r2 R2 (30,23) cubizh [X]
 R2 U r' U' r2 U' R2 U r2 U r U r2 U r2 U R2 U' r2 R2 U' r2 R2 (30,23) cubizh [X]
 R2 U R U R' U' R' U' r2 R2 U2 r' U2 r2 U2 r2 U2 r' U2 r2 R U R' (31,23) cubizh [X]
 R2 U R U R' U' R' U' r2 R2 U2 r U2 r2 U2 r2 U2 r U2 r2 R U R' (31,23) cubizh [X]
 U r2 U2 R U' R' U r' U2 r2 U2 r2 U2 r' R' U' R U R' U' R2 U' r2 (32,33) cubizh [X]
 U r2 U' R2 U' R' U R U' r' R' U2 r2 U2 r2 U2 r' U R' U' R U2 r2 (32,33) cubizh [X]
 R2 U R U R2 U' r2 R2 U2 r' U2 r2 U2 r2 U2 r' U2 r2 R U' R' U2 R'//Safe (33,23) cubizh [X]
 R2 U R U R2 U' r2 R2 U2 r U2 r2 U2 r2 U2 r U2 r2 R U' R' U2 R'//Safe (33,23) cubizh [X]

#### Zigzag 4-cycle of Dedges (W Perm/8 Perm)

 PKF04 (2R2 U2 2R2 u2 2R2 2U2) (F2 U M' U2 M U F2) (22,13) Per Kristen Fredlund [X]
 CG04 R2 u2 B2 R2 u2 B2 R2 U R2 B2 R2 U B2 u2 (26,14) Clément Gallet [X]
 u2 F2 U R2 F2 R2 U R2 F2 u2 R2 F2 u2 R2 (26,14) Clément Gallet [X]
 SP04 R' U R' U' R' U' R' U R U' u2 2R2 u2 2R2 U2 r2 (21,16) Stefan Pochmann [X]
 R2 U R' U' R2 U R U r2 U2 R' r U2 R' r2 U2 r2 U2 r U2 r2//Safe (30,20) cubizh [X]
 R2 U R' U' R2 U R U R' r2 U2 R' r U2 r2 U2 r2 U2 r U2 r2//Safe (30,20) cubizh [X]
 R2 U R' U' R2 U R U R' r2 U2 R' r' U2 r2 U2 r2 U2 r' U2 r2//Safe (31,21) cubizh [X]
 y R2 U' R' U' R U2 r U R r' U' r' U' r U r U' r' U' R r' U R r y' (24,22) cubizh [X]
 (M2 U f2 2R2 2U 2R2 u2 S' U' B R B' R2 U) M' (U' R2 B R' B' U S u2 2R2 2U' 2R2 f2 U' M2) (41,29) Christopher Mowla [X]

## Two Corner Swaps

 PKF10 y' B2 L U L' B2 R D' 2R2 F2 2R2 f2 2R2 2F2 R D R2 y (25,16) Per Kristen Fredlund [X]
 R U' R B2 L' D L B2 R2 U 2R2 F2 2R2 f2 2R2 2F2 (25,16) [X]
 z 2R2 U2 R' U2 R' U2 R x U2 r2 U2 B2 L U2 L' U2 r2 U2 z' y' (29,17) Christopher Mowla [X]
 z' r2 x U2 R' U2 x' U2 R' U2 R U2 L' x U2 r2 U2 r2 U2 r2 x' U2 r2 z U (33,19) Christopher Mowla [X]
 2R2 U2 2R2 u2 2R2 2U2 y' R U R' U' R' F R2 U' R' U' R U R' F' y (27,20) [X]
 z' x r2 R' U2 r' U2 r2 U2 r2 U2 L r' U2 R' U2 R U2 L' R U2 R U2 r2 x' z (35,22) Christopher Mowla [X]
 z' x r2 R' U2 r U2 r2 U2 r2 U2 L r U2 R' U2 R U2 L' R U2 R U2 r2 x' z (35,22) Christopher Mowla [X]

### Opposite/Diagonal

 CG14 r2 2F2 U2 f2 D r2 U2 f2 U' f2 L2 U2 B2 l2 U (27,15) Clément Gallet [X]
 MF14 r2 2F2 U2 f2 U' r2 U2 f2 U f2 R2 U2 F2 r2 U (27,15) Michael Fung [X]
 r2 F2 U2 y r2 U' r2 U D l2' U' l2' y' 2R2 U2 F2 r2 U (27,16) Christopher Mowla [X]
 r2 F2 U2 y l2' U l2' U' D' r2 U r2 y' 2R2 U2 F2 r2 U (27,16) Christopher Mowla [X]
 reParity y' z (R2' u2' R2 U R2') y (R2 U2 R2' U' R2) y' (R2' u2' R2 (d' 2U) R2) u2 U2 x (29,16) reThinking the Cube [X]
 y' 2R2 F2 R' F2 U2 R2 U2 R' F2 R2 U2 r' 2R' U2 F2 r2 U2 y (30,17) Christopher Mowla [X]
 y' 2R2 B2 R' U2 r2 U2 B2 R' B2 2R2 U2 r2 B2 U2 R' U2 r2 y (31,17) Christopher Mowla [X]
 FB14 y' R r2 F2 L U L' B D L D' B2 U B 2R2 U2 F2 r2 R' U2 y (25,19) Frédérick Badie [X]
 y' 2R2 U2 2R2 u2 2R2 u2 L' U R' U2 L U' L' R U R' U2 L U' R U y (29,21) [X]
 r2 R U2 R U2 R' U2 r U2 r2 U2 r2 U2 L r U2 R' U2 R U2 L' r2 (35,22) Christopher Mowla [X]
 r2 R U2 R U2 R' U2 r' U2 r2 U2 r2 U2 L r' U2 R' U2 R U2 L' r2 (35,22) Christopher Mowla [X]
 2L2 U2 F2 2L R U2 R' U2 L2 F2 L F2 l F2 U2 l2 U2 F2 U2 F2 L' U2 (38,22) Christopher Mowla [X]

## Two Corner Swaps (Only 2 X-center Piece Exchange on the Supercube)

 U r2 b2 r F r' b2 r F' L2 2R U' 2B2 U L' F' L U' 2B2 U L' F M x (29,23) Christopher Mowla [X]
 z' f' L2 u F' U R U' F 2R2 F' U R' U' F 2R2 f u' L u f' u' L f L z (27,24) Christopher Mowla [X]
 z' f' L2 u 2R2 F' U R U' F 2R2 F' U R' U' F f u' L u f' u' L f L z (27,24) Christopher Mowla [X]
 y' (r' R2 U' R' U R' r U' R U2' R' U' R' r U R r' U' r' F r2 U' r' U') (r U r' F') y (27,25) Robert Yau [X]
 y' r' U' R U 2R U' R' U R2 U R' U' 2R U 2R' F' r U r' U' r' F r2 U' r' U' y (28,26) Robert Yau [X]
 y' r' U' R U 2R U' R' U R2 U R' U' 2R U 2R' U' r' F r2 U' r' U' r U r' F' y (28,26) Robert Yau [X]
 y' r' U' R U 2R U' R' U R2 U R' U' 2R U R U' l u2 r' U' r d2 r' U r' F' x y' (29,26) Robert Yau [X]
 U r2 b2 r F r' b2 r F' D' 2F' D F2 D' 2F D F2 r L F' L' 2F' L F L' 2F (31,26) Chris Hardwick [X]
 y' r U R' U' 2R' U R U' R2 U' R U 2R' U' R' l U l' U' r U l U' l' r' U r U y (29,28) Christopher Mowla [X]

### Opposite/Diagonal

 F 2R2 F' U R' U' F 2R2 f u' L u f' u' L f L f' L2 u F' U R U' (27,24) Christopher Mowla [X]
 F r' F r2 U' r' U' r' U' R U 2R U' R' U R2 U R' U' 2R U 2R' F' r U r' U' F' (30,28) Robert Yau [X]

### Two X-Center Piece Swap Algorithms

• These algorithms, and other algorithms like them, are used to make two corner swap algorithms which only swap two X-center pieces on the 4x4x4 supercube.

 y r U' l u2 r' U r u2 x r2 U y' (13,10) [X]
 f' U' r f r' U' r f' r' U2 f U' (13,12) Christopher Mowla [X]
 r U r' U' r' F r2 U' r' U' r U r' F' (15,14) [X]
 r' U' l U l' U' r U l U' l' r' U r U (15,15) Christopher Mowla [X]

## The Shortest PLL Parity Fixes in SQTM

• Note that these are the only PLL parity algorithms on this page which only use two inner layer slice quarter turns.
• This alg set is for theoretical purposes only, as they scramble the pseudo 3x3x3 state of the 4x4x4.

 r U D L2 U D S2 r (10,8) Christopher Mowla [X]
 r' U D L2 U D S2 r' (10,8) Christopher Mowla [X]
 r U' D' L2 U' D' S2 r (10,8) Christopher Mowla [X]
 r' U' D' L2 U' D' S2 r' (10,8) Christopher Mowla [X]
 r U D R2 U D S2 r (10,8) Christopher Mowla [X]
 r' U D R2 U D S2 r' (10,8) Christopher Mowla [X]
 r U' D' R2 U' D' S2 r (10,8) Christopher Mowla [X]
 r' U' D' R2 U' D' S2 r' (10,8) Christopher Mowla [X]

• Taking the first algorithm, for example, and solving back the outer layers, we see that these short parity fixes were made from a supercube safe unoriented 4-cycle of dedges in M:
 2R U D L2 U D S2 2R S2 D' U' L2 D' U' (18,14) Christopher Mowla [X]

# Pure Flips

• OLL parity algorithms in this section preserve more than any other OLL parity algorithm form, escpecially algorithms marked as "Safe", which are also supercube safe.

## One Dedge Flip

 y' r u2 x' U D 2R y u2 r2 u 2R u' r2 d 2R' 2U D' x u2 r' y (22,17) Christopher Mowla [X]
 Holy Grail z d' M D l' u' 2R' u l u' l2' b' 2R' b r' R' 2U y' M' u x2 z' (19,18) Christopher Mowla [1]
 r' E u2 f u r' 2R' f' 2R f r u' f' u 2R u E' r (19,18) Christopher Mowla [X]
 l' S b2 d 2R d' b d 2R' d' 2R' b' 2R b 2R b S' l (19,18) Christopher Mowla [X]
 l' F' 2B' r' b' r 2U r' b r 2U2 b' 2U b 2U 2B F l (19,18) Christopher Mowla [X]
 r2 B 2R' U R' U 2R U' R U' 2R B U2 2R U2 2R' B2 r2 (23,18) Christopher Mowla [X]
 z r f2 3u U y' 2R' u2 y' l2 u' 2R' u l2 u' 2R 3u' 2u' U' x' u2 r' z (23,18) Christopher Mowla [X]
 2R2 U2 2R2 U2 2R U2 2R U2 2R' U2 B2 U' 2R' U B2 U' 2R U' (27,18) Christopher Mowla [X]
 2R2 U2 2R2 U2 2R U2 2R U2 2R' U2 B2 U 2R' U' B2 U 2R U (27,18) Christopher Mowla [X]
 r' E u 2B' u 2R u 2R' u' 2B u 2R' u' 2R u 2R u E' r (19,19) Christopher Mowla [X]
 r' E u' 2R' u l' u' f r' f' 2R' f 2R r f' u l E' r (19,19) Christopher Mowla [X]
 r' D' 2U' u' r 2F r' u r 2F' r' 2F' u' 2F u 2F 2U D r (19,19) Christopher Mowla [X]
 d' F' z r2 B2 x' 2L' d u r2 d' 2L' d r2 d' 2L 2U' U r2 D r z' (24,19) Christopher Mowla [X]
 u f' D 2R' d 2B d 2B' d' 2R d 2B' d' 2B d 2B d2 D' f u' (21,20) Christopher Mowla [X]
 f' U' 2L 2U2 l b' l' 2U l u b 2U' b' u' b l' 2U 2L' U f (21,20) Christopher Mowla [X]
 r U L F' U 2L' d u r2 d' 2L' d r2 d' 2L u' U' F L' U' r' (23,20) Christopher Mowla [X]
 b' R' 2U r' 2R' u b' u' 2R u r b 2R' b' r' b u' r 2U' R b (21,21) Christopher Mowla [X]
 b' R' 2U r' 2L' d b' d' 2L d l b 2L' b' l' b d' r 2U' R b (21,21) Christopher Mowla [X]
 z' f' u 2R' u' f u 2R' f' U L 2U 2L u' 2R' u 2L' 2U' L' U' f 2R2 u' z (23,22) Christopher Mowla [X]
 z' f' u 2R' u' f u 2R' f' 2F' U L U' 2F 2R' 2F' U L' U' 2F f 2R2 u' z (23,22) Christopher Mowla [X]
 z' f' u 2R' u' f u 2R' f' 2F' L f R' F' 2R' F R f' L' 2F f 2R2 u' z (23,22) Christopher Mowla [X]
 2R U2 2R' U2 2R U2 2R U 2R U 2R U2 2R2 U2 2R U2 2R U 2R' U' 2R2 U2 (31,22) [X]
 2R U2 2R' U2 2R U2 2R U' 2R U' 2R U2 2R2 U2 2R U2 2R U' 2R' U 2R2 U2 (31,22) [X]
 2R2 U 2R' U' 2R U2 2R U2 2R2 U2 2R U' 2R U' 2R U2 2R U2 2R' U2 2R U2 (31,22) [X]
 2R2 U' 2R' U 2R U2 2R U2 2R2 U2 2R U 2R U 2R U2 2R U2 2R' U2 2R U2 (31,22) [X]
 2R' U2 2R' U' 2R U 2R U 2R' D' 2F2 2R 2F2 2R' D U2 2R' U 2R U 2R U 2R//Safe (27,23) [X]
 r U2 r2 U L' U 2R U' L U2 L' U 2R' U' L U r' U2 2R U2 r' U2 r' (29,23) Christopher Mowla [X]
 r2 U2 2R' U2 2R' U' 2L 2R 2U2 2L' 2R' U 2L 2R 2U2 x U2 2L' U2 2R U2 2R' U2 r2 x'//Safe (33,23) Per Kristen Fredlund [X]
 z' 2F' u r' f' 2R' f r u' 2F 2R' u r' f' U L U' 2R' U L' U' f r u' 2R2 z (25,24) Christopher Mowla [X]
 y' r' U' r U' r U r2 U' r2 U2 r U2 r' U2 r U2 r U' r U r' U' r U2 r' U2 y (34,26) Bruce Norskog [X]
 y' r' U' r U' r U' r2 U2 r2 U' r U2 r' U2 r U2 r U r U r' U' r U2 r' U2 y (34,26) Bruce Norskog [X]
 r U2 r U r U2 r U2 r' U2 r' U 2R' U' r U2 r U2 r' U2 r' U 2R2 U2 r' U2 r' (37,27) Christopher Mowla [X]
 r U2 r U' r U2 r U2 r' U2 r' U' 2R' U r U2 r U2 r' U2 r' U' 2R2 U2 r' U2 r' (37,27) Christopher Mowla [X]
 y r U r2' U' r U' r' U r U' r2' U2' r' U2' r U2' r' U r2' U' r U r' U' r U2' r U2' r2' U r' U' r' y' (42,33) Ben Whitmore [X]

## One Dedge Flip + PLL Parity (Double Parity)

 x 2R F2 l' F 2R' F' U2 F 2R F' U2 l x' E2 2L' U2 2L y2 (21,16) Christopher Mowla [X]
 r2 B2 2L F2 2R' F2 U2 2R' U2 2R U2 M' 2R U2 B2 r2 (26,16) [X]
 Alg(v1) r2 U B2 U' 2R' U B2 U' r 2R U B2 U 2R' U' B2 U' r (23,18) Christopher Mowla [X]
 Alg(v2) r2 3d L2 3d' 2R' 3d L2 3d' r 2R 3d L2 3d 2L' 3d' L2 3d' r (23,18) Christopher Mowla [X]
 Alg(v1) 2R2 U' B2 U 2R' U' B2 U 2R2 U' B2 U' 2R' U B2 U 2R (23,18) Christopher Mowla [X]
 Alg(v2) 2R2 3d' R2 3d 2R' 3d' R2 3d 2R2 3d' R2 3d' 2L' 3d R2 3d 2R (23,18) Christopher Mowla [X]
 (F 2D' S 2R F r2 2F2 2U' 2R2 u2 S') 2R (S u2 2R2 2U 2F2 r2 F' 2R' S' 2D F') (31,23) Christopher Mowla [X]
 (B l2 U' L' U 2R 2U2 2B' 2R2 b2 E) 2R' (E' b2 2R2 2B 2U2 2R' U' L U l2 B') (31,23) Christopher Mowla [X]
 2R' U' R' U' 2R' U R U 2R U2 2R' U' R' U' 2R2 U R U 2R U2 2R U2 2R' U2 (29,24) Christopher Mowla [X]
 (z' l u2 U' r L2 U2 l' U' l R F2 U' R' U) 2R' (U' R U F2 R' l' U l U2 L2 r' U u2 l' z) (33,29) Christopher Mowla [X]
 (z' l u2 U' r L2 U2 l' U' r R U2 3l U' R2 U) 2R' (U' R2 U 3l' U2 R' r' U l U2 L2 r' U u2 l' z) (37,31) Christopher Mowla [X]

 (y' 2R2 F r' F') 2R2 U2 2R' U2 2R' U2 2R' U2 (F r F' 2R2 y) (23,16) Christopher Mowla [X]
 (y r' z' L' U 2R F) 2R2 U2 2R' U2 2R' U2 2R' U2 (F' 2R' U' L z r y') (23,18) Christopher Mowla [X]
 (x' 2R2 U' r U) M' U2 2R' U2 2R' U2 2R' U2 2L 2R (U' r' U 2R2 x) (24,18) Christopher Mowla [X]
 R B y2 2R U2 2R' E2 F2 2L F2 2L' F2 2R F2 2R' D2 2L B' R' (25,18) Kenneth Gustavsson [X]
 R B' 2R2 B2 2R B2 2R' B2 2R B2 U2 2R U2 2R B2 2R2 B' R' (27,18) [X]
 x r F2 U' r R U' 2R U2 2R U2 2R U2 2R2 U' r' R' U F2 r' x' (25,19) Christopher Mowla [X]
 y r2 U x' 2L U' L' U' 2L' U L U F2 2L' F2 2R U2 2R' U2 x U' r2 y' (25,19) Christopher Mowla [X]
 2R2 F' r' F' U2 F 2R F 2L D2 2R' D2 F2 2L' U2 F R F 2R2 (26,19) Christopher Mowla [X]
 2R' B L' B 2L' B' L B' 2L U 2L B2 2L' F2 2L' F2 2L B2 U' M' (24,20) Christopher Mowla [X]
 (y r2 U x') 2L U' L' U' 2L' U L U M' U2 2L' U2 2L U2 2R' U2 (x U' r2 y') (26,20) Christopher Mowla [X]
 (y r2 F L) F U2 2L F2 D2 2R D2 2L' F 2R' F U2 F' 2R (L' F' r2 y') (27,20) Christopher Mowla [X]
 (z' r' U' r' 2L' U L') U' 2R2 U2 2R' U2 2R' U2 2R' U' (L U' 2L r U r z) (25,21) Christopher Mowla [X]
 (l2 B' R B 2L 2B2 2U' 2L2 u2 S) 2L' (S' u2 2L2 2U 2B2 2L' B' R' B l2) (29,21) Christopher Mowla [X]
 (y x' l2 F' L2 F 2R 2F2 2U' 2R2 u2 S') 2R' (S u2 2R2 2U 2F2 2R' F' L2 F l2 x y') (31,21) Christopher Mowla [X]
 x u F U 2R S' L2 S 2R S' L2 S 2R U R' U 2R U' R U' 2R U' F' u' x' (25,23) Christopher Mowla [X]
 x2 2R U2 2L U2 2R' F R F' U 2L' F2 2L U B2 2L B2 2L' U' F2 U' F R' F' x2 (29,23) Christopher Mowla [X]
 (y b' R U 2B' R' b' u2 r' 2U2 2L2 S') 2U (S 2L2 2U2 r u2 b R 2B U' R' b y') (29,23) Christopher Mowla [X]
 (2R2 z B' u R u M' u2 2L' 2U2 r2 S) 2U (S' r2 2U2 2L u2 M u' R' u' B z' 2R2) (31,23) Christopher Mowla [X]
 (z r' u2 U l' R2 U2 r U l' U2 F' R2 F) 2L' (F' R2 F U2 l U' r' U2 R2 l U' u2 r z') (33,27) Christopher Mowla [X]
 (l U r2 F r U' r U' F2 U r2 U' R' U') 2R (U R U r2 U' F2 U r' U r' F' r2 U' l') (35,29) Christopher Mowla [X]
 u b' 2L b u' b' 2L u2 R' u' L' B' L B U' L U 2L U' L' U B' L' B L u R u2 2L2 b (33,30) Christopher Mowla [X]
 u b' 2L b u' b' 2L u2 R' u' L' U' B' L B L U 2L U' L' B' L' B U L u R u2 2L2 b (33,30) Christopher Mowla [X]
 (l' u2 U l' U2 r U l' U' L U B' U' R' B U') 2L (U B' R U B U' L' U l U' r' U2 l U' u2 l) (37,33) Christopher Mowla [X]
 (r u2 l r u2 r2 U2 r U' l U2 F L2 U F' U') 2L' (U F U' L2 F' U2 l' U r' U2 r2 u2 l' r' u2 r') (45,33) Christopher Mowla [X]

## Three Flips

### OLL Parity (Only)

 y M U 2R B2 2L' D2 2L' B2 2R2 B2 U2 2L' D2 2L U2 2L2 B2 U' M' y' (29,19) [X]
 y M U 2R2 D2 2R' U2 2R D2 B2 2L2 B2 2R U2 2R B2 2L' B2 U' M' y' (29,19) [X]
 M2' U' x' 2R2 U2 M 2R' U2 2R F2 U2 2R2 U2 M' 2R x U2 2R U2 2R' U' M2' (31,20) [X]
 y M U 2L 2R F2 2L B 2U' B D2 B' 2U B 2R' D2 B2 2L2 F2 2R' U' M' y' (27,21) Christopher Mowla [X]
 y M U 2L 2R F2 2L B' 2D' B' D2 B 2D B' 2R' D2 B2 2L2 F2 2R' U' M' y' (27,21) Christopher Mowla [X]
 y M U 2R' D2 2L' 2R' B2 D2 2L' B 2U B' D2 B 2U' B 2L D2 2R2 U' M' y' (27,21) Christopher Mowla [X]
 y M U 2R' D2 2L' 2R' B2 D2 2L' B' 2D B D2 B' 2D' B' 2L D2 2R2 U' M' y' (27,21) Christopher Mowla [X]
 y M U 2L' 2R' U2 2L' D' 2F D' B2 D 2F' D' 2R B2 D2 2L2 U2 2R U' M' y' (27,21) Christopher Mowla [X]
 y M U 2L' 2R' U2 2L' D 2B D B2 D' 2B' D 2R B2 D2 2L2 U2 2R U' M' y' (27,21) Christopher Mowla [X]
 Alg(v1) r U r U2 r U2 r U r' U2 2R' U2 r' U r U2 r U2 r U r (27,21) Kåre Krig [X]
 Alg(v2) r' U' r' U2 r' U2 r' U' r U2 2R U2 r U' r' U2 r' U2 r' U' r' (27,21) Kåre Krig [X]
 u2 F2 l' U' L U l U2 l' U' L' U' 2L' U2 l' F2 2L F2 l' U2 l' 2L' F2 u2 (33,24) Christopher Mowla [X]
 u2 B2 l U L' U' l' U2 l U L U 2L U2 l B2 2L' B2 l U2 l 2L B2 u2 (33,24) Christopher Mowla [X]

Other Cases

 (r' U' F' R' F R2 U') 2R (U R2 F' R F U r)(D' F2 D)(2R' F2 2R F2)(F2 2R2 F2 2R2 F2 2R2 F2)(D' F2 D) (39,29) Christopher Mowla [X]
 r U2 r U2 r U R U r U2 r' U' R' U' r' R' U R U r U2 r' U' R' U' r U2 r' U2 r' (36,30) Christopher Mowla [X]

### OLL Parity + PLL Parity (Double Parity)

 r' U2 2R U2 r' x' U2 2R' U' R' U' r' U2 r U R U' r R U2 x (24,19) Christopher Mowla [X]
 y M U 2R U2 2R2 U2 2R' U2 F2 2R' F2 2R2 U2 2L F2 2L' F2 2R2 U' M' y' (31,20) [X]
 (y' 2R' U2 M 2L U) (2R' U' F' R' F R2 U') 2R' (U R2 F' R F U 2R) 2R U2 2R' U2 (U' M' 2L' U2 2R y) (33,27) Christopher Mowla [X]

# Pure Flips/OLL Parity Algorithms which Don't Preserve the Last Layer

• If all inner slice turns of the algorithms in this section are converted to wide turns, the algorithms will simply affect more pieces in the last layer.
• The maximum number of wide turns are used in each algorithm to do pure flips to allow the easiest execution possible, but no extra outer face turns are included as to make any algorithm longer (in STM or SQTM) than what it is without any inserted face turns at all.
• Of course, all wide turns may be converted into inner slice turns to achieve the same pure flip result.

### OLL Parity (Only)

and

#### 15 STM Solutions

• A complete list of all 248 possible (25,15) solutions is in this post: http://www.speedsolving.com/forum/showthread.php?46925-Announcing-New-4x4x4-Brute-Force-Solver&p=983757&viewfull=1#post983757
• The following 26 (25,15) solutions can be used to generate all possible (25,15) solutions if we do the following to any of the (25,15) algorithms listed:
• Take the mirror.
• Take the inverse.
• Take the mirror and the inverse.
• Rotate the cube so that the single flipped dedge is in the desired location on the cube.
• For (25,15) solutions which end with "B2", conjugate these solutions with B2 and then apply all of the adjustments above to the resulting algorithms.
• These 26 solutions have been carefully sorted into 4 groups. An algorithm in its group can be transformed into any other algorithm in its group, and thus there are technically only 4 distinct (25,15) paths.

Group 1 (non-symmetrical algorithms)

 Alg.1(v1) 2R' U2 2L F2 2L' F2 2R2 U2 2R U2 2R' U2 F2 2R2 F2 (25,15) Frédérick_Badie [X]
 Alg.1(v2) 2R' U2 2R U2 2L' U2 2L2 F2 2R F2 2L' U2 F2 2R2 F2 (25,15) Frédérick_Badie [X]
 Alg.1(v3) 2R B2 2R' U2 2R U2 2R2 F2 2R' D2 2L D2 B2 2R2 F2 (25,15) Frédérick_Badie [X]
 Alg.1(v4) 2R B2 2R' U2 2R U2 2L2 B2 2R' U2 2L U2 F2 2L2 F2 (25,15) Frédérick_Badie [X]
 Alg.1(v5) 2R B2 2L' B2 2L U2 2R2 F2 2L' F2 2R U2 F2 2L2 F2 (25,15) Frédérick_Badie [X]

Group 2 (non-symmetrical algorithms)

 Alg.2(v1) 2R B2 2R' U2 2L B2 2L2 B2 U2 2L U2 2L' U2 2L2 B2 (25,15) [X]
 Alg.2(v2) 2R B2 2R' U2 2R D2 2R2 U2 F2 2L F2 2R' D2 2R2 B2 (25,15) [X]
 Alg.2(v3) 2R B2 2R' U2 2R D2 2L2 D2 B2 2L B2 2R' U2 2L2 B2 (25,15) [X]
 Alg.2(v4) 2R B2 2L' B2 2L D2 2L2 D2 B2 2R D2 2L' D2 2R2 B2 (25,15) [X]
 Alg.2(v5) 2R' U2 2R U2 2L' D2 2R2 D2 B2 2L' D2 2R D2 2L2 B2 (25,15) [X]

Group 3 (symmetrical algorithms)

 Old Standard r2 B2 U2 2L U2 2R' U2 2R U2 F2 2R F2 2L' B2 r2 (25,15) [X]
 Alg.3(v2) r2 B2 U2 2L U2 2L' B2 2R B2 U2 2R U2 2R' B2 r2 (25,15) [X]
 Alg.3(v3) r2 F2 D2 2R' D2 2R F2 2L' F2 D2 2L' D2 2L F2 r2 (25,15) [X]
 Alg.3(v4) r2 F2 D2 2R' D2 2L D2 2L' D2 B2 2L' B2 2R F2 r2 (25,15) [X]
 Alg.3(v5) r2 B2 U2 2R B2 2R' B2 2L U2 F2 2R F2 2L' B2 r2 (25,15) [X]
 Alg.3(v6) r2 F2 D2 2L' F2 2L F2 2R' D2 B2 2L' B2 2R F2 r2 (25,15) [X]
 Alg.3(v7) r2 B2 U2 2R B2 2L' D2 2R D2 B2 2L U2 2R' B2 r2 (25,15) [X]
 Alg.3(v8) r2 F2 D2 2L' F2 2R U2 2L' U2 F2 2R' D2 2L F2 r2 (25,15) [X]

Group 4 (symmetrical algorithms)

 Alg.4(v1) r2 B2 U2 2L' U2 2R U2 2R' U2 B2 2R' B2 2L B2 r2 (25,15) [X]
 Alg.4(v2) r2 B2 U2 2L' U2 2L F2 2R' F2 U2 2R' U2 2R B2 r2 (25,15) [X]
 Alg.4(v3) r2 F2 D2 2R D2 2R' B2 2L B2 D2 2L D2 2L' F2 r2 (25,15) [X]
 Alg.4(v4) r2 F2 D2 2R D2 2L' D2 2L D2 F2 2L F2 2R' F2 r2 (25,15) [X]
 Alg.4(v5) r2 B2 U2 2R' F2 2R F2 2L' U2 B2 2R' B2 2L B2 r2 (25,15) [X]
 Alg.4(v6) r2 B2 U2 2R' F2 2L D2 2R' D2 F2 2L' U2 2R B2 r2 (25,15) [X]
 Alg.4(v7) r2 F2 D2 2L B2 2R' U2 2L U2 B2 2R D2 2L' F2 r2 (25,15) [X]
 Alg.4(v8) r2 F2 D2 2L B2 2L' B2 2R D2 F2 2L F2 2R' F2 r2 (25,15) [X]

#### 23 Single Slice Quarter Turn Solutions

• The 53 23 single slice quarter turn solutions listed below contain at least some quarter face turns as opposed to the (25,15) solutions above.
• Additional unique 23 single slice quarter turn pure one dedge flip solutions can be created from all these by doing one of the following, but note that no additional inner slice turns may be used in the resulting algorithms more than what is already used in these solutions.
• If the algorithm contains face quarter turns with inner l and r slices in between them, invert all face quarter turns in the algorithm.
• If the algorithm contains face quarter turns with u, d, f, or b slices in between them,
• Invert all face quarter turns in the algorithm
• Convert u inner slice turns to d inner slice turns (and vice versa) OR convert f inner slice turns to b inner slice turns (and vice versa). Do not invert these turns, just substitute.
• Like the (25,15) solutions, these solutions have been grouped next to their transformations. To save space, we label each group's algorithm as Alg(v1),Alg(v2),Alg(v3), and so forth.
• These are most likely not all of the possible 23 single slice quarter turn solutions. Perhaps in the future, we will have efficient optimal solvers in the single slice quarter turn metric for which we can use to extract all possible solutions.
• Since there is currently one 23 single slice quarter turn algorithm listed in the previous category, and since we can create an additional pure dedge flip algorithm from each of the following 53 solutions, we effectively show 107 unique 23 single slice quarter turn solutions on this page.
• Lastly, it's worthy to note that using the classic setup through depth 18 of single slice half turns, the following 21 slice quarter turn algorithm was the only 21 slice quarter turn 3x3x3 algorithm which was closest to being a single dedge flip algorithm. Perhaps if the classic setup is used up to depth 21, some 21 slice quarter turn solutions may be found, but, as of yet, 23 is the current upperbound.
 2R' F 2L' F2 2R F 2R' F2 D F' D 2L D' F D' 2L F2 2R (21,18) Christopher Mowla [X]

 cmowlaparity x' r2 U2 l' U2 2R U2 l F2 U 2R U' F2 U 2R' U r2 x (23,16) Christopher Mowla [X]
 cmowlaparity(v2) x' r2 U2 r' F2 2R F2 r F2 U 2R U' F2 U 2R' U r2 x (23,16) Christopher Mowla [X]
 Alg(v1) x r2 U2 r U2 2L' U2 r' F2 U' 2L' U F2 U' 2L U' r2 x' (23,16) Christopher Mowla [X]
 Alg(v2) x r2 U2 l F2 2L' F2 l' F2 U' 2L' U F2 U' 2L U' r2 x' (23,16) Christopher Mowla [X]
 Alg(v1) r' U2 B 2R' B' U2 F2 l' B2 2R' B2 l B 2R F2 B' r (22,17) Christopher Mowla [X]
 Alg(v2) r' U2 B 2R' B' U2 F2 r' U2 2R' U2 r B 2R F2 B' r (22,17) Christopher Mowla [X]
 2R' U' 2R' U B2 U' 2R U' 2R' U2 2R' B2 2L U2 2L' U2 2R2 (23,17) Christopher Mowla [X]
 Alg(v1) 2R B 2L B' D2 B 2L' B 2L B2 2R F2 2L' F2 2L D2 2R2 (23,17) Christopher Mowla [X]
 Alg(v2) 2R B 2L B' D2 B 2L' B 2L B2 2L D2 2L' D2 2R D2 2R2 (23,17) Christopher Mowla [X]
 Alg(v1) r D B2 D' 2L D B2 U2 r D2 2L D2 r' D' 2L' U2 r' (23,17) Christopher Mowla [X]
 Alg(v2) r D B2 D' 2L D B2 U2 l B2 2L B2 l' D' 2L' U2 r' (23,17) Christopher Mowla [X]
 Alg(v1) r' F2 D2 B 2L B' D2 B 2L' B' r' B2 2L B2 r F2 r (23,17) Christopher Mowla [X]
 Alg(v2) r' F2 D2 B 2L B' D2 B 2L' B' l' D2 2L D2 l F2 r (23,17) Christopher Mowla [X]
 2R2 D2 2R D2 2L' D 2L D' F2 D 2L' D 2L D2 2L F2 2R (23,17) Christopher Mowla [X]
 Alg(v1) r' B' U2 B 2R' B' U2 F2 l' B2 2R' B2 l B 2R F2 r (23,17) Christopher Mowla [X]
 Alg(v2) r' B' U2 B 2R' B' U2 F2 r' U2 2R' U2 r B 2R F2 r (23,17) Christopher Mowla [X]
 Alg(v1) 2L' 2R' D' 2R D' B2 D 2R' D' B2 2L' D2 2L' F2 2L F2 2L2 (23,17) Christopher Mowla [X]
 Alg(v2) 2L 2R U' 2R U' F2 U 2R' U' F2 2L' U2 2L' B2 2L B2 2R2 (23,17) Christopher Mowla [X]
 Alg(v1) r U2 F2 D' 2R' D F2 D' 2R D l D2 2R' D2 l' U2 r' (23,17) Christopher Mowla [X]
 Alg(v2) r U2 F2 D' 2R' D F2 D' 2R D r F2 2R' F2 r' U2 r' (23,17) Christopher Mowla [X]
 Alg(v1) 2R2 B' 2R B' U2 B 2R' B' U2 2L' B2 l' D2 2L D2 l 2R (23,17) Christopher Mowla [X]
 Alg(v2) 2L2 F' 2R F' D2 F 2R' F' D2 2L' F2 2L' U2 2L U2 2L' 2R' (23,17) Christopher Mowla [X]
 Alg(v1) 2R2 U2 2L' U2 2L F' 2R' F U2 F' 2R F' 2R' F2 2R' U2 2R' (23,17) Christopher Mowla [X]
 Alg(v2) 2R2 U2 2R' F2 2L D' 2R' D F2 D' 2R D' 2R' D2 2L' U2 2R' (23,17) Christopher Mowla [X]
 2R2 B2 D2 2R D 2B' D F2 D' 2B D 2R' B2 2L 2R F2 2R (23,17) Christopher Mowla [X]
 2R2 F2 U2 2L' U' 2F U' B2 U 2F' U' 2R' B2 2R 2L F2 2R (23,17) Christopher Mowla [X]
 2R2 D 2F D' B2 D 2F' D 2R D2 F2 2R' B2 2L 2R F2 2R (23,17) Christopher Mowla [X]
 2R2 U' 2B' U F2 U' 2B U' 2L' U2 B2 2R' B2 2R 2L F2 2R (23,17) Christopher Mowla [X]
 Alg(v1) 2R2 D2 2R D 2B' D F2 D' 2B D 2R' B2 2L B2 2R F2 2R (23,17) Christopher Mowla [X]
 Alg(v2) 2R2 D2 2R D 2B' D F2 D' 2B D 2L' D2 2L D2 2L F2 2R (23,17) Christopher Mowla [X]
 Alg(v1) 2R2 F2 2R' B2 2L D 2F D' B2 D 2F' D 2R D2 2R F2 2R (23,17) Christopher Mowla [X]
 Alg(v2) 2R2 F2 2L' D2 2L F 2U F' D2 F 2U' F 2R F2 2L F2 2R (23,17) Christopher Mowla [X]
 Alg(v3) 2R2 B2 2R U2 2R' B' 2D' B U2 B' 2D B' 2R' U2 2R' U2 2R' (23,17) Christopher Mowla [X]
 l' B' 2R' B D2 B' 2R B' 2R' B2 2R' D2 2L B2 2L' B2 2R l (23,18) Christopher Mowla [X]
 Alg(v1) 2R l F2 2L' F2 2L D' 2R' D F2 D' 2R D' 2R' D2 2R' F2 l' (23,18) Christopher Mowla [X]
 Alg(v2) r 2L F2 2L' F2 2R F' 2R' F U2 F' 2R F' 2R' F2 2R' U2 r' (23,18) Christopher Mowla [X]
 Alg(v3) 2R l F2 2R' D2 2L B' 2R' B D2 B' 2R B' 2R' B2 2L' F2 l' (23,18) Christopher Mowla [X]
 Alg(v1) r' U' 2R' U B2 U' 2R U' 2R' U2 2L' D2 2R D2 2R' B2 r 2L (23,18) Christopher Mowla [X]
 Alg(v2) r' U' 2R' U B2 U' 2R U' 2R' U2 2R' B2 2R B2 2L' B2 r 2L (23,18) Christopher Mowla [X]
 Alg(v1) r' F2 D 2B' D 2L D2 2L' D' 2B D r' B2 2L B2 r F2 r (23,18) Christopher Mowla [X]
 Alg(v2) r' F2 D 2B' D 2L D2 2L' D' 2B D l' D2 2L D2 l F2 r (23,18) Christopher Mowla [X]
 2R2 B' D2 2R D 2B' D F2 D' 2B D 2R' B2 2L B' 2R F2 2R (23,18) Christopher Mowla [X]
 2R2 B D2 2R D 2B' D F2 D' 2B D 2R' B2 2L B 2R F2 2R (23,18) Christopher Mowla [X]
 2R' l' B' 2D' B U2 B' 2D B' 2L' B2 D2 2R' D2 2R 2L U2 l (23,18) Christopher Mowla [X]
 r 2L D2 F2 2L' F' 2D F' U2 F 2D' F' 2L D2 2R' 2L' U2 r' (23,18) Christopher Mowla [X]
 r 2L F' 2U' F D2 F' 2U F' 2L' F2 U2 2L D2 2R' 2L' U2 r' (23,18) Christopher Mowla [X]
 Alg(v1) r 2L F2 2L' F' 2D F' U2 F 2D' F' l D2 2R' D2 l' U2 r' (23,18) Christopher Mowla [X]
 Alg(v2) r 2L F2 2L' F' 2D F' U2 F 2D' F' r F2 2R' F2 r' U2 r' (23,18) Christopher Mowla [X]
 2R' l' U2 B2 2L' B' 2U B' D2 B 2U' B' 2R' D2 2R 2L U2 l (23,18) Christopher Mowla [X]
 Alg(v1) r 2L U2 2R F2 2R' U' 2B' U F2 U' 2B U' 2L' U2 2R' U2 r' (23,18) Christopher Mowla [X]
 Alg(v2) 2R l U2 2R F2 2R' U' 2B' U F2 U' 2B U' 2R' F2 2R' F2 l' (23,18) Christopher Mowla [X]
 Alg(v1) r U2 F' 2D F' 2R' F2 2R F 2D' F' l D2 2R' D2 l' U2 r' (23,18) Christopher Mowla [X]
 Alg(v2) r U2 F' 2D F' 2R' F2 2R F 2D' F' r F2 2R' F2 r' U2 r' (23,18) Christopher Mowla [X]

#### Algorithms of this category which are not optimal (in either single slice metric)

• The following list of algorithms are all unique solutions generated with the classic setup which use only 2 different face turns which are at most 27 single slice quarter turns and 18 single slice half turns.
• There is only one algorithm which just uses the face turn, U2.
• Like the (25,15) solutions, all solutions ending in B2 may be conjugated with B2 to create a handful more "unique" algorithms.
• All other algorithms can be modified by inverses, mirrors, cube rotations, or a combination of any of these to create every possible algorithm which uses 2 faces which are generated from the classic setup.
• Since this is a large set of algorithms,
• None of these algorithms have any wide turns in them (despite that some algorithms can have some).
• Even though it is very likely that algorithms related by transformations are listed next to each other, no official sorting by transformations was done.
• Since these algorithms use either U and F faces or U and B faces, all algorithms which affect one or the other have been grouped together for convenience of personal preference.

 2R' U2 2R U2 2L F2 2L' 2R' F2 2R' U2 2R' U2 F2 2R2 F2 (25,16) [X]
 2R' U2 2R U2 2L' U2 2L 2R U2 2R U2 2R' U2 F2 2R2 F2 (25,16) [X]
 2R' U2 2L F2 2L' F2 2L 2R F2 2R F2 2L' U2 F2 2R2 F2 (25,16) [X]

 2R B2 2R' U2 2L B2 2L2 B2 U2 2R B2 2L' B2 2L' 2R' B2 (25,16) [X]
 2L' 2R' B2 2R' B2 2L U2 B2 2R2 B2 2R U2 2L' B2 2L B2 (25,16) [X]

 2R' U2 2R U2 2R U2 2R2 F2 2L 2R2 F2 2L' U2 F2 2R2 F2 (27,16) [X]
 2R' U2 2L F2 2R F2 2L2 U2 2L 2R2 U2 2R' U2 F2 2R2 F2 (27,16) [X]

 2R' U2 2R' U2 2R2 U2 2R U2 2R2 B2 2R U2 2R U2 2R2 B2 (27,16) [X]
 2R' U2 2L' B2 2R2 B2 2L U2 2R2 B2 2R U2 2R U2 2R2 B2 (27,16) [X]

 2R U2 2R U2 2R' U2 2L F2 2L' F2 2R F2 2R' F2 2R' U2 2R' (25,17) [X]
 2R U2 2R U2 2R' U2 2R U2 2L' U2 2L F2 2R' F2 2R' U2 2R' (25,17) [X]
 2R U2 2L F2 2L' U2 2R U2 2R' F2 2R F2 2R' F2 2R' U2 2R' (25,17) [X]
 2R U2 2L F2 2R' F2 2R F2 2L' F2 2R F2 2R' F2 2R' U2 2R' (25,17) [X]
 2R' F2 2L' F2 2R U2 2R' F2 2R F2 2L' F2 2L F2 2L U2 2L (25,17) [X]
 2R' F2 2L' F2 2R U2 2L' U2 2R U2 2R' F2 2L F2 2L U2 2L (25,17) [X]
 2L 2R U2 F2 2L' F2 2R F2 2R' F2 U2 2R' U2 2L U2 2L' 2R' (25,17) [X]
 2L 2R U2 2R F2 2R' F2 U2 2R' U2 2L F2 2L' F2 U2 2L' 2R' (25,17) [X]

 2R' U2 2R' U2 2L F2 2R' U2 2R' U2 2R2 F2 2R' U2 2L' U2 2R2 (27,17) [X]
 2R' U2 2R U2 2R' F2 2L 2R F2 2L' 2R2 U2 2R' U2 F2 2R2 F2 (27,17) [X]
 2R' U2 2R U2 2R U2 2L' 2R' U2 2L 2R2 U2 2R' U2 F2 2R2 F2 (27,17) [X]
 2R' U2 2L F2 2L' F2 2R2 U2 2L F2 2R' F2 2L' 2R F2 2R2 F2 (27,17) [X]
 2R' U2 2L F2 2R F2 2L' 2R' F2 2L 2R2 F2 2L' U2 F2 2R2 F2 (27,17) [X]
 2R' U2 2L F2 2L2 2R U2 2L 2R U2 2L F2 2L' U2 F2 2R2 F2 (27,17) [X]
 2R' U2 2L F2 2R' U2 2R2 U2 F2 2R' F2 2R F2 2R2 U2 2L' 2R (27,17) [X]
 2R2 U2 2R' U2 2L F2 U2 2R2 U2 2R F2 2L' U2 2L U2 2L' 2R (27,17) [X]
 2R2 U2 2R' F2 2R F2 2L' U2 2L F2 2R' U2 2R' U2 2R2 F2 2R (27,17) [X]
 2R2 U2 2L' U2 2L F2 2R' F2 2R F2 2R' U2 2R' U2 2R2 F2 2R (27,17) [X]
 2R2 U2 2L' U2 2R U2 2R' U2 2L F2 2R' U2 2R' U2 2R2 F2 2R (27,17) [X]
 2R2 F2 2R F2 2L2 2R U2 F2 2L' F2 2L F2 2R' F2 U2 2L 2R (27,17) [X]
 2R2 F2 2R' U2 2L F2 2L' U2 2R U2 2R' F2 2R' U2 2R2 F2 2R (27,17) [X]
 2R2 F2 2R' U2 2L F2 2R' F2 2R F2 2L' F2 2R' U2 2R2 F2 2R (27,17) [X]
 2R2 F2 2R' U2 2R U2 2R' U2 2L F2 2L' F2 2R' U2 2R2 F2 2R (27,17) [X]
 2R2 F2 2R' U2 2R U2 2R' U2 2R U2 2L' U2 2R U2 2L2 F2 2L' (27,17) [X]
 2L 2R2 F2 2R F2 2R F2 2L' 2R' F2 2R' U2 2R' U2 F2 2R2 F2 (27,17) [X]

 2R B2 2R' U2 2R U2 2L2 B2 2L' B2 2L B2 U2 2L2 U2 2L 2R' (27,17) [X]
 2R B2 2L' B2 2R B2 2R2 U2 2L' U2 2R B2 U2 2L2 U2 2L 2R' (27,17) [X]
 2R' U2 2R U2 2R' B2 2R2 B2 U2 2R' U2 2L' 2R2 B2 2L 2R B2 (27,17) [X]
 2R' U2 2R U2 2R' B2 2R2 B2 U2 2L' B2 2R B2 2L2 U2 2L' 2R (27,17) [X]
 2R2 U2 2R B2 2R U2 2R2 B2 2R B2 2R U2 2L' B2 2L U2 2R (27,17) [X]
 2R2 U2 2R' U2 2R U2 B2 2R2 B2 2R U2 2R' U2 2L U2 2L' 2R (27,17) [X]
 2R2 U2 2R' U2 2R' B2 2R2 U2 2R' U2 2L' 2R' B2 2R B2 2L B2 (27,17) [X]
 2R2 U2 2R' U2 2R' B2 2L' 2R' B2 2R' B2 2R2 B2 2R B2 2L B2 (27,17) [X]
 2R2 U2 2L' U2 2R U2 2R' U2 2R U2 2R' B2 2R' B2 2R2 U2 2L (27,17) [X]
 2R2 U2 2L' B2 2R B2 2L 2R' B2 2R2 B2 2R U2 2L' B2 2L B2 (27,17) [X]
 2R2 B2 U2 2L' U2 2R U2 2R' U2 B2 2L 2R2 U2 2L U2 2L' 2R' (27,17) [X]
 2R2 B2 U2 2L' U2 2R U2 2R' U2 2L 2R' U2 2R' U2 2R B2 2R2 (27,17) [X]
 2R2 B2 2R U2 2L' B2 2L 2R' B2 2R' B2 2R B2 2R' U2 B2 2R2 (27,17) [X]
 2R2 B2 2R U2 2R' B2 2L' B2 2L U2 2L' B2 2L U2 2R2 U2 2R' (27,17) [X]
 2R2 B2 2R U2 2R' U2 B2 2R' B2 2R B2 2L' B2 2L 2R' B2 2R2 (27,17) [X]
 2R2 B2 2R U2 2R' U2 2L 2R' U2 2L' B2 2R B2 2R' U2 B2 2R2 (27,17) [X]
 2R2 B2 2R U2 2R' U2 2L 2R' U2 2R' U2 2R U2 2L' U2 B2 2R2 (27,17) [X]
 2R2 B2 2L' 2R B2 2L' B2 2L U2 2R' U2 B2 2R' B2 2L B2 2R2 (27,17) [X]

 2R U 2R U' F2 U 2R' U' 2L F2 2L' F2 U2 2R' U2 2R2 F2 2R (25,18) [X]
 2R2 U2 2L U2 2R' U2 2R F2 2R F 2R' F U2 F' 2R F 2L' 2R2 (25,18) [X]
 2R2 F2 2R F2 2L2 2R U2 2R' F2 U' 2L' U F2 U' 2L U' 2L 2R (25,18) [X]
 2R' F2 2L' F2 2L F2 2L' U' 2L' U F2 U' 2L U F2 2L2 F2 2R (25,18) [X]

 2R U2 2R U2 2R' U2 2R B 2R B' U2 B 2R' B' U2 2R2 U2 2R' (25,18) [X]
 2R2 U2 2L' U2 2R U' 2R' U B2 U' 2R U' 2R' U2 2R' B2 2L 2R2 (25,18) [X]
 2R2 B2 2R U2 2R' U2 B 2R' B' 2L U2 2L' U2 2L' B' 2L B' 2R2 (25,18) [X]
 2R2 B2 2R U2 2R' U2 B 2R' B' 2R B2 2L' B2 2R' B' 2L B' 2R2 (25,18) [X]
 2R2 B' 2R B' U2 B 2R' B' U2 2L' B2 2L 2R2 U2 2L U2 2L' 2R' (25,18) [X]

 2R U2 2R U2 2R' U2 2R U2 2L' U2 2R U2 2R' U2 2L 2R2 U2 2R' (27,18) [X]

 2R U2 2R2 U2 2R' U2 2L F2 2L' U2 2L U2 2R F2 2R' U2 2L' 2R' (27,18) [X]
 2R U2 2R U2 2R' U2 2R U2 2R' F2 2R F2 2R' F2 2L 2R2 F2 2L' (27,18) [X]
 2R U2 2R U2 2R' U2 2R U2 2R' F2 2R F2 2L' U2 2L 2R2 U2 2R' (27,18) [X]
 2R U2 2L F2 2R' F2 2L' 2R2 U2 2L' U2 2L F2 2R' F2 2R' U2 2R' (27,18) [X]
 2R U2 2L F2 2L' U2 2R U2 2L' U2 2R U2 2L 2R2 F2 2R' U2 2R' (27,18) [X]
 2R' U2 2R U2 2R U2 2R2 F2 2R' U2 2R' U2 F2 2L' 2R' U2 2L 2R' (27,18) [X]
 2R' U2 2R U2 2R' F2 2R2 U2 2R U2 2R' U2 2L' 2R U2 2L 2R F2 (27,18) [X]
 2R' U2 2R U2 2R' F2 2L 2R F2 2R F2 2R' F2 2L' 2R F2 2R2 F2 (27,18) [X]
 2R' U2 2R U2 2L' U2 2L 2R U2 2L F2 2R' F2 2L' 2R F2 2R2 F2 (27,18) [X]
 2R' U2 2L F2 2R' U2 2R2 U2 F2 2L' U2 2R U2 2L 2R U2 2L' 2R (27,18) [X]
 2R' U2 2L F2 2L' F2 2R2 U2 2L F2 2L' U2 2L' 2R U2 2L 2R F2 (27,18) [X]
 2R' F2 2R2 U2 2R U2 2R F2 2R' F2 2R F2 2R' F2 2L F2 2L' 2R' (27,18) [X]
 2R' F2 2L' F2 2R U2 2L' U2 2R U2 2L' U2 2L U2 2L2 2R' U2 2L (27,18) [X]
 2R' F2 2L' F2 2R U2 2L' U2 2L F2 2L' F2 2R U2 2L2 2R' U2 2L (27,18) [X]
 2R' F2 2L' F2 2R U2 2L' U2 2L F2 2L' F2 2L F2 2L2 2R' F2 2R (27,18) [X]
 2R' F2 2L' F2 2L F2 2L' F2 2R F2 2L' F2 2R U2 2L2 2R' U2 2L (27,18) [X]
 2R' F2 2L' F2 2L F2 2L' F2 2R F2 2L' F2 2L F2 2L2 2R' F2 2R (27,18) [X]
 2R' F2 2L' F2 2L F2 2L' F2 2L' 2R2 U2 2R' F2 2L F2 2L U2 2L (27,18) [X]
 2R' F2 2L2 F2 2R U2 2R' F2 2R U2 2R' U2 2L' F2 2L U2 2L 2R (27,18) [X]
 2R2 U2 2R' F2 2R F2 2R' F2 2R F2 2R' U2 2R' U2 2L' 2R' U2 2L (27,18) [X]
 2R2 U2 2R' F2 2R F2 2R' F2 2R F2 2R' U2 2R F2 2L 2R F2 2L' (27,18) [X]
 2R2 F2 U2 2L' U2 2L F' 2R' F U2 F' 2R F 2R2 U2 2R2 F2 2R (27,18) [X]
 2R2 F2 2R' U2 2R U2 2R' U2 2R U2 2R' F2 2R' U2 2L' 2R' U2 2L (27,18) [X]
 2R2 F2 2R' U2 2R U2 2R' U2 2R U2 2R' F2 2R F2 2L 2R F2 2L' (27,18) [X]
 2R2 F2 2L' 2R F2 2L' U2 2L U2 2L' U2 F2 2L' F2 2L U2 2L 2R (27,18) [X]
 2L 2R' U2 2L' U2 2L F2 2R' U2 2R2 U2 F2 2R' F2 2R F2 2L' 2R' (27,18) [X]

 2R U2 2R U2 2L' B2 2R B2 2R' U2 2R U2 2R' U2 2L 2R2 U2 2R' (27,18) [X]
 2R U2 2R U2 2L' B2 2R B2 2L' B2 2L U2 2R' U2 2R' B2 2L 2R2 (27,18) [X]
 2R U2 2R U2 2L' B2 2L U2 2L' U2 2R U2 2R' U2 2R' B2 2L 2R2 (27,18) [X]
 2R U2 2L2 B2 2L' B2 2L' U2 2L U2 2L' U2 2R B2 2R' B2 2L' 2R' (27,18) [X]
 2R U2 2L2 B2 2L' B2 2L' U2 2R B2 2L' B2 2L B2 2R' B2 2L' 2R' (27,18) [X]
 2R U2 2L2 B2 2L' B2 2L' U2 2R B2 2R' U2 2L U2 2L' B2 2L' 2R' (27,18) [X]
 2R U2 2L2 B2 2L' U2 2L' B2 2L B2 2R' U2 2R B2 2L' U2 2L' 2R' (27,18) [X]
 2R U2 2L2 B2 2L' U2 2R' U2 2L U2 2L' U2 2R B2 2L' U2 2L' 2R' (27,18) [X]
 2R U2 2L2 B2 2L' U2 2R' U2 2R B2 2L' B2 2L B2 2L' U2 2L' 2R' (27,18) [X]
 2R U2 2L2 B2 2R' B2 2L B2 2R' B2 2R B2 2R' B2 2R U2 2L 2R (27,18) [X]
 2R B2 2R' U2 2R U2 2L' 2R' U2 2L' U2 2R B2 U2 2L2 U2 2L 2R' (27,18) [X]
 2R B2 2R' U2 2L B2 2L2 B2 U2 2R B2 2R' U2 2L' 2R' U2 2L' 2R (27,18) [X]
 2R B2 2L B2 2R' U2 2L B2 2L B2 2L2 U2 2L B2 2R B2 2L' 2R' (27,18) [X]
 2R B2 2L' B2 2R B2 2L' 2R' B2 2L' B2 2L B2 U2 2L2 U2 2L 2R' (27,18) [X]
 2R B2 2L' B2 2R' U2 2L 2R U2 2L B2 2L B2 U2 2L2 U2 2L 2R' (27,18) [X]
 2R' U2 2R U2 2R' B2 2L 2R U2 2L' 2R U2 2R' U2 2R U2 2R2 B2 (27,18) [X]
 2R' U2 2R U2 2R' B2 2R2 B2 2L' 2R B2 2R' B2 2R B2 2L 2R B2 (27,18) [X]
 2R' U2 2R U2 2R' B2 2R2 B2 U2 2R' U2 2R' B2 2L' 2R' B2 2L 2R (27,18) [X]
 2R' U2 2R U2 2R' B2 2R2 B2 U2 2R' U2 2R U2 2L 2R U2 2L' 2R (27,18) [X]
 2R' U2 2R' U2 2L' 2R' B2 2R B2 2L 2R B2 2R U2 2R U2 2R2 B2 (27,18) [X]
 2R' U2 2R' U2 2R U2 2R' B2 2R' B2 2R2 U2 2R' B2 2L' B2 2L 2R (27,18) [X]
 2R' U2 2R U2 2R U2 2L' 2R' U2 2R' B2 2R' B2 U2 2R2 U2 2L 2R' (27,18) [X]
 2L 2R' U2 2L' U2 2R U2 2R' B2 2R2 B2 U2 2L' B2 2R B2 2L 2R (27,18) [X]
 2R2 U2 2L' B2 2L U2 2L 2R' U2 2L' 2R' B2 2R U2 2L' B2 2L B2 (27,18) [X]
 2R2 B2 2R U2 2R' U2 B2 2R' B2 2R B2 2R' U2 2L 2R' U2 2L' 2R' (27,18) [X]

 r U2 r2 U L U 2R U' L' U2 L U 2R' U' L' U r' U2 2R U2 r' U2 r' (29,23) Christopher Mowla [X]

### OLL Parity + PLL Parity (Double Parity)

and

 2R U2 2R' E2 F2 2L F2 2L' F2 2R F2 2R' D2 2L y2 (21,14) Kenneth Gustavsson [X]
 2R2 B2 2R' U2 2R' U2 x' U2 2R' U2 2R U2 2R' U2 2R2 U2 x (25,15) Frédérick Badie [X]
 x 2R F2 l' F' 2R' F U2 F' 2R F U2 l x' E2 2L' U2 2L y2 (21,16) Christopher Mowla [X]
 r' U2 2R' U2 2L U2 r' U2 2R U2 3r U2 2R2 U2 2R' U2 r (26,17) [X]
 2R2 U2 3r' U2 2R' U2 r U2 2L' U2 r U2 2R U2 r' U2 2R (26,17) [X]
 r U2 l' U2 2L U2 l U2 r' U2 2L U2 2L' U2 x U2 2L2 U2 x' U2 (29,18) [X]

# OLL Parity algorithms Which Don't Preserve the Last Layer

## OLL Parity (Only)

• Note that, unlike the previous section, none of these are pure flips if wide turns are converted to inner slice turns.

### 1 Flip

• With the exception of Kåre Krig's 2 gen algorithms, all algorithms listed in this section were derived from algorithms in the previous section by using 4x4x4 wide turn equivalencies such as r = (l x), l2 = (r2 x2), etc.
• Specifically, these algorithms were derived from the classic setup's solutions which are at most 18 single slice half turns. They only involve 2 face turns and thus they are additional transformations which are "legal" in wide turns, but "illegal" transformations to keep these algorithms pure dedge flips.
• Therefore we have more 15 STM solutions which preserve F3L in wide turns than we have for pure dedge flips, for example.
• Most of these algorithms need to be adjusted to work on 5x5x5.
• Lastly, it's interesting to note that the following algorithm cannot even be used to solve OLL parity if all of its wide turns are converted into single slice turns.
 r2 F2 U2 r' F' u L' U2 L u' F' r' U2 r2 F2 r (23,16) Christopher Mowla [X]

 r' U2 r U2 r U2 r2 F2 r' U2 r' U2 F2 r2 F2 (25,15) [X]
 r' U2 r U2 r' F2 r2 U2 r U2 r' U2 F2 r2 F2 (25,15) [X]
 r' U2 l F2 r' U2 r2 U2 F2 l' U2 l F2 r2 U2 x (25,15) [X]
 r' U2 r U2 r' B2 r2 B2 U2 l' B2 l U2 r2 B2 (25,15) [X]
 r' U2 r U2 r' B2 r2 B2 U2 r' U2 r U2 r2 B2 (25,15) [X]
 r' U2 r U2 l' U2 r2 B2 l U2 l' B2 U2 r2 U2 x' (25,15) [X]
 r2 U2 r' U2 r U2 B2 r2 B2 r U2 r' U2 r B2 (25,15) [X]
 x' r2 B2 l B2 r' U2 B2 l2 B2 l' U2 l U2 l' B2 (25,15) [X]

 l r U2 F2 l' F2 r F2 r' F2 U2 r' U2 l U2 l2 x' (25,16) [X]
 l r U2 F2 l' F2 r F2 r' F2 U2 r' U2 l U2 r2 x (25,16) [X]
 r' U2 r U2 r' B2 r2 B2 U2 r' U2 r U2 l r U2 x (25,16) [X]
 r' U2 r U2 r U2 l' r' U2 r' B2 r' B2 U2 r2 U2 x' (25,16) [X]

 r' U2 r U2 l r2 U2 r2 U2 F2 l' U2 l F2 r2 U2 x (27,16) [X]
 r' U2 r U2 l r2 U2 r2 U2 F2 l' U2 r U2 l2 F2 x2 (27,16) [X]
 r' U2 r U2 l2 r B2 l2 U2 r U2 l' B2 U2 r2 U2 x' (27,16) [X]
 r' U2 r U2 r' B2 r2 B2 U2 r' U2 r' B2 r2 U2 r2 (27,16) [X]
 r' U2 r U2 r U2 l2 B2 l r2 B2 r' B2 U2 r2 U2 x' (27,16) [X]
 r' U2 r' U2 r2 U2 r' B2 r2 U2 l' B2 l U2 r2 B2 (27,16) [X]
 r' U2 r' U2 r2 U2 r' B2 r2 U2 r' U2 r U2 r2 B2 (27,16) [X]
 r2 U2 B2 r B2 r B2 U2 r U2 r' U2 r' U2 r2 U2 B2 (27,16) [X]
 r2 U2 r' U2 r' B2 r2 U2 r' U2 r2 U2 r U2 l U2 x (27,16) [X]
 x' r2 B2 l2 r' U2 l' U2 B2 l2 B2 l' U2 l U2 l' B2 (27,16) [X]

 r' U' r' U B2 U' r U' r' U2 r' x' U2 r U2 l' U2 r2 (23,17) Christopher Mowla [X]
 r2 U2 l' U2 r U' r' U B2 U' r U' r' U2 r' x' U2 r' (23,17) Christopher Mowla [X]
 l r U' r U' F2 U r' U' x U2 r' U2 r' U2 l U2 r2 x (23,17) Christopher Mowla [X]

 lucasparity r U2 x r U2 r U2 r' U2 l U2 r' U2 r U2 r' U2 r' (25,17) Lucas Garron & Stefan Pochmann [X]
 lucasparity r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 x' r' U2 r' (25,17) Lucas Garron & Stefan Pochmann [X]
 r U2 r U2 l' B2 l U2 l' U2 r U2 r' U2 r' x' U2 r' (25,17) [X]
 r U2 r U2 r' U2 r U2 r' F2 r F2 r' F2 r' U2 r' (25,17) [X]

 r' U2 r U2 r U2 l' r' U2 l2 r F2 l' U2 F2 r2 F2 (27,17) [X]
 r' U2 r' U2 r2 U2 r U2 l r U2 r F2 r F2 r2 U2 x (27,17) [X]
 r' U2 r' U2 r U2 r' B2 r' B2 r2 U2 r' B2 l' B2 l2 x (27,17) [X]
 r' U2 r' U2 r U2 r' B2 r' B2 r2 U2 r' B2 r' U2 r2 (27,17) [X]
 r2 U2 l' U2 r U2 l' B2 l U2 r' B2 r' B2 r2 U2 l (27,17) [X]
 l r U2 l' r2 U2 r' U2 B2 r' B2 l U2 l' U2 B2 r2 (27,17) [X]

 r U2 r U2 r' U2 l U r U' F2 U r' U' x U2 r2 U2 r' (25,18) Christopher Mowla [X]

 r' U' r2 U' r' U r' U' r' U2 r U r U' r U r2 U r2 U2 r U2 r2 U2 r (33,25) Kåre Krig [X]
 r' U2 r' U2 r U r' U r' U2 r2 U r2 U' r2 U' r' U' r U r' U2 r2 U2 r (34,25) Kåre Krig [X]
 r' U2 r' U2 r U r' U r U' r2 U' r2 U2 r2 U' r U' r U r' U2 r2 U2 r (34,25) Kåre Krig [X]
 r' U2 r' U2 r U r' U r U r2 U2 r2 U' r2 U r U' r U r' U2 r2 U2 r (34,25) Kåre Krig [X]
 r' U r' U' r' U2 r2 U2 r2 U' r U2 r U2 r' U2 r' U' r2 U r2 U r' U2 r (35,25) Kåre Krig [X]
 r' U r' U' r' U2 r2 U2 r2 U r U2 r U2 r' U2 r' U r2 U r2 U r' U2 r (35,25) Kåre Krig [X]
 r' U r' U' r' U2 r2 U2 r2 U r U2 r U2 r' U2 r U' r2 U' r2 U' r U2 r (35,25) Kåre Krig [X]
 r' U r' U' r U' r2 U2 r2 U2 r' U2 r U2 r' U2 r' U' r2 U r2 U r' U2 r (35,25) Kåre Krig [X]
 r' U r' U' r U r2 U2 r2 U2 r' U2 r U2 r' U2 r' U r2 U r2 U r' U2 r (35,25) Kåre Krig [X]
 r' U r' U' r U r2 U2 r2 U2 r' U2 r U2 r' U2 r U' r2 U' r2 U' r U2 r (35,25) Kåre Krig [X]

### 3 Flip

 r F U2 F' r U2 r U2 r U' L' U2 L F 3l' U' R U' x' U F r (25,21) Christopher Mowla [X]
 r U' R U2 R' U2 R' U' r U2 r R U2 r U' R' U2 R' U2 R U' r (28,22) Kåre Krig [X]
 r U' R U2 R' U2 R' U' r U2 r R U2 r U' r' U2 R' U2 r U' r (28,22) Kåre Krig [X]
 r U' r U2 R' U2 r' U' r U2 r R U2 r U' r' U2 R' U2 r U' r (28,22) Kåre Krig [X]

## OLL Parity + PLL Parity (Double Parity)

### 1 Flip

 Alg(v1) r' U2 r' U2 l U2 r' U2 r U2 x U2 r2 U2 r' U2 r (25,16) [X]
 Alg(v2) l U2 l U2 r' U2 l U2 l' U2 x U2 l2 U2 l U2 l' (25,16) [X]
 Alg(v1) r U2 r' U2 r U2 r U2 l' U2 r U2 r' U2 x' U2 r2 (25,16) [X]
 Alg(v2) l' U2 l U2 l' U2 l' U2 r U2 l' U2 l U2 x' U2 l2 (25,16) [X]
 Alg(v1) r2 U2 x' U2 r' U2 r U2 l' U2 r U2 r U2 r' U2 r (25,16) [X]
 Alg(v2) l2 U2 x' U2 l U2 l' U2 r U2 l' U2 l' U2 l U2 l' (25,16) [X]
 Alg(v1) r U2 r' U2 r2 U2 x U2 r U2 r' U2 l U2 r' U2 r' (25,16) [X]
 Alg(v2) l' U2 l U2 l2 U2 x U2 l' U2 l U2 r' U2 l U2 l (25,16) [X]
 Alg(v1) r U2 r U2 r U R r U2 R2 U r U r2 R U r U' r (23,19) Kåre Krig [X]
 Alg(v2) r' U2 r' U2 r' U' R' r' U2 R2 U' r' U' r2 R' U' r' U r' (23,19) Kåre Krig [X]
 r' U2 r' U2 r R U r2 R' U2 r' R' U R U r' U2 r2 U' r2 (26,20) Kåre Krig [X]
 z' U2 l' R2 x U' F2 U l' U2 2R' U2 r F' L2 F 2R2 U2 x' U' F2 U x U2 l' U2 z (32,22) Christopher Mowla [X]
 r' U' r' U' r U r U2 r2 U2 r U2 r U r' U r U r2 U2 r' U2 r U2 r' (33,25) Kåre Krig [X]
 r' U r2 U r2 U2 r U2 r' U2 r U r U r U2 r' U2 r' U2 r2 U' r2 U r' (35,25) Kåre Krig [X]

### 3 Flip

 r U' r U' r2 U r' U2 r U2 r' U r' U' r U2 r U2 r U2 r U' r' U2 r' U2 r (35,27) Kåre Krig [X]

• You can make a 3 flip from a one flip, for example, by conjugating a 1 flip with B' R'. For example,
 (B' R') r' U' r' U B2 U' r U' r' U2 r' x' U2 r U2 l' U2 r2 (R B) (27,21) Christopher Mowla [X]

# OLL Parity Algorithms Which Don't Preserve F3L

• One advantage to having access to algorithms which don't preserve the first three layers of the 4x4x4 is that the shortest algorithms that this category of algorithms has to offer have fewer moves than algorithms which preserve the first three layers, which might be of interest to certain cubers.
• Unlike algorithms which preserve the first three layers, which are at minimum 14 block half turns (double parity) and 19 block quarter turns (one dedge flip), some algorithms in this section are just 13 block half turns and some are just 15 block quarter turns.

## Just Corners are Permuted (Most are also Just FR F3L Slot Destroyers)

OLL Parity (Only)

 2R' U2 F' U2 F U2 2R' U2 2R' U2 2R' F' U2 F 2R' (21,15) Christopher Mowla & Bruce Norskog [X]
 2L U2 F' U2 F U2 2L U2 2L U2 2L F' U2 F 2L (21,15) Christopher Mowla & Bruce Norskog [X]
 2R' U R U2 R' U' 2R' U2 2R' U2 2R' U' R U2 R' U 2R' (21,17) Christopher Mowla & Bruce Norskog [X]
 2L' U' R U2 R' U 2L' U2 2L' U2 2L' U R U2 R' U' 2L' (21,17) Christopher Mowla & Bruce Norskog [X]
 2R2 S' 2R U' R U' 2R U2 2R U2 2R U' R' U' 2R S 2R2 (21,17) Christopher Mowla [X]
 2R2 S 2R U' R U' 2R U2 2R U2 2R U' R' U' 2R S' 2R2 (21,17) Christopher Mowla [X]
 r U' R' U2 R U 2R U2 2R U2 2R U R' U2 R U' r R2 (22,18) Kåre Krig [X]
 r U' R' U2 R U 2R U2 r R U2 2R U R U2 R' U' r (22,18) Kåre Krig [X]
 r U' R U2 R' U 2R U2 r R U2 2R U R' U2 R U' r (22,18) Kåre Krig [X]
 r R U' R2 U2 R2 U 2R U2 r U2 2R U R U2 R' U' r (24,18) Kåre Krig [X]
 r U' R' U2 R U r R2 U2 2R U2 r U R' U2 R U' r R2 (24,19) Kåre Krig [X]
 r U' R' U2 R U r U2 2R U2 r R2 U R' U2 R U' r R2 (24,19) Kåre Krig [X]
 r U' R' U2 R U' R U2 2R U2 r U2 2R U R U2 R' U' r (24,19) Kåre Krig [X]
 (2R' U' R U 2R)(2R U2 2R2 U2 2R' U2 2R U2 2R' U2 2R2 U2 2R)(2R U R' U' 2R') (31,21) Christopher Mowla [X]

## FR F3L Slot Destroyers

OLL Parity (Only)

 2L U2 2L' F U R U' l' D2 2R D2 2R U2 r' U2 l F (22,17) Christopher Mowla [X]
 r U2 l' U2 2L D2 2L D2 r' U' L U F 2R' U2 2R F (22,17) Christopher Mowla [X]
 r U R' U' r2 R U2 r2 U r U2 r' U2 r U' r2 U2 r' (24,18) Kåre Krig [X]
 r U R' U' r2 U2 r2 R U r U2 r' U2 r U' r2 U2 2R' (24,18) Kåre Krig [X]
 r U2 R U R' U2 R U 2R U2 r U2 r U R U2 R' U' r (24,19) Kåre Krig [X]

OLL Parity + PLL Parity (Double Parity)

 r U2 r U2 r U r U2 R' U r U r2 R U r U' r (21,18) Kåre Krig [X]
 r U2 r U2 r' U' R' r2 U2 2R U' R' U' r U2 2R2 U r2 (24,18) Kåre Krig [X]
 (2R' U' R U 2R)(2R2 F2 u2 2R u2 F2 2R2)(2R U R' U' 2R')(f2 2L' u2 2L 2R u2 2R' f2) (31,23) Christopher Mowla [X]

## Petrus (They Destroy 2 Adjacent Faces)

OLL Parity (Only)

 Alg(v1) r B' U2 B r B2 l B2 r B D2 B' r (17,13) Bruce Norskog [X]
 Alg(v2) x' r' U F2 U' l' U2 r' U2 r' F' U2 F r' (17,13) Bruce Norskog [X]
 Alg(v3) x' r' U F2 U' l' U2 r' U2 l' U' B2 U r' x' (17,13) Bruce Norskog [X]
 Alg(v4) x' r' U F2 U' l' U2 r' 3d2 r' U' F2 U l' x (17,13) Bruce Norskog [X]
 r' S' L2 S r' U2 r' U2 r' F U2 F' U' L' U r' (20,16) Christopher Mowla [X]
 r' S R2 S' r' U2 r' U2 r' F' U2 F U R U' r' (20,16) Christopher Mowla [X]
 r U2 r' U2 r D2 2R D2 2L' U L' U' B' 2L' U2 2L x' (21,16) Christopher Mowla [X]
 r' U R' U2 R U' r' U2 r' U2 r' U' R U2 R' U r' (21,17) Kåre Krig, Bruce Norskog, & Christopher Mowla [X]
 r U' R' U2 R U 2R U2 2R U2 2R U R' U2 R U' r (21,17) Kåre Krig [X]
 r U' R' U2 R U r U2 r U2 2R U R U2 R' U' r (21,17) Kåre Krig [X]
 r' U' L' U2 L U r' U2 r' U2 r' U L' U2 L U' r' (21,17) Bruce Norskog & Christopher Mowla [X]
 r U2 l' U2 2L D2 2L D2 r' U' L U F r' U2 r F (22,17) Christopher Mowla [X]
 r U' R2 U2 R2 U 2R U2 r U2 2R U R U2 R' U' r (23,17) Kåre Krig [X]
 r U2 r2 U' r' U2 r U2 r' U r2 U2 r2 U R2 U' r' (25,17) Kåre Krig & Christopher Mowla [X]
 r U2 r2 U r' U2 r U2 r' U' r2 U2 2R2 U R2 U' r' (25,17) Kåre Krig & Christopher Mowla [X]
 r U' R2 U2 R2 U 2R U2 2R U2 2R U R2 U2 R2 U' r (25,17) Kåre Krig [X]
 r' U R U2 R' U' 2R' U2 r' U2 r' F' U' R' U' R F r' (21,18) Kåre Krig [X]
 r' U R U2 R' U' r' U2 r' U2 r' R' U' R U2 R' U r' (22,18) Kåre Krig [X]
 r U' R' U2 R U r U2 2R U2 r R2 U R' U2 R U' r (23,18) Kåre Krig [X]
 r U' R U2 R' U r U2 2R U2 r R2 U R U2 R' U' r (23,18) Kåre Krig [X]
 r U' R U2 R' U r U2 r R2 U2 r U R' U2 R U' r (23,18) Kåre Krig [X]
 r U R U' r2 R U2 r2 U' r U2 r' U2 r U r2 U2 r' (24,18) Kåre Krig [X]
 r' U R U2 R' U' r' U2 r' U2 r' R2 U' R2 U2 R2 U r' (24,18) Kåre Krig [X]
 r U R' U' r2 U2 r2 R U r U2 r' U2 r U' r2 U2 r' (24,18) Kåre Krig [X]
 r U R' U r U2 2R U2 2R U' r' U R U R' U2 r U' r (22,19) Kåre Krig [X]
 r U R' U 2R U2 2R U2 r U' r' U R U R' U2 r U' r (22,19) Kåre Krig [X]
 r U' r U2 R' U R U r' U' 2R U2 r U2 r U R' U r (22,19) Kåre Krig [X]
 r U' R' U2 R U r R U2 2R U2 r R U R' U2 R U' r (23,19) Kåre Krig [X]
 r U' r U2 R' U R U r' U' r U2 2R U2 r U R2 U r (23,19) Kåre Krig [X]

OLL Parity + PLL Parity (Double Parity)

 Alg(v1) l' U2 F2 l' U' F R' F l2 U2 r U2 l2 U' x' (20,14) Bruce Norskog [X]
 Alg(v2) l' U2 F2 l' U' F 3l' U l2 x' U2 r U2 l2 U' x' (20,14) Bruce Norskog [X]
 Alg(v1) r U2 F2 r U' F R' F r2 U2 l' U2 r2 U' x' (20,14) Bruce Norskog [X]
 Alg(v2) r U2 F2 r U' F 3l' U r2 x' U2 l' U2 r2 U' x' (20,14) Bruce Norskog [X]
 2R' U R U' 2R' U2 2R U R' U 2R U2 2R F2 l' U2 l (21,17) Christopher Mowla [X]
 r' U R U (r' U2)3 r2 U R' U' r2 U' R' U r' (24,19) Stefan Pochmann [X]

## More than 1 F3L Slot Destroyed (Not Petrus)

OLL Parity (Only)

 r' U2 F U2 F' U2 r' U2 r' U2 r' F U2 F' r' (21,15) Bruce Norskog & Christopher Mowla [X]
 r U D' R2 U' R2 D x' U2 l U2 l U2 l F' U2 F l (22,16) Christopher Mowla [X]
 r F U2 F' r U2 r U2 r R U2 F U2 F' U2 2R (22,16) Christopher Mowla [X]
 r' U R U2 R' U' r' U2 r' U2 r' F U' B' U' S d' z' (20,17) Christopher Mowla [X]
 l U' L' R U2 L U l U2 l U2 l U' R' U2 R U 2R x' (22,18) Christopher Mowla [X]
 l U' L' R' U2 L U l U2 l U2 l U' R U2 R' U l R (23,19) Christopher Mowla [X]

## Affect M Layer Only

OLL Parity (Only)

 2R B2 2L' D2 2L' B2 2R2 B2 U2 2L' D2 2L U2 2L2 B2 (25,15) [X]
 2R2 D2 2R' U2 2R D2 B2 2L2 B2 2R U2 2R B2 2L' B2 (25,15) [X]
 2L 2R F2 2L B 2U' B D2 B' 2U B 2R' D2 B2 2L2 F2 2R' (23,17) Christopher Mowla [X]
 2L 2R F2 2L B' 2D' B' D2 B 2D B' 2R' D2 B2 2L2 F2 2R' (23,17) Christopher Mowla [X]
 2R' D2 2L' 2R' B2 D2 2L' B 2U B' D2 B 2U' B 2L D2 2R2 (23,17) Christopher Mowla [X]
 2R' D2 2L' 2R' B2 D2 2L' B' 2D B D2 B' 2D' B' 2L D2 2R2 (23,17) Christopher Mowla [X]
 2L' 2R' U2 2L' D' 2F D' B2 D 2F' D' 2R B2 D2 2L2 U2 2R (23,17) Christopher Mowla [X]
 2L' 2R' U2 2L' D 2B D B2 D' 2B' D 2R B2 D2 2L2 U2 2R (23,17) Christopher Mowla [X]
 2R F2 2R2 F2 U2 D 2L D' F2 D 2L' D' 2R F2 2R' U2 2R2 (24,17) Christopher Mowla [X]
 2R F2 2R2 F2 U2 D' 2L D F2 D' 2L' D 2R F2 2R' U2 2R2 (24,17) Christopher Mowla [X]
 2R F2 2R2 F2 U2 D' 2L D F2 D' 2L' D 2R F2 2R' U2 2R2 (24,17) Christopher Mowla [X]
 2R F2 2R2 F2 U2 D 2L D' F2 D 2L' D' 2R F2 2R' U2 2R2 (24,17) Christopher Mowla [X]
 2R' D2 2L2 D2 F2 2L' D 2B D' F2 D 2B' D M 2L D2 2R2 (24,17) Christopher Mowla [X]
 2R' D2 2L2 D2 F2 2L' D' 2F D F2 D' 2F' D' M 2L D2 2R2 (24,17) Christopher Mowla [X]
 2R2 U2 M' 2R U2 2R' B2 U2 2R2 U2 M 2R' B2 2R' B2 2R B2 (27,17) [X]
 2R2 F2 2R U2 2R U2 F2 2R2 U2 F2 2R' U2 2R U2 F2 2R U2 (29,17) [X]
 2R2 U2 F2 2R U2 F2 2R U2 2R2 U2 F2 2R U2 2R F2 2R' U2 (29,17) [X]
 (r' U' F' R' F R2 U' l' E' R2 E) 2R' (E' R2 E l U R2 F' R F U r) (2R U2 2R' U2) (U2 2R2 U2 2R2 U2 2R2 U2) (41,31) Christopher Mowla [X]
 (r' U' F' R' F R2 U') 2R (U R2 F' R F U r) (D' U' F2 U D) (2R' F2 2R F2) (F2 2R2 F2 2R2 F2 2R2 F2) (D' U' F2 U D) (43,33) Christopher Mowla [X]

OLL Parity + PLL Parity (Double Parity)

 2R U2 2R' U2 2R' D2 2R D2 2R' B2 2R B2 2R' (19,13) Christopher Mowla [X]
 2R U2 2R' U2 M' 2L' U2 2R U2 2L' U2 2R U2 2L' (20,14) Christopher Mowla [X]
 reParity(v2) 2R U2 2L' U2 2L 2R2 U2 2L U2 2R' U2 2L U2 2L 2R2 (23,15) reThinking the Cube [X]
 (2R' U' F' R' F R2 U' l2 E L2 E') 2R' (E L2 E' l2 U R2 F' R F U 2R) (2R U2 2R' U2) (35,26) Christopher Mowla [X]

## Complete 3x3x3 Scrambles

OLL Parity (Only)

 r B U2 B' 2R B2 2L B2 2R B' D2 B r (17,13) Bruce Norskog [X]
 r U F2 U' l F2 2R U2 r U B2 U' r (17,13) Bruce Norskog [X]
 r U' F2 U l F2 2R U2 r U' B2 U r (17,13) Bruce Norskog [X]
 u M R D R' b 2F M 2D' M' f 2B M' F u (15,15) Bruce Norskog [X]
 r' S' L2 S r' U2 r' U2 r' F' U2 F U R U' r' (20,16) Christopher Mowla [X]
 r' S R2 S' r' U2 r' U2 r' F U2 F' U' L' U r' (20,16) Christopher Mowla [X]
 r U2 r' U2 r D2 r D2 r' F L' F' U' r' D2 r (21,16) Christopher Mowla [X]
 2R U' M' U M' U' M' U' M u2 2R u2 U' M' U2 M' U M U2 M U M' U M U M' U' 2L' (31,28) Christopher Mowla [X]

OLL Parity + PLL Parity (Double Parity)

 r F2 U2 l F U' R U' r2 B2 r' B2 r2 (19,13) Bruce Norskog [X]
 r F2 U2 l F' U L' U r2 B2 r' B2 r2 (19,13) Bruce Norskog [X]
 r F2 U2 l F U' (L' R) U' r2 B2 r' B2 r2 (19,13) Bruce Norskog [X]
 r F2 U2 l F' U (L' R) U r2 B2 r' B2 r2 (19,13) Bruce Norskog [X]
 reParity r U2 l' U2 x' (r' U2 l U2)2 l' (19,13) reThinking the Cube [X]
 r U' 2B M 2R' D' 2B D 2L' 2B2 l E' B' r (15,14) Bruce Norskog [X]
 u M F R' B 2U 2D 2L' 2U 2D F R' B M u (15,15) Bruce Norskog [X]
 u M F' L B' 2U 2D 2L' 2U 2D F' L B' M u (15,15) Bruce Norskog [X]
 2R' U M' U' M' U M' U M u2 2L' u2 M' U M U' M' U M' U M U2 M' U2 2R (29,25) Christopher Mowla [X]

• It's interesting to note that the long (u,U,2R,2L,M) algorithms have the same structure as Bruce's (15,15) OLL parity (only) solution.

# Non Dedge-Preserving Last Layer 2-Cycle Cases

• This section contains all of the 2-cycle cases that can occur in the last layer besides the one dedge flip.
• These are additional cases which arise in the K4 Method and other direct-solving methods.
• Algorithms marked as "Safe" are supercube safe.

## In Opposite Dedges

 Alg(v1) 2R U2 2R U2 x U2 2R U2 2L' x' U2 2L U2 2R2 (19,12) [X]
 Alg(v2) 2L' B2 2R' B2 D2 2R' D2 2R D2 2L' D2 2L2 (19,12) [X]
 2R U2 2R U2 M' U2 2R U2 2R' U2 2L U2 2R2 (20,13) [X]
 r' U2 2R' U2 3r' U2 2R' U2 2R U2 2L' U2 r2 (20,13) [X]
 z r' U 2L' u' r u 2L' f' 2L' f 2L2 u' r' 2U r z' (16,15) Christopher Mowla [X]
 2L F2 2R F2 2L D 2F' D 2R D2 2R' D' 2F D 2L2 (19,15) Christopher Mowla [X]
 2B 2U b' r' 2F' r b r' 2F2 u' 2F' u 2F' r 2U' 2B' (17,16) Christopher Mowla [X]
 r U2 2R U2 r U2 r2 F 2R F' r2 U2 r2 F 2R' F' (23,16) Christopher Mowla [X]
 2L' B2 2L D2 2R' D2 2L' B2 2L' D2 2R' D2 2L2 D2 2R D2 (25,16) Christopher Mowla [X]
 2L' B2 2L D 2R' D' 2L' B2 2L D 2L 2R F2 2R' F2 2L' D' (21,17) Christopher Mowla [X]
 2R' U2 2R D 2R' D' 2R' U2 2L' B2 2R' B2 2L 2R D 2R D' (21,17) Christopher Mowla [X]
 2R' U2 2L D2 2L' U2 2L D2 2L' U2 2R' U2 2R' U2 2R' U2 2R' (25,17) Christopher Mowla [X]
 d' f 2L' f' u f 2L' u' 2U' R 2U 2L' 2U' R' u 2U 2L2 f' y' (19,18) Christopher Mowla [X]
 z' f' u 2R' u' f u 2R' f2 L 2F L F 2R' F' L' 2F' L' f2 2R2 u' z (23,20) Christopher Mowla [X]
 2L U 2L U' 2L' U' 2L U 2L U 2L' D' 2B2 2L 2B2 2L' U2 D 2L' U//Safe (23,20) [X]
 2R' U' R U 2R' U R' U' 2R U R U2 R' U' 2R' U2 2R' U2 2R' U2 2R' (25,21) Christopher Mowla [X]
 U2 2L' U2 2R U2 2R' U2 x' U2 2R' U2 2R' U' 2L 2R 2U2 2L' 2R' U 2L 2R 2U2 x//Safe (29,21) Per Kristen Fredlund [X]
 x' 2L u' r' u 2L2 f2 r f' U L' U' 2L U L U' f r' f2 2L u' r u x (25,22) Christopher Mowla [X]
 r U' R U r U2 r' U' R' U' 2R U R U r U2 r' U' R' U' 2R2 U2 r' (27,23) Christopher Mowla [X]

### Opposite/diagonal 2-Swap

 2L' S2 U2 2L U2 2L' U2 2R U2 2R' F2 2L B2 2R z2 (21,14) Kenneth Gustavsson [X]
 2L2 B2 U2 2L U2 2L2 B2 2L U2 2L2 U2 B2 2L B2 (25,14) Nicholas Ho [X]
 2R2 B2 U2 2R U2 2R2 B2 2R' U2 2R2 U2 B2 2R' B2 (25,14) Nicholas Ho [X]
 u l' u' 2L' u l f' l2 u' 2L' u l' L' f u' (16,15) Christopher Mowla [X]
 u l' u' 2L' u l2 d' l u' 2L' u L' d l' u' (16,15) Christopher Mowla [X]
 u l' b' 2R' b l2 b' r f' 2R' f R' b l' u' (16,15) Christopher Mowla [X]
 f' L2 u 2B' u' l2 u L' u 2B' u' l u' 2L2 f (18,15) Christopher Mowla [X]
 f' u B2 r' 2U r b2 l2 2D l 2D' l 2B2 u' f (19,15) Christopher Mowla [X]
 u r' F2 r' 2F r b2 l2 2U l 2U' r 2U2 r u' y2 (19,15) Christopher Mowla [X]
 r2 F2 U2 2R U2 x U2 r2 U2 2R' U2 2L 2R U2 2L' U2 x' (25,15) Christopher Mowla [X]
 f' L U' 2F u2 l2 d 2R d' l2 u 2F' u U L' f (19,16) Christopher Mowla [X]
 2L' U2 2L U2 2L U2 2R' U2 2L U2 2L' U2 F2 2L2 F2 2R (25,16) [X]
 2L' F2 D2 2R' D2 F' 2L2 U2 2L2 F' 2L F 2L2 U2 2L2 F' (25,16) Christopher Mowla [X]
 l' U2 2L U2 2L U2 2R' U2 2L U2 2L' U2 M' U2 2L2 U2 l (26,17) [X]
 u r f' L f 2D f' L' f r2 f2 l 2U l' f2 r 2B' u' (21,18) Christopher Mowla [X]
 x' 2L' u r' u' 2L u f u B' 2L2 B 2R' B' 2L2 B u' f' r u' x (21,19) Christopher Mowla [X]
 x' U2 2R' U' 2R U 2R U 2R' D' 2F2 2R 2F2 2R' D U2 2R' U 2R U 2R U x//Safe (25,21) [X]
 z f u' 2L u f' u' 2L f2 R' f' F' L2 F 2L F' L2 f F R f2 2L2 u z' (27,22) Christopher Mowla [X]
 x 2L U2 2L' U2 2L U2 2L2 U' 2L' U 2L U2 2L U2 2L2 U2 2L U 2L U 2L U2 x' (31,22) [X]
 x 2L U2 2L' U2 2L U2 2L2 U 2L' U' 2L U2 2L U2 2L2 U2 2L U' 2L U' 2L U2 x' (31,22) [X]
 r2 U2 2R' U R U 2R U2 2R' U' R' U' 2R2 U R U 2R U2 2R' U' R' U r2 (29,23) Christopher Mowla [X]
 l' U2 2L U2 2L U 2L' 2R' 2U2 2L 2R U' 2L' 2R' 2U2 x U2 2R U2 2L' U2 2L U2 l x'//Safe (31,23) Per Kristen Fredlund [X]
 r' U' r U2 r U2 r' U2 r' U' 2R U r U2 r U2 r' U2 r' U' 2R2 U2 r (31,23) Christopher Mowla [X]
 r U r' U' r U2 r' U2 r' U' r U' r U r2 U' r2 U2 r U2 r' U2 r U2 r U' (34,26) Bruce Norskog [X]
 r U r' U' r U2 r' U2 r' U' r U' r U' r2 U2 r2 U' r U2 r' U2 r U2 r U (34,26) Bruce Norskog [X]

### Case 1 (Close Adjacent Unoriented)

 2L2' U l U2 2R' U2 r' U2 x U' 2R' U x' U2 x U' M' 2L' (20,15) Christopher Mowla [X]
 x' 2L2 F U2 2L U2 2R' U2 2R U2 F2 2R F2 2L' F' 2L2 x (23,15) reThinking the Cube [X]
 2R U2 2R U2 F' L F' 2R F L' F 2L' U2 2L U2 2R2 (21,16) Christopher Mowla [X]
 x' 2L2 F U2 2L U2 2R' U2 2R U2 3r U2 2R U2 r' F' 2L2 x (24,16) Christopher Mowla [X]
 z F' r2 2B' u l' u' 2B u 2B l 2B' l' 2B' l u' r2 F z' (19,17) Christopher Mowla [X]
 z F' l2 u b' 2L b u' b' l u 2L u' l' 2L' b l2 F z' (19,17) Christopher Mowla [X]
 z F' r2 z' r b' 2L b r' b' l d 2L d' l' 2L' f r2 F z' (19,17) Christopher Mowla [X]
 l' 2B u2 l u 2F' u' l' u 2F' l 2F' l' 2F2 u 2B' l (19,17) Christopher Mowla [X]
 2R2 U2 2L U2 2R' U2 2R U2 F' L F' 2R F L' F 2L' 2R2 (23,17) Christopher Mowla [X]
 2L2 U M U2 2L U2 2R' U2 2R U2 M' U2 2R U2 2L' U' 2L2 (25,17) Christopher Mowla [X]
 2L U 2L U 2L' U' 2L' U' 2L D 2F2 2L' 2F2 2L U2 D' 2L U' 2L' U' 2L2//Safe (25,21) [X]
 x' 2L' r' U2 r U' L' U' 2R' U L U r' U2 r U' L' U' 2R2 U L U 2L x (25,22) Christopher Mowla [X]
 z' f' u 2R' u' f u 2R' f2 L f U' R2 U 2R' U' R2 U f' L' f2 2R2 u' z (27,22) Christopher Mowla [X]
 2L' F2 2L F2 U' r U2 2L U2 r' B' M U' l2 F2 2R2 F2 l2 U M' B U (31,22) Christopher Mowla [X]

### Case 2 (Far Adjacent Unoriented)

 2L2' U' l U2 2R' U2 r' U2 x U 2R' U' x' U2 x U M' 2L' (20,15) Christopher Mowla [X]
 x r2 F' U2 2L U2 2R' U2 2R U2 F2 2R F2 2L' F r2 x' (23,15) Christopher Mowla [X]
 2R U2 2R U2 F R' F 2R F' R F' 2L' U2 2L U2 2R2 (21,16) Christopher Mowla [X]
 x r2 F' U2 2L U2 2R' U2 2R U2 3r U2 2R U2 r' F r2 x' (24,16) Christopher Mowla [X]
 r 2R z 2L f' u f 2L' f' 2L' u' 2L u 2L u' f z' r' 2R' (17,17) Christopher Mowla [X]
 r 2R y r' d 2L' d' r d l' b' 2L' b l 2L u' r' 2R' (17,17) Christopher Mowla [X]
 l 2R u' f 2L' f' u f l' u' 2L' u l 2L f' l' 2R' (17,17) Christopher Mowla [X]
 r' 2L' x' U2 2L U2 2R' U2 2R U2 x U R U 2R U' R' U' r (21,17) Christopher Mowla [X]
 2R2 U2 2L U2 2R' U2 2R U2 F R' F 2R F' R F' 2L' 2R2 (23,17) Christopher Mowla [X]
 l2 U' M U2 2L U2 2R' U2 2R U2 M' U2 2R U2 2L' U l2 (25,17) Christopher Mowla [X]
 y z d' 3r' D l' u' 2R' u l u' l2' b' 2R' b r' R' 2U y' 3r u y x (19,18) Christopher Mowla [X]
 x' U2 2L' U2 2L U2 2R' U2 2R B2 2R' F D2 F' 2L' F D2 F' 2R B2 x (27,19) Christopher Mowla [X]
 z' f' u 2R' u' f u 2R' f2 U L U' f 2R' f' U L' U' f2 2R2 u' z (23,20) Christopher Mowla [X]
 r' U R U 2R2 U' R' U' r' U2 r U R U 2R U' R' U' r' U2 r2 (25,21) Christopher Mowla [X]
 2L U' 2L U' 2L' U 2L' U 2L D' 2B2 2L' 2B2 2L U2 D 2L U 2L' U 2L2//Safe (25,21) [X]
 z' f' u 2R' u' f u 2R' f2 L f U' R U 2R' U' R' U f' L' f2 2R2 u' z (25,22) Christopher Mowla [X]
 r 2R U' R' U 2R' U' R U2 R' U 2R U2 2R U2 2R U2 2R2 U R U' r' (27,22) Christopher Mowla [X]
 r2 R U2 r U 2R U' r' U2 r' U2 r U2 r U 2R2 U' r' U2 r' U2 r' R' (30,23) Christopher Mowla [X]

### Case 3 (Oriented Case)

 y' f2 2L u' b u 2L' u' 2L' b' 2L b 2L b' u f2 y (17,15) Christopher Mowla [X]
 r2 u' b 2R' b' u b r' u' 2R' u r 2R b' r2 (17,15) Christopher Mowla [X]
 y' z' f2 r' d 2L' d' r d l' b' 2L' b l 2L d' f2 z y (17,15) Christopher Mowla [X]
 r2 2F u' b d 2R' d' 2R' b' 2R b 2R b' u r2 (17,15) Christopher Mowla [X]
 y' z' f' 2L' u' b2 u' 2R u b2 u2 2R u 2R' u2 2L f z y (19,15) Christopher Mowla [X]
 u r' u' 2R' u r b' r' 2R' u' 2R' u R 2R' b u' (16,16) Christopher Mowla [X]
 y' b' R d 2R' d' r' b r' R' d 2R' d' r2 f' 2U f y (17,16) Christopher Mowla [X]
 y' u' f 2L' f' u f 2L' u2 R u 2L' u' R' u2 2L2 f' y (19,16) Christopher Mowla [X]
 2R' u l u' 2R2 f2 l' f 2R' f' l f2 2R' u l' u' (19,16) Christopher Mowla [X]
 z x2 r' u 2L' u' r u 2L' f2 R f 2L' f' R' f2 2L2 u' x2 z' (19,16) Christopher Mowla [X]
 y' b' u2 2R' d 2L' d' 2R u' l2 u' 2R' u l2 u' 2R b y (19,16) Christopher Mowla [X]
 y' 2L' 2R' U r U2 2L' U2 l' U2 x' U' 2L' U F2 U' M' 2L' y (20,16) Christopher Mowla [X]
 z u' r u2 2L' u 2R' u' 2L u2 R' u b2 u' 2R' u b2 z' (20,16) Christopher Mowla [X]
 x 2L' U L U 2L' U' L' U' F2 2L' F2 2R U2 2R' U2 2L2 x' (21,16) Christopher Mowla [X]
 L' B' 2L' U2 2L' U2 x U2 2L' U2 x' 2R U2 2R' U2 2L2 B L (23,16) [X]
 x 2L' U L U 2L' U' L' U' x U2 2L' U2 2L U2 2R' U2 2L 2R x2 (21,17) Christopher Mowla [X]
 x 2L' U L U 2L' U' L' U' M' U2 2L' U2 2L U2 2R' U2 2L2 x' (22,17) Christopher Mowla [X]
 x 2L U2 2L U2 M U L U 2L U' L' U' 2L' U2 2R U2 2L2 x' (22,17) Christopher Mowla [X]
 r U2 2L U2 3l U L' U 2L U' L U' 2L' U2 2R U2 l2 x' (22,17) Christopher Mowla [X]
 x' 2F' u r' f' 2R' f r u' 2F 2R' u r' f' 2R' f r u' 2R2 x (19,18) Christopher Mowla [X]
 y' 2L U F2 U' 2L U F2 2L F L F 2L F' L' F' 2L' U' 2L2 y (21,18) Christopher Mowla [X]
 x l' L' U L' U 2L' U' L U' M' U2 2L' U2 2L U2 2R' U2 l2 x' (23,18) Christopher Mowla [X]
 z' 2R' u b u' b L b 2R b' L' b' u b' l u' 2R u l' u' z (19,19) Christopher Mowla [X]
 2R u' l u 2R' u' f' u' B r2 B' 2L B r2 B' u f l' u (21,19) Christopher Mowla [X]
 y' x l' U' R' U' 2L2 U R U l' U2 l U' R' U' 2L U R U l' U2 l2 x' y (25,21) Christopher Mowla [X]
 2R' U' 2R U' 2R' D 2B2 2R 2B2 2R' U2 D' 2R' U' 2R U' 2R U 2R' U 2R2//Safe (25,21) [X]
 x 2L2 U2 l U 2L U' l' U2 l' U2 l U2 l U 2L2 U' l' U2 l' U2 l' L2 x' (30,22) Christopher Mowla [X]
 r2 U r' U r' U r U2 r U2 r' U r U' r' U' r' U2 r' U2 r U2 r' U r2 U2 (34,26) Bruce Norskog [X]
 y r' U2 2R' r2 U2 r U2 r U 2R U' r' U2 r' U2 r U2 r U 2R2 U' r' U2 r 2R U2 r y' (37,27) Christopher Mowla [X]
 U' r' U r U' r2 U2 r' U2 r U2 r' U r2 U' r U r' U' r U2 r U2 r2 U r U' r (36,28) Ben Whitmore [X]

# Non Dedge-Preserving Last Layer 4-Cycle Cases in Two Dedges

• These are all of the other 4-cycles that can occur in two dedges in the last layer besides double parity and adjacent double parity.
• These are additional cases which arise in the K4 Method and other direct-solving methods.

## In Opposite Dedges

### Checkerboard

 r2 U2 2R' E2 2R E2 2R' U2 r2 (15,9) Tom Rokicki & Ed Trice [X]
 (2F2 2U' 2R2 u2 S') 2R (S u2 2R2 2U 2F2) (17,11) Christopher Mowla [X]
 (2F2 2U 2R2 u2 S') 2R (S u2 2R2 2U' 2F2) (17,11) Christopher Mowla [X]
 2R2 U2 2R S2 2R S2 2R' U2 2L2 2F2 M2 2F2 (21,12) [X]
 2R2 U2 2R S2 2R S2 2R' U2 2L2 2B2 M2 2B2 (21,12) [X]
 2R S' L2 S 2R U2 2R U2 2R S' L2 S 2R (17,13) Christopher Mowla [X]
 2R S R2 S' 2R U2 2R U2 2R S R2 S' 2R (17,13) Christopher Mowla [X]
 (r2 u U S l2 2U2) 2R (2U2 l2 S' U' u' r2) (19,13) Christopher Mowla [X]
 (r2 u' U' S l2 2U2) 2R (2U2 l2 S' U u r2) (19,13) Christopher Mowla [X]
 r2 S 2R' U2 2R' U2 2R' U2 2R' U2 2R' S' r2 (19,13) Christopher Mowla [X]
 r2 S' 2R' U2 2R' U2 2R' U2 2R' U2 2R' S r2 (19,13) Christopher Mowla [X]
 Alg(v1) 2R' U2 2R2 U2 2R U2 2R' U2 2R U2 2R2 U2 2R' (21,13) [X]
 Alg(v2) 2L' U2 2L2 U2 2L U2 2L' U2 2L U2 2L2 U2 2L' (21,13) [X]

### Bowtie/Hourglass

• It might be best to think of this case as adjacent 2-swap + PLL Parity, because if we combine the proper algorithms for each, we produce the shortest algorithm in STM.
The algorithms ending in u2 are the result of combing algorithms for those two cases.

 2R U2 2R U2 F2 2R F2 2L' U2 M 2R' u2 2R2 2U2 (22,14) [X]
 2R U2 2R U2 M' U2 2R U2 2R' U2 M 2R' u2 2R2 2U2 (23,15) [X]
 2R2 U2 2L' U2 2L F2 U2 2R2 U2 2R F2 2R' F2 2R' F2 (25,15) [X]
 2R' U2 2R U2 2L' U2 2R U2 2L F2 2R' F2 2R F2 2R2 F2 (25,16) [X]
 (B' R u' 2U' 2R' 2F2 r2 E) 2F' (E' r2 2F2 2R 2U u R' B) (21,17) Christopher Mowla [X]
 (B' R u' 2U' 2R 2F2 r2 E) 2F' (E' r2 2F2 2R' 2U u R' B) (21,17) Christopher Mowla [X]
 2R U2 M' U2 2R2 U2 2R' U2 2R U2 2R' U2 2R' U2 2L U2 2R' (26,17) [X]

### Checkboard

 F2 u' F2 r2 u2 R2 z' S 2U F' B' u2 r2 u' F2 (22,14) Tom Rokicki [X]
 (F U' F' r2 U2) 2R' E2 2R E2 2R' (U2 r2 F U F') (21,15) Tom Rokicki & Ed Trice [X]
 (R2 u' 2U' 2R' 2F2 r2 E) 2F (E' r2 2F2 2R 2U u R2) (21,15) Christopher Mowla [X]
 (R2 u' 2U' 2R 2F2 r2 E) 2F (E' r2 2F2 2R' 2U u R2) (21,15) Christopher Mowla [X]
 (r' F2 M2 F) U2 2L U2 2L2 U2 2L U2 2L (F' M2 F2 r) (25,16) Christopher Mowla [X]
 (l' U2 M2 U) M U2 2L U2 2L2 U2 2L U2 2R (U' M2 U2 l) (26,17) Christopher Mowla [X]
 (R B) 2R U2 M' U2 2R2 U2 2R' U2 2R U2 2R' U2 2R' U2 2L U2 2R' (B' R') (30,21) [X]

### Bowtie/Hourglass

 (R B) r2 U2 2R' E2 2R E2 2R' U2 r2 (B' R') (19,13) Tom Rokicki & Ed Trice [X]
 (R f 2F 2R 2U2 r2 S) 2U' (S' r2 2U2 2R' f' 2F' R') (19,15) Christopher Mowla [X]
 (R f 2F 2R' 2U2 r2 S) 2U' (S' r2 2U2 2R f' 2F' R') (19,15) Christopher Mowla [X]
 F' l U D l2 b2 r' R' U' D B2 l2 f2 D2 l' F (21,15) Tom Rokicki [X]
 (l' U2 M2 U' x) 2R' U2 2R' U2 2R2 U2 2R' U2 (x' U M2 U2 l) (25,16) Christopher Mowla [X]
 R B 2R U2 2R2 U2 2R' U2 2R U2 2R' U2 2R2 U2 2R B' R' (25,17) [X]
 (2R U2 M2 U') M' U2 2L' U2 2L2 U2 2L' U2 2R' (U M2 U2 2R') (26,17) Christopher Mowla [X]
 y2 f' 2R2 u2 L' u' U' R U F' R' F 2R' F' R F U' R' u U L u2 2R' f u f' 2R' f u' y2 (31,28) Christopher Mowla [X]

# Summary of Last Layer 2-cycles and 4-cycles (in two dedges) Movecounts

• Since there is disagreement among cubers about whether lowest slice quarter turn moves (SQTM) or lowest slice half turn moves (STM) defines the "optimal algorithm", the table below categorizes all 2-cycle and 4-cycle cases (in two dedges) on the 4x4x4 by the average of the two.
• Algorithms optimal in STM need not be the algorithm with the lowest average of SQTM and STM, and algorithms optimal in SQTM turns need not be the algorithm with the lowest average of SQTM and STM. Therefore the average for a given case might be from an algorithm optimal in SQTM moves, optimal in STM moves or optimal in both.
• Of course this ranking is based off of required moves. It is not based off of the amount of time it takes to solve a case.
• These results will change in the future if shorter algorithms are found for any of these cases.
• Lastly, note that term "optimal average" below does not necessarily mean the average of the minimum SQTM and STM (if an algorithm has both the fewest SQTM and STM, then yes). It's an average of the SQTM and STM of one algorithm. Should one algorithm have the least SQTM but not the least STM, we do not average the shortest numbers from two different algorithms.
• The table shows that the "worst case" is Adjacent Double Parity, and that the "easiest case" is the Checkerboard Pattern (in two opposite edges).

# Algorithms Which Don't Preserve the Centers

• These algorithms are for fixing the wing edges, but they do not preserve the centers. Many of these algorithms can be used in solving methods such as the Cage Method.

## One Dedge Flip

 r2 U 2R' U' B2 U 2R U 2R' U2 B2 r2 (17,12) Christopher Mowla [X]
 r2 U' 2R' U B2 U' 2R U' 2R' U2 B2 r2 (17,12) Christopher Mowla [X]
 r2 3d 2L U' R2 U 2L' U y 2R U2 F2 r2 (17,12) Christopher Mowla [X]
 r2 F 2R F' u2 B 2L' B 2L B2 u2 r2 (17,12) Christopher Mowla [X]
 l2 3d' 2R' U L2 U' 2R D' 2R' U D L2 y' l2 (17,13) Christopher Mowla [X]
 r2 U' 2R' U B2 U' 2R U B2 D2 2L D2 r2 (19,13) Christopher Mowla [X]
 r2 3d 2L U' R2 U 2L' U' D2 y 2L D2 F2 r2 (19,13) Christopher Mowla [X]
 r2 U2 2R U2 2L' U2 2L U2 2R' F2 2R F2 r2 (21,13) [X]
 r2 U2 2R U2 2L' U2 2L U2 2R' B2 2L B2 r2 (21,13) [X]
 2L U2 2L' U2 l2 F2 2R' F2 r2 U2 2L' U2 2R x2 (21,13) [X]
 2R2 u b' u' 2R u r b 2R' b' r' b u' 2R (15,14) Christopher Mowla [X]
 2R2 U2 2R' U2 2R U2 2L' U2 2R U2 2R' U2 2L 2R (21,14) Lucas Garron [X]
 2R2 U2 2R U2 2L' U2 2L U2 2L' U2 2R U2 2L 2R (21,14) Nicholas Ho [X]
 x 2R R' U' r U' 2R U r' 2R' U R U' 2R U 2R' x' (15,15) [X]

## One Dedge Flip + PLL Parity (Double Parity)

• Most of these algorithms are exactly the same as the one dedge flip ones except that the extra quarter turn is inverted.
 r' F' 2U F' D2 F 2U' F' 2L D2 F2 r (15,12) Christopher Mowla [X]
 r2 U 2R' U' B2 U 2R U 2R U2 B2 r2 (17,12) Christopher Mowla [X]
 r2 U' 2R' U B2 U' 2R U' 2R U2 B2 r2 (17,12) Christopher Mowla [X]
 r2 3d 2L U' R2 U 2L' U y 2R' U2 F2 r2 (17,12) Christopher Mowla [X]
 l2 3d' 2R' U L2 U' 2R D' 2R U D L2 y' l2 (17,13) Christopher Mowla [X]
 r2 U' 2R' U B2 U' 2R U B2 D2 2L' D2 r2 (19,13) Christopher Mowla [X]
 r2 3d 2L U' R2 U 2L' U' D2 y 2L' D2 F2 r2 (19,13) Christopher Mowla [X]
 x' r2 F2 2R F2 2R U2 2L' U2 2L U2 2R' U2 r2 x (21,13) [X]
 2L' y' u2 r2 u' 2L' u r2 u' 2L 2U' U y 2L U2 2L' (18,14) Christopher Mowla [X]
 r2 2B2 D' 2F' D B2 D' 2F U' 2F U D b2 r2 (19,14) Christopher Mowla [X]
 2L 2R U2 2L U2 2R' U2 2R U2 2R' U2 2L' U2 2L2 (21,14) Nicholas Ho [X]
 2R U2 2R' U2 b' 2R f 2R' b r2 f' 2R f r2 f' (19,15) Christopher Mowla [X]

 (2R2 U' r U' R' U') 2R' (U R U r' U 2R2) (15,13) Christopher Mowla [X]
 (2R2 U F2 U' 2R' F2) 2R (F2 2R U F2 U' 2R2) (19,13) Christopher Mowla [X]
 r' U r U F2 U' 2R' U F2 2R' U' R' U' r (16,14) Christopher Mowla [X]
 (x u F U 2R' U R' U) 2R (U' R U' 2R U' F' u' x') (15,15) Christopher Mowla [X]
 x' 2R2 U' r U' 2L' U2 2L' U2 2L' U2 2L2 U' r' U 2R2 x (21,15) Christopher Mowla [X]

## Three Flips

### OLL Parity (Only)

 (2R2 F2 U) 2R U2 2R' U2 2R U2 2R' U2 2R (U' F2 2R2) (23,15) Christopher Mowla [X]
 (2R2 U2 F' M' U) 2R U2 2R' U2 2R U2 2R' U2 2R (U' M F U2 2R2) (27,19) Christopher Mowla [X]

### OLL Parity + PLL Parity (Double Parity)

 (2R' U' F' R' F R2 U') 2R' (U R2 F' R F U 2R) (2R U2 2R' U2) (23,18) Christopher Mowla [X]

## 2-Cycles In Two Adjacent Edges (in the M ring)

 2L U' 2L' U' F2 U 2L U' F2 U2 (13,10) Christopher Mowla [X]
 2L U 2L' U F2 U' 2L U F2 U2 (13,10) Christopher Mowla [X]
 z2 y' U2 R2 U 2L U' R2 U 2L' U y 2R z2 (13,10) Christopher Mowla [X]
 2L F2 2R' U2 2L U2 2L' U2 2R U2 F2 (17,11) [X]
 2L' U l' U L U' 2L' U L' U' l 2L U' (13,13) [X]

### Opposite/Diagonal 2-Swap

• Some of these are nearly the same as the one dedge flip algorithms.
 l' U' 2L U B2 U' 2L' U' 2L U2 B2 l (15,12) Christopher Mowla [X]
 l' U 2L U' B2 U 2L' U 2L U2 B2 l (15,12) Christopher Mowla [X]
 l' 3d' 2R' U L2 U' 2R U' y' 2L' U2 F2 l (17,12) Christopher Mowla [X]
 x' U r' U 2R' U' r 2R U' R' U 2R' U' R x (13,13) Marc Waterman [X]
 x' R U r' U 2R' U' r2R U' R' U 2R' U' x (13,13) Marc Waterman [X]
 x' 2R' u b' u' 2R u r b 2R' b' r' b u' x (13,13) Christopher Mowla [X]
 l' 3d' 2R' U L2 U' 2R D' 2R' U D L2 y' l (15,13) Christopher Mowla [X]
 l' U 2L U' B2 U 2L' U' B2 D2 2R' D2 l (17,13) Christopher Mowla [X]
 l' 3d' 2R' U L2 U' 2R U D2 y' 2R' D2 F2 l (17,13) Christopher Mowla