4x4x4 parity algorithms

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Parity (also known as Orientation Parity and Permutation Parity) on the 4x4x4 is a common situation (occurring in 3/4 of all solves) where two or four edge pieces need to be cycled. This is considered difficult because in most methods the centers are solved before the edges, and there is no efficient or intuitive way to swap those edges (and no others) without affecting the centers. Because of how common and difficult this case is, many algorithms have been developed. This page attempts to list all efficient algorithms. 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:
Alg (r U2)4 r/(13,9)

For those who are familiar with commutators and conjugates, this quick parity fix can be represented as [r: [U2, r] [r2 U2: r] ]. In fact, we can do the same 4-cycle of wing edges with just one conjugate [r2 d' r2 Uw2 S': r'].


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

  • Algorithms with a lower btm (block half turn) move count are listed first in each category.
  • All algorithms can be applied to the 6x6x6 if instead of turning the outer 2 layers, you turn the outer 3 layers. For slice moves, instead of turning 1 slice layer, you turn 2 slices.


PLL Parity (WCA Notation)

PLL Parity in Two Opposite Edges ("Oriented")

Alg r2 U2 r2 Uw2 r2 u2/(12,6)
Alg r2 U2 r2 Uw2 r2 Uw2/(12,6)
Alg (Rw2 R2') U2 (Rw2 R2') Uw2 (Rw2 R2') Uw2/(12,6)
Alg Uw2 Rw2 U2 (Rw2 R2') U2 Rw2 Uw2/(14,7)
Alg Uw2 Rw2 U2 r2 U2 Rw2 Uw2/(14,7)
Alg Rw2 B2 U2 r2 U2 B2 Rw2/(14,7)


PLL Parity in Two Adjacent Edges ("Oriented")

Alg R2 D' x r2 U2 r2 Uw2 r2 u2 x' D R2/(18,10)
Alg R2 D' x Rw2 F2 U2 r2 U2 F2 Rw2 x' D R2/(20,11)
Alg F2 U Rw2 U2 F2 r2 F2 U2 Rw2 U' F2/(20,11)
Alg y' R' F l E F2 E' l' r' E F2 E' r F' R y/(16,14)
Alg y' R' F l E' F2 E l' r' E' F2 E r F' R y/(16,14)
Alg (Rw' U R U Lw' U2 Rw' U2) r2 (U2 Rw U2 Lw U' R' U' Rw)/(22,17)


PLL Parity in Two Opposite Edges ("Unoriented")

Alg F2 l E F2 E' l' r' E F2 E' r F2/(16,12)
Alg F2 l E' F2 E l' r' E' F2 E r F2/(16,12)
Alg y R' U F' r2 U2 r2 Uw2 r2 u2 F U' R y'/(18,12)
Alg r U2 l D2 l' U2 M U2 r D2 r' U2 l'/(19,13)
Alg y' R' F U' Rw2 U2 F2 r2 F2 U2 Rw2 U F' R y/(20,13)
Alg y R' U F' Rw2 F2 U2 r2 U2 F2 Rw2 F U' R y'/(20,13)


PLL Parity in Two Adjacent Edges ("Unoriented")

Alg R B r2 U2 r2 Uw2 r2 u2 B' R'/(16,10)
Alg (R B Rw2 F2 U2) r2 (U2 F2 Rw2 B' R')/(18,11)
Alg ((Lw r') U Rw2 U2 F2) r2 (F2 U2 Rw2 U' (Lw' r))/(18,11)


Adjacent Two Corner Swap

Alg R U' R B2 L' D L B2 R2 U r2 F2 r2 Fw2 r2 f2/(25,16)
Alg B2 L U L' B2 R D' r2 F2 r2 Fw2 r2 f2 R D R2/(25,16)
Alg z r2 U2 R' U2 R' U2 R x U2 Rw2 U2 B2 L U2 L' U2 Rw2 U2 z' y'/(29,17)
Alg z' Rw2 x U2 R' U2 x' U2 R' U2 R U2 L' x U2 Rw2 U2 Rw2 U2 Rw2 x' U2 Rw2 z U/(33,19)
Alg r2 U2 r2 Uw2 r2 u2 y' R U R' U' R' F R2 U' R' U' R U R' F' y/(27,20)
Alg z' x Rw2 R' U2 Rw' U2 Rw2 U2 Rw2 U2 L Rw' U2 R' U2 R U2 L' R U2 R U2 Rw2 x' z/(35,22)
Alg z' x Rw2 R' U2 Rw U2 Rw2 U2 Rw2 U2 L Rw U2 R' U2 R U2 L' R U2 R U2 Rw2 x' z/(35,22)


Adjacent Two Corner Swap (Only Two X-Center Piece Exchange on the Supercube)

Alg U Rw2 Bw2 Rw F Rw' Bw2 Rw F' L2 r U' b2 U L' F' L U' b2 U L' F M x/(29,23)
Alg z' Fw' L2 Uw F' U R U' F r2 F' U R' U' F r2 Fw Uw' L Uw Fw' Uw' L Fw L z/(27,24)
Alg z' Fw' L2 Uw r2 F' U R U' F r2 F' U R' U' F Fw Uw' L Uw Fw' Uw' L Fw L z/(27,24)
Alg Rw' U' R U r U' R' U R2 U R' U' r U r' F' Rw U Rw' U' Rw' F Rw2 U' Rw' U'/(28,26)
Alg Rw' U' R U r U' R' U R2 U R' U' r U r' U' Rw' F Rw2 U' Rw' U' Rw U Rw' F'/(28,26)
Alg Rw' U' R U r U' R' U R2 U R' U' r U R U' Lw Uw2 Rw' U' Rw Dw2 Rw' U Rw' F' x y2/(29,26)
Alg U Rw2 Bw2 Rw F Rw' Bw2 Rw F' D' f' D F2 D' f D F2 Rw L F' L' f' L F L' f/(31,26)
Alg Rw U Rw' U' Rw' F Rw2 U' Rw' U' Rw U Rw' R' F' r' F R F' r2 B' R B r' B' R' B/(29,27)
Alg Rw U R' U' r' U R U' R2 U' R U r' U' R' Lw U Lw' U' Rw U Lw U' Lw' Rw' U Rw U/(29,28)


Opposite/Diagonal Two Corner Swap

Alg Rw2 f2 U2 Fw2 U' Rw2 U2 Fw2 U Fw2 R2 U2 F2 Rw2 U/(27,15)
Alg Rw2 F2 U2 y Rw2 U' Rw2 U D Lw2' U' Lw2' y' r2 U2 F2 Rw2 U/(27,16)
reParity z (R2' Uw2' R2 U R2') y (R2 U2 R2' U' R2) y' (R2' Uw2' R2 (Dw' u) R2) Uw2 U2 y' z' /(29,16)
Alg r2 F2 R' F2 U2 R2 U2 R' F2 R2 U2 Rw' r' U2 F2 Rw2 U2/(30,17)
Alg r2 B2 R' U2 Rw2 U2 B2 R' B2 r2 U2 Rw2 B2 U2 R' U2 Rw2/(31,17)
Alg r2 U2 r2 Uw2 r2 Uw2 L' U R' U2 L U' L' R U R' U2 L U' R U/(29,21)
Alg Rw2 R U2 R U2 R' U2 Rw U2 Rw2 U2 Rw2 U2 L Rw U2 R' U2 R U2 L' Rw2/(35,22)


Opposite/Diagonal (Only Two X-Center Piece Exchange on the Supercube)

Alg F r2 F' U R' U' F r2 Fw Uw' L Uw Fw' Uw' L Fw L Fw' L2 Uw F' U R U'/(27,24)
Alg F Rw' F Rw2 U' Rw' U' Rw' U' R U r U' R' U R2 U R' U' r U r' F' Rw U Rw' U' F'/(30,28)



Impure Dedge Flips (OLL Parity)(Turn the inner and outer layers simultaneously)

  • Unless mentioned otherwise, the algorithms in this section are written in SiGN notation in which lowercase letters mean to move the inner and outer layer at the same time (wide turns).


OLL Parity

cmowlaparity x' r2 U2 l' U2 r U2 r U2 x' U r U' F2 U r' U r2 x/(23,16)
Alg r U2 x U2 F' r' F U2 F' r F r U2 r' U2 x' r' U2 r'/(23,17)
lucasparity r U2 x r U2 r U2 r' U2 l U2 r' U2 r U2 r' U2 r'/(25,17)
lucasparity r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 x' r' U2 r'/(25,17)
Alg r U2 r2 U2 x' U r U' F2 U r' U' l' U2 r U2 r' U2 r'/(25,18)
Alg r2 B2 3d' r' 3d r U2 r' 3d' r U' y' r' U2 r' U2 r B2 r2/(25,18)


OLL Parity + PLL Parity (Double Parity)

Alg_1(v1) r' U2 r' U2 l U2 r' U2 r U2 x U2 r2 U2 r' U2 r/(25,16)
Alg_1(v2) l U2 l U2 r' U2 l U2 l' U2 x U2 l2 U2 l U2 l'/(25,16)
Alg_2(v1) r U2 r' U2 r U2 r U2 l' U2 r U2 r' U2 x' U2 r2/(25,16)
Alg_2(v2) l' U2 l U2 l' U2 l' U2 r U2 l' U2 l U2 x' U2 l2/(25,16)
Alg_3(v1) r2 U2 x' U2 r' U2 r U2 l' U2 r U2 r U2 r' U2 r/(25,16)
Alg_3(v2) l2 U2 x' U2 l U2 l' U2 r U2 l' U2 l' U2 l U2 l'/(25,16)
Alg_4(v1) r U2 r' U2 r2 U2 x U2 r U2 r' U2 l U2 r' U2 r'/(25,16)
Alg_4(v2) l' U2 l U2 l2 U2 x U2 l' U2 l U2 r' U2 l U2 l/(25,16)
Alg_5(v1) Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U Rw2 R U Rw U' Rw/(23,19)(WCA)
Alg_5(v2) Rw' U2 Rw' U2 Rw' U' R' Rw' U2 R2 U' Rw' U' Rw2 R' U' Rw' U Rw'/(23,19)(WCA)


Impure Dedge Flips with Wide Turns/Pure Flips with Inner Layer Turns (SiGN Notation)

OLL Parity

  • If the following group of algorithms is executed using WCA notation (where lowercase = inner layer turns), pure one dedge flips will result.
  • Note that all 26 possible (25,15) solutions (excluding mirrors and inverses) of this category are listed.
Frédérick_Badie(v1) r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2/(25,15)
Frédérick_Badie(v2) r' U2 r U2 l' U2 l2 F2 r F2 l' U2 F2 r2 F2/(25,15)
Frédérick_Badie(v3) r B2 r' U2 r U2 r2 F2 r' D2 l D2 B2 r2 F2/(25,15)
Frédérick_Badie(v4) r B2 r' U2 r U2 l2 B2 r' U2 l U2 F2 l2 F2/(25,15)
Frédérick_Badie(v5) r B2 l' B2 l U2 r2 F2 l' F2 r U2 F2 l2 F2/(25,15)
Alg.2(v1) r B2 r' U2 l B2 l2 B2 U2 l U2 l' U2 l2 B2/(25,15)
Alg.2(v2) r B2 r' U2 r D2 r2 U2 F2 l F2 r' D2 r2 B2/(25,15)
Alg.2(v3) r B2 r' U2 r D2 l2 D2 B2 l B2 r' U2 l2 B2/(25,15)
Alg.2(v4) r B2 l' B2 l D2 l2 D2 B2 r D2 l' D2 r2 B2/(25,15)
Alg.2(v5) r' U2 r U2 l' D2 r2 D2 B2 l' D2 r D2 l2 B2/(25,15)
Alg.3(v1) r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2/(25,15)
Alg.3(v2) r2 B2 U2 l U2 l' B2 r B2 U2 r U2 r' B2 r2/(25,15)
Alg.3(v3) r2 F2 D2 r' D2 r F2 l' F2 D2 l' D2 l F2 r2/(25,15)
Alg.3(v4) r2 F2 D2 r' D2 l D2 l' D2 B2 l' B2 r F2 r2/(25,15)
Alg.3(v5) r2 B2 U2 r B2 r' B2 l U2 F2 r F2 l' B2 r2/(25,15)
Alg.3(v6) r2 F2 D2 l' F2 l F2 r' D2 B2 l' B2 r F2 r2/(25,15)
Alg.3(v7) r2 B2 U2 r B2 l' D2 r D2 B2 l U2 r' B2 r2/(25,15)
Alg.3(v8) r2 F2 D2 l' F2 r U2 l' U2 F2 r' D2 l F2 r2/(25,15)
Alg.4(v1) r2 B2 U2 l' U2 r U2 r' U2 B2 r' B2 l B2 r2/(25,15)
Alg.4(v2) r2 B2 U2 l' U2 l F2 r' F2 U2 r' U2 r B2 r2/(25,15)
Alg.4(v3) r2 F2 D2 r D2 r' B2 l B2 D2 l D2 l' F2 r2/(25,15)
Alg.4(v4) r2 F2 D2 r D2 l' D2 l D2 F2 l F2 r' F2 r2/(25,15)
Alg.4(v5) r2 B2 U2 r' F2 r F2 l' U2 B2 r' B2 l B2 r2/(25,15)
Alg.4(v6) r2 B2 U2 r' F2 l D2 r' D2 F2 l' U2 r B2 r2/(25,15)
Alg.4(v7) r2 F2 D2 l B2 r' U2 l U2 B2 r D2 l' F2 r2/(25,15)
Alg.4(v8) r2 F2 D2 l B2 l' B2 r D2 F2 l F2 r' F2 r2/(25,15)
cmowlaparity x' r2 U2 l' U2 r U2 l F2 U r U' F2 U r' U r2 x/(23,16)
Alg r U2 F2 D' r' D F2 D' r D r F2 r' F2 r' U2 r'/(23,17)
lucasparity r U2 r U2 r' U2 r U2 l' U2 l F2 r' F2 r' U2 r'/(25,17)
Alg r2 U2 l' U2 r U2 r' U2 r U2 r' x' U2 r' U2 r2 F2 l x/(27,17)
Alg r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 l r2 U2 r'/(27,18)
Alg r2 F2 r' U2 r U2 r' U2 r U2 r' F2 r' U2 l' r' U2 l/(27,18)


OLL Parity + PLL Parity (Double Parity)(SiGN Notation)

  • If the following group of algorithms is executed using WCA notation (where lowercase = inner layer turns), pure one dedge flips + PLL Parity will result.
Alg r2 B2 r' U2 r' U2 x' U2 r' U2 r U2 r' U2 r2 U2 x/(25,15)
Alg r U2 r' U2 r U2 r U2 r' x U2 r U2 r' U2 x' U2 r2 U2/(27,17)


Pure Flips (WCA Notation)

(Algorithms in SiGN Notation are labeled.)

OLL Parity: One Dedge Flip

cmowlaparity x' Rw2 U2 Lw' U2 r U2 Rw U2 x' U r U' F2 U r' U Rw2 x/(23,16)
Alg x' Rw2 U2 Lw' U2 r U2 Rw U2 x' U' r U F2 U' r' U' Rw2 x/(23,16)
Alg Rw Uw2 x' U D r y Uw2 Rw2 Uw r Uw' Rw2 Dw r' u D' x Uw2 Rw'/(22,17)
Alg Rw U2 F2 D' r' D F2 D' r D Rw F2 r' F2 Rw' U2 Rw'/(23,17)
Alg Rw U2 F2 D r' D' F2 D r D' Rw F2 r' F2 Rw' U2 Rw'/(23,17)
lucasparity Rw U2 Rw U2 r' U2 r U2 l' U2 l F2 r' F2 Rw' U2 Rw'/(25,17)
Holy_Grail z Dw' M D Lw' Uw' r' Uw Lw Uw' Lw2' Bw' r' Bw Rw' R' u y' M' Uw x2 z'/(19,18) [1]
Alg Rw' E Uw2 Fw Uw Rw' r' Fw' r Fw Rw Uw' Fw' Uw r Uw E' Rw/(19,18)
Alg Lw' S Bw2 Dw r Dw' Bw Dw r' Dw' r' Bw' r Bw r Bw S' Lw/(19,18)
Alg Lw' F' b' Rw' Bw' Rw u Rw' Bw Rw u2 Bw' u Bw u b F Lw/(19,18)
Alg Rw2 F' r U' R U' r' U R' U r' F' U2 r' U2 r F2 Rw2/(23,18)
Alg z r f2 3u U y' 2R' u2 y' l2 u' 2R' u l2 u' 2R 3u' u' U' x' u2 r' z/(23,18)(SiGN)
Alg Rw2 B2 U' b' U r U2 r' U' b U' Rw' U2 r' U2 Rw B2 Rw2/(25,18)
Alg Rw U2 Rw2 U2 x' U r U' F2 U r' U' Lw' U2 r U2 Rw' U2 Rw'/(25,18)
Alg Rw U2 Rw2 U2 x' U' r U F2 U' r' U Lw' U2 r U2 Rw' U2 Rw'/(25,18)
Alg Rw U2 Rw U2 r' U2 Rw U2 l' U2 r U2 r' U2 (Rw' l) Rw' U2 Rw'/(27,18)
Alg r2 U2 r2 U2 r U2 r U2 r' U2 B2 U' r' U B2 U' r U'/(27,18)
Alg r2 U2 r2 U2 r U2 r U2 r' U2 B2 U r' U' B2 U r U/(27,18)
Alg Rw' E Uw b' Uw r Uw r' Uw' b Uw r' Uw' r Uw r Uw E' Rw/(19,19)
Alg Rw' E Uw' r' Uw Lw' Uw' Fw Rw' Fw' r' Fw r Rw Fw' Uw Lw E' Rw/(19,19)
Alg Rw' D' u' Uw' Rw f Rw' Uw Rw f' Rw' f' Uw' f Uw f u D Rw/(19,19)
Alg Rw' E R Uw' Fw' u' Fw Uw2 Bw' Uw Fw' u' Fw U' Bw Uw' R' E' Rw/(20,19)
Alg Dw' F' z Rw2 B2 x' l' Dw Uw Rw2 Dw' l' Dw Rw2 Dw' l u' U Rw2 D Rw z'/(24,19)
Alg Rw' E r' f' Dw' f' Dw Fw Lw' f Fw Dw' f' Dw Fw' Lw F' r E' Rw/(20,20)
Alg Uw Fw' D r' Dw b Dw b' Dw' r Dw b' Dw' b Dw b Dw2 D' Fw Uw'/(21,20)
Alg Fw' U' l u2 Lw Bw' Lw' u Lw Uw Bw u' Bw' Uw' Bw Lw' u l' U Fw/(21,20)
Alg Rw U L F' U l' Dw Uw Rw2 Dw' l' Dw Rw2 Dw' l Uw' U' F L' U' Rw'/(23,20)
Alg Bw' R' u Rw' r' Uw Bw' Uw' r Uw Rw Bw r' Bw' Rw' Bw Uw' Rw u' R Bw/(21,21)
Alg Bw' R' u Rw' l' Dw Bw' Dw' l Dw Lw Bw l' Bw' Lw' Bw Dw' Rw u' R Bw/(21,21)
Alg z' Fw' Uw r' Uw' Fw Uw r' Fw' U L u l Uw' r' Uw l' u' L' U' Fw r2 Uw' z/(23,22)
Alg z' Fw' Uw r' Uw' Fw Uw r' Fw' f' U L U' f r' f' U L' U' f Fw r2 Uw' z/(23,22)
Alg z' Fw' Uw r' Uw' Fw Uw r' Fw' f' L Fw R' F' r' F R Fw' L' f Fw r2 Uw' z/(23,22)


One Dedge Flip + PLL Parity (Double Parity)

Alg_1 r U2 r' E2 F2 l F2 l' F2 r F2 r' D2 l y2/(21,14)
Alg_2(v1) r2 B2 r' U2 r' U2 B2 r' B2 r B2 r' B2 r2 B2/(25,15)
Alg_2(v2) r2 F2 r U2 r U2 F2 r F2 r' F2 r F2 r2 F2/(25,15)
Alg_3 Rw2 B2 U2 l r2 U2 r' U2 r U2 F2 r F2 l' B2 Rw2/(26,16)
Alg_4 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 U2 r' U2 r/(26,17)
Alg_5 r' U2 r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2/(26,17)
Alg_6 (F d' S r F Rw2 f2 u' r2 Uw2 S') r (S Uw2 r2 u f2 Rw2 F' r' S' d F')/(31,23)
Alg_7 (B Lw2 U' L' U r u2 b' r2 Bw2 E) r (E' Bw2 r2 b u2 r' U' L U Lw2 B')/(31,23)
Alg_8 r' U' R' U' r' U R U r U2 r' U' R' U' r2 U R U r U2 r U2 r' U2/(29,24)


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

Alg (r2 F Rw' F') U2 r U2 r U2 r U2 r2 (F Rw F' r2)/(23,16)
Alg (Rw' z' L' U r F) U2 r U2 r U2 r U2 r2 (F' r' U' L z Rw)/(23,18)
Alg (x' r2 U' Rw U) M' U2 r' U2 r' U2 r' U2 l r (U' Rw' U r2 x)/(24,18)
Alg R B r U2 r' E2 F2 l F2 l' F2 r F2 r' D2 l y2 B' R'/(25,18)
Alg (R B r2 B2) r' U2 r' U2 B2 r' B2 r B2 r' (B2 r2 B R')/(27,18)
Alg Rw U2 x U' Rw R U' r U2 r U2 r U2 r2 U' Rw' R' U x' U2 Rw'/(25,19)
Alg Rw2 U x' U2 r U2 r' F2 l F2 U' L' U' l U L U l' x U' Rw2/(25,19)
Alg r2 F' R' F' U2 l F2 D2 r D2 l' F' r' F' U2 F Rw F r2/(26,19)
Alg M' U F2 r' B2 r B2 r F2 r' U' r' F R' F r F' R F' l/(24,20)
Alg (Rw2 U x') U2 r U2 l' U2 l U2 M U' L' U' l U L U l' (x U' Rw2)/(26,20)
Alg Rw2 F L F U2 l F2 D2 r D2 l' F r' F U2 F' r L' F' Rw2/(27,20)
Alg (z' Rw' U' Rw' l' U L') U r U2 r U2 r U2 r2 U (L U' l Rw U Rw z)/(25,21)
Alg (Rw2 F' L F r f2 u' r2 Uw2 S') r (S Uw2 r2 u f2 r' F' L' F Rw2)/(29,21)
Alg (x' Lw2 F' L2 F r f2 u' r2 Uw2 S') r (S Uw2 r2 u f2 r' F' L2 F Lw2 x)/(31,21)


3 Flips

OLL Parity

Alg Rw U Rw U2 Rw U2 Rw U Rw' U2 r' U2 Rw' U Rw U2 Rw U2 Rw U Rw/(27,21)

OLL Parity + PLL Parity (Double Parity)

Alg Rw' U2 r U2 Rw' x' U2 r' U' R' U' Rw' U2 Rw U R U' Rw R U2 x/(24,19)



Other ELL Cases (WCA Notation)

4-Cycles in Two Opposite Edges

Checkerboard pattern (Adj. 2-swap 2-Swap + Two Flip)

Alg_1 (f2 u' r2 Uw2 S') r (S Uw2 r2 u f2)/(17,11)
Alg_2 f2 M2 f2 l2 U2 r S2 r' S2 r' U2 r2/(21,12)
Alg_2 r S' L2 S r U2 r U2 r S' L2 S r/(17,13)
Alg_2 r S R2 S' r U2 r U2 r S R2 S' r/(17,13)
Alg_3(v1) r U2 r2 U2 r' U2 r U2 r' U2 r2 U2 r/(21,13)
Alg_3(v2) l' U2 l2 U2 l U2 l' U2 l U2 l2 U2 l'/(21,13)
Alg_3(v3) r' U2 r2 U2 r U2 r' U2 r U2 r2 U2 r'/(21,13)
Alg_3(v4) l U2 l2 U2 l' U2 l U2 l' U2 l2 U2 l/(21,13)


Bowtie/Hourglass (Adj. 2-swap + Pll Parity)

Alg_1 r2 U2 l' U2 l F2 U2 r2 U2 r F2 r' F2 r' F2/(25,15)
Alg_2(v1) r' U2 r U2 l' U2 r U2 l F2 r' F2 r F2 r2 F2/(25,16)
Alg_2(v2) l U2 l' U2 r U2 l' U2 r' F2 l F2 l' F2 l2 F2/(25,16)
Alg_3 (B' R Uw' u' r' f2 Rw2 E) f (E' Rw2 f2 r u Uw R' B)/(21,17)
Alg_4 r U2 l' U2 r U2 r U2 r' U2 r U2 r2 U2 M U2 r'/(26,17)
Alg_5 r U2 M' U2 r2 U2 r' U2 r U2 r' U2 r' U2 l U2 r'/(26,17)


4-Cycles in Two Adjacent Edges

Checkboard Pattern

Alg (R2 Uw' u' r' f2 Rw2 E) f (E' Rw2 f2 r u Uw R2)/(21,15)
Alg (Rw' F2 M2 F) l' U2 l' U2 l2 U2 l' U2 (F' M2 F2 Rw)/(25,16)
Alg (Rw U2 M2 U') M U2 r' U2 r2 U2 r' U2 l' (U M2 U2 Rw')/(26,17)


Bowtie/Hourglass

Alg (L' Fw' f' l' u2 Lw2 S') u (S Lw2 u2 l f Fw L)/(19,15)
Alg (Lw' U2 M2 U' x) r' U2 r' U2 r2 U2 r' U2 (x' U M2 U2 Lw)/(25,16)
Alg R B r' U2 r2 U2 r U2 r' U2 r U2 r2 U2 r' B' R'/(25,17)
Alg (l' U2 M2 U) M' U2 r U2 r2 U2 r U2 l (U' M2 U2 l)/(26,17)


2-Cycles in Two Opposite Edges

Adjacent 2-swap

Alg_1(v1) r U2 r U2 x U2 r U2 l' x' U2 l U2 r2/(19,12)
Alg_1(v2) l' U2 l' U2 x U2 l' U2 r x' U2 r' U2 l2/(19,12)
Alg_1(v3) r2 D2 r' D2 l D2 l' D2 B2 l' B2 r'/(19,12)
Alg_2(v1) r U2 r U2 M' U2 r U2 r' U2 l U2 r2/(20,13)
Alg_2(v2) l' U2 l' U2 M' U2 l' U2 l U2 r' U2 l2/(20,13)
Alg Rw' U2 r' U2 (Rw' l) U2 r' U2 r U2 l' U2 Rw2/(20,13)
Alg z' Lw U' r Uw Lw' Uw' r Fw r Fw' r2 Uw Lw u' Lw' z/(16,15)
Alg l2 D' f' D r D2 r' D' f D' l' F2 r' F2 l'/(19,15)
Alg b u Bw' Rw' f' Rw Bw Rw' f2 Uw' f' Uw f' Rw u' b'/(17,16)
Alg Rw U2 r U2 Rw U2 Rw2 F r F' Rw2 U2 Rw2 F r' F'/(23,16)
Alg Fw' Uw r' Uw' Fw Uw r' Fw' f' L f r' f' L' f Fw r2 Uw'/(19,18)


Opposite/diagonal 2-swap

Alg_1(v1) l' S2 U2 l U2 l' U2 r U2 r' F2 l B2 r z2/(21,14)
Alg_1(v2) r S2 U2 r' U2 r U2 l' U2 l F2 r' B2 l' z2/(21,14)
Alg r2 B2 U2 r' U2 r2 B2 r' U2 r2 U2 B2 r' B2/(25,14)
Alg r2 B2 U2 r U2 r2 B2 r' U2 r2 U2 B2 r' B2/(25,14)
Alg Uw Lw' Uw' l' Uw Lw2 Dw' Lw Uw' l' Uw L' Dw Lw' Uw'/(16,15)
Alg Uw Lw' Uw' l' Uw Lw Fw' Lw2 Uw' l' Uw Lw' L' Fw Uw' /(16,15)
Alg Uw Lw' Bw' r' Bw Lw2 Bw' Rw Fw' r' Fw R' Bw Lw' Uw'/(16,15)
Alg Fw' L2 Uw b' Uw' Lw2 Uw L' Uw b' Uw' Lw Uw' l2 Fw/(18,15)
Alg Fw Uw' B2 Lw u' Lw' Bw2 Rw2 d' Rw' d Rw' b2 Uw Fw'/(19,15)
Alg Uw Rw' F2 Rw' f Rw Bw2 Lw2 u Lw u' Rw u2 Rw Uw' y2/(19,15)
Alg Rw2 F2 U2 r U2 x U2 Rw2 U2 r' U2 l r U2 l' U2 x'/(25,15)
Alg Fw' L U' f Uw2 Lw2 Dw r Dw' Lw2 Uw f' Uw U L' Fw/(19,16)
Alg_2(v1) l' U2 l U2 l U2 r' U2 l U2 l' U2 F2 l2 F2 r/(25,16)
Alg_2(v2) r U2 r' U2 r' U2 l U2 r' U2 r U2 F2 r2 F2 l'/(25,16)
Alg r F2 D2 l D2 F r2 U2 r2 F r' F' r2 U2 r2 F/(25,16)
Alg r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 U2 r'/(26,17)
Alg Uw' Lw' Fw R' Fw' d' Fw R Fw' Lw2 Fw2 Rw' u' Rw Fw2 Lw' b Uw/(21,18)
Alg x' l' Uw Rw' Uw' l Uw Fw Uw B' l2 B r' B' l2 B Uw' Fw' Rw Uw' x/(21,19)


2-Cycles in Two Adjacent Edges

Case 1 (Close Adjacent "unoriented")

Alg l2 r' U B2 U' r U B2 Lw U2 r U2 Lw' U' l2/(21,15)
Alg x' l2 F U2 l U2 r' U2 r U2 F2 r F2 l' F' l2 x/(23,15)
Alg r U2 r U2 F' L F' r F L' F l' U2 l U2 r2/(21,16)
Alg x' l2 F U2 l U2 r' U2 r U2 (l' Rw) U2 r U2 Rw' F' l2 x/(24,16)
Alg z' F Lw2 b Uw' Rw Uw b' Uw' b' Rw' b Rw b Rw' Uw Lw2 F' z/(19,17)
Alg z' F Rw2 Uw' Bw r' Bw' Uw Bw Rw' Uw' r' Uw Rw r Bw' Rw2 F' z/(19,17)
Alg z' F Lw2 z Lw' Bw r' Bw' Lw Bw Rw' Dw' r' Dw Rw r Fw' Lw2 F' z/(19,17)
Alg Rw b' Uw2 Rw' Uw' f Uw Rw Uw' f Rw' f Rw f2 Uw' b Rw'/(19,17)
Alg r2 U2 l U2 r' U2 r U2 F' L F' r F L' F l' r2/(23,17)


Case 2 (Far Adjacent "unoriented")

Alg l2 r' U' B2 U r U' B2 Lw U2 r U2 Lw' U l2/(21,15)
Alg x Rw2 F' U2 l U2 r' U2 r U2 F2 r F2 l' F Rw2 x'/(23,15)
Alg r U2 r U2 F R' F r F' R F' l' U2 l U2 r2/(21,16)
Alg x Rw2 F' U2 l U2 r' U2 r U2 (l' Rw) U2 r U2 Rw' F Rw2 x'/(25,16)
Alg Lw l b Uw' Rw Uw b' Uw' b' Rw' b Rw b Rw' Uw Lw' l'/(17,17)
Alg Lw l z Lw' Bw r' Bw' Lw Bw Rw' Dw' r' Dw Rw r Fw' Lw' l'/(17,17)
Alg Rw l Uw' Bw r' Bw' Uw Bw Rw' Uw' r' Uw Rw r Bw' l' Rw'/(17,17)
Alg Rw' l' x' U2 l U2 r' U2 r U2 x U R U r U' R' U' Rw/(21,17)
Alg r2 U2 l U2 r' U2 r U2 F R' F r F' R F' l' r2/(23,17)
Alg z Dw' (M R') D Lw' Uw' r' Uw Lw Uw' Lw2' Bw' r' Bw Rw' R' u y' (M' R) Uw x2 z'/(19,18)


Case 3 ("Oriented" Case)

Alg Lw2 b Uw' Rw Uw b' Uw' b' Rw' b Rw b Rw' Uw Lw2/(17,15)
Alg Rw2 Uw' Bw r' Bw' Uw Bw Rw' Uw' r' Uw Rw r Bw' Rw2/(17,15)
Alg Lw2 z Lw' Bw r' Bw' Lw Bw Rw' Dw' r' Dw Rw r Fw' Lw2/(17,15)
Alg Rw2 f Uw' Bw Dw r' Dw' r' Bw' r Bw r Bw' Uw Rw2/(17,15)
Alg z' Fw' l' Uw' Bw2 Uw' r Uw Bw2 Uw2 r Uw r' Uw2 l Fw z/(19,15)
Alg Uw Rw' Uw' r' Uw Rw Bw' Rw' r' Uw' r' Uw R r' Bw Uw'/(16,16)
Alg Bw L' Dw' l Dw Lw Bw' Lw L Dw' l Dw Lw2 Fw u' Fw'/(17,16)
Alg Uw' Fw l' Fw' Uw Fw l' Uw2 R Uw l' Uw' R' Uw2 l2 Fw'/(19,16)
Alg r' Uw Lw Uw' r2 Fw2 Lw' Fw r' Fw' Lw Fw2 r' Uw Lw' Uw'/(19,16)
Alg z' Lw Uw' r Uw Lw' Uw' r Fw2 L' Fw' r Fw L Fw2 r2 Uw z/(19,16)
Alg Bw' Uw2 r' Dw l' Dw' r Uw' Lw2 Uw' r' Uw Lw2 Uw' r Bw/(19,16)
Alg z' Uw Lw' Uw2 r Uw' l Uw r' Uw2 L Uw' Bw2 Uw l Uw' Bw2 z/(20,16)
Alg x l2 U2 r U2 r' F2 l F2 U L U l U' L' U' l x'/(21,16)
Alg l2 r' U F2 U' l U F2 Rw U2 l U2 Rw' U' l r/(21,16)
Alg R B r U2 r U2 x U2 r U2 l' x' U2 l U2 r2 B' R'/(23,16)
Alg x l' U L U l' U' L' U' x U2 l' U2 l U2 r' U2 l r x2/(21,17)
Alg x r U' R' U' r U R U M' U2 r U2 r' U2 l U2 r2 x'/(22,17)
Alg x' f' Uw Rw' Fw' r' Fw Rw Uw' f r' Uw Rw' Fw' r' Fw Rw Uw' r2 x/(19,18)
Alg r' U' F2 U r' U' F2 r' F' R' F' r' F R F r U r2/(21,18)
Alg z' r' Uw Bw Uw' Bw L Bw r Bw' L' Bw' Uw Bw' Lw Uw' r Uw Lw' Uw' z/(19,19)
Alg l' Uw Rw' Uw' l Uw Fw Uw B' Rw2 F r' F' Rw2 B Uw' Fw' Rw Uw'/(21,19)



Summary of 2-Cycle and 4-Cycle ELL Case Movecounts

  • Since there is disagreement among cubers about whether lowest q moves or lowest btm moves defines the "optimal algorithm", the table below categorizes all 2-cycle and 4-cycle ELL cases on the 4x4x4 by the average of q moves and btm.
  • Algorithms optimal in btm need not be the algorithm with the lowest average of q and btm, and algorithms optimal in q turns need not be the algorithm with the lowest average of q and btm. Therefore, the average for a given case might be from an algorithm optimal in q moves, optimal in btm 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.
  • The table shows that the "worst case" is Adjacent Double Parity, and that the "easiest case" is the Checkerboard Pattern (in two opposite edges).


Arranged by Rank (Worst Last Layer Cases Are First)
Last Layer Case Minimum q Turns Minimum btm Optimal Average Rank
Adjacent Double Parity 23 16 19.5 1
Bowtie/Hourglass 21 15 19 2
Single Dedge Flip 19 15 18.5 3
Adjacent Checkerboard 21 15 18 4
Adjacent 2-Cycle Case 1 (Close "Unoriented" Case) 19 15 18 4
Double Parity 21 14 17.5 5
Adjacent Bowtie/Hourglass 19 15 17 6
Adjacent 2-Cycle Case 2 (Far Adjacent "Unoriented" Case) 17 15 17 6
Adjacent 2-Cycle Case 3 (“Oriented" Case) 16 15 16 7
Opposite/diagonal 2-swap 16 14 15.5 8
Adjacent 2-swap 16 12 15.5 8
Checkerboard Pattern 17 11 14 9


OLL Parity Algorithms Which Don't Preserve F3L (WCA Notation)

  • One advantage of having access to algorithms which don't preserve the first three layers of the 4x4x4 is that the optimal algorithms require less 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 and 19 block quarter turns, some algorithms in this section are 13 block half turns and some are just 15 block quarter turns.
  • All of the 13 btm algorithms in this section have many variations, but yet all variations are the same as the few algorithms listed below. The following "translations" of the 13 btm algorithms work on all cube sizes (the first version, at least, for those with more than one version). For a list of more variations of the 13 btm algorithms (but definitely not all possible variations), see http://www.speedsolving.com/forum/showthread.php?26564-4x4x4-edge-parity-is-there-a-shorter-alg-that-doesn-t-preserve-corners&p=531938&viewfull=1#post531938
  • A few of the algorithms are in SiGN Notation: they are labeled.

OLL Parity

Complete 3x3x3 Scrambles

Alg Rw B U2 B' r B2 l B2 r B' D2 B Rw/(17,13)
Alg r U F2 U' l F2 2R U2 r U B2 U' r/(17,13)(SiGN)
Alg r U' F2 U l F2 2R U2 r U' B2 U r/(17,13)(SiGN)
Alg Uw M R D R' Fw f E' l' E Fw b E F Lw/(15,15)
Alg Rw' S' L2 S Rw' U2 Rw' U2 Rw' F' U2 F U R U' Rw'/(20,16)
Alg Rw' S R2 S' Rw' U2 Rw' U2 Rw' F U2 F' U' L' U Rw'/(20,16)
Alg r U2 r' U2 r D2 r D2 r' F L' F' U' r' D2 r/(21,16)(SiGN)


More than 1 F3L Slot Destroyed (Not Petrus)

Alg r' U2 F U2 F' U2 r' U2 r' U2 r' F U2 F' r'/(21,15)(SiGN)
Alg Rw U D' R2 U' R2 D x' U2 Lw U2 Lw U2 Lw F' U2 F Lw/(22,16)
Alg Rw F U2 F' Rw U2 Rw U2 Rw R U2 F U2 F' U2 r/(22,16)
Alg Rw' U R U2 R' U' Rw' U2 Rw' U2 Rw' F U' B' U' S Dw' z'/(20,17)
Alg Lw U' L' R U2 L U Lw U2 Lw U2 Lw U' R' U2 R U r x'/(22,18)
Alg Lw U' L' R' U2 L U Lw U2 Lw U2 Lw U' R U2 R' U Lw R/(23,19)


FR F3L Slot Destroyers

Alg r' U2 F' U2 F U2 r' U2 r' U2 r' F' U2 F r'/(21,15)
Alg l U2 F' U2 F U2 l U2 l U2 l F' U2 F l/(21,15)
Alg r' U R U2 R' U' r' U2 r' U2 r' U' R U2 R' U r'/(21,17)
Alg l' U' R U2 R' U l' U2 l' U2 l' U R U2 R' U' l'/(21,17)
Alg l U2 l' F U R U' Lw' D2 r D2 r U2 Rw' U2 Lw F/(22,17)
Alg Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F r' U2 r F/(22,17)
Alg Rw U R' U' Rw2 R U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw'/(24,18)
Alg Rw U R' U' Rw2 U2 Rw2 R U Rw U2 Rw' U2 Rw U' Rw2 U2 r'/(24,18)
Alg Rw U2 R U R' U2 R U r U2 Rw U2 Rw U R U2 R' U' Rw/(24,19)


Just Corners are Permuted

Alg Rw U' R' U2 R U r U2 r U2 r U R' U2 R U' Rw R2/(22,18)
Alg Rw U' R' U2 R U r U2 Rw R U2 r U R U2 R' U' Rw/(22,18)
Alg Rw U' R U2 R' U r U2 Rw R U2 r U R' U2 R U' Rw/(22,18)
Alg Rw R U' R2 U2 R2 U r U2 Rw U2 r U R U2 R' U' Rw/(24,18)
Alg Rw U' R' U2 R U Rw R2 U2 r U2 Rw U R' U2 R U' Rw R2/(24,19)
Alg Rw U' R' U2 R U Rw U2 r U2 Rw R2 U R' U2 R U' Rw R2/(24,19)
Alg Rw U' R' U2 R U' R U2 r U2 Rw U2 r U R U2 R' U' Rw/(24,19)


Petrus Parities (Destroy 2 Adjacent Faces)

Alg_1 r B' U2 B r B2 l B2 r B D2 B' r/(17,13)(SiGN)
Alg_1(v2) x' r' U F2 U' l' U2 r' U2 r' F' U2 F r'/(17,13)(SiGN)
Alg_1(v3) x' r' U F2 U' l' U2 r' U2 l' U' B2 U r' x'/(17,13)(SiGN)
Alg_1(v4) x' r' U F2 U' l' U2 r' 3d2 r' U' F2 U l' x/(17,13)(SiGN)
Alg Rw' S' L2 S Rw' U2 Rw' U2 Rw' F U2 F' U' L' U Rw'/(20,16)
Alg Rw' S R2 S' Rw' U2 Rw' U2 Rw' F' U2 F U R U' Rw'/(20,16)
Alg Rw U2 Rw' U2 Rw D2 r D2 l' U L' U' B' l' U2 l x'/(21,16)
Alg Rw' U R' U2 R U' Rw' U2 Rw' U2 Rw' U' R U2 R' U Rw'/(21,17)
Alg Rw U' R' U2 R U r U2 r U2 r U R' U2 R U' Rw/(21,17)
Alg Rw U' R' U2 R U Rw U2 Rw U2 r U R U2 R' U' Rw/(21,17)
Alg Rw' U' L' U2 L U Rw' U2 Rw' U2 Rw' U L' U2 L U' Rw'/(21,17)
Alg Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F Rw' U2 Rw F/(22,17)
Alg Rw U' R2 U2 R2 U r U2 Rw U2 r U R U2 R' U' Rw/(23,17)
Alg Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw'/(25,17)
Alg Rw U2 Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U2 r2 U R2 U' Rw'/(25,17)
Alg Rw U' R2 U2 R2 U r U2 r U2 r U R2 U2 R2 U' Rw/(25,17)
Alg Rw' U R U2 R' U' r' U2 Rw' U2 Rw' F' U' R' U' R F Rw'/(21,18)
Alg Rw' U R U2 R' U' Rw' U2 Rw' U2 Rw' R' U' R U2 R' U Rw'/(22,18)
Alg Rw U' R' U2 R U Rw U2 r U2 Rw R2 U R' U2 R U' Rw/(23,18)
Alg Rw U' R U2 R' U Rw U2 r U2 Rw R2 U R U2 R' U' Rw/(23,18)
Alg Rw U' R U2 R' U Rw U2 Rw R2 U2 Rw U R' U2 R U' Rw/(23,18)
Alg Rw U R U' Rw2 R U2 Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw'/(24,18)
Alg Rw' U R U2 R' U' Rw' U2 Rw' U2 Rw' R2 U' R2 U2 R2 U Rw'/(24,18)
Alg Rw U R' U' Rw2 U2 Rw2 R U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw'/(24,18)
Alg Rw U R' U Rw U2 r U2 r U' Rw' U R U R' U2 Rw U' Rw/(22,19)
Alg Rw U R' U r U2 r U2 Rw U' Rw' U R U R' U2 Rw U' Rw/(22,19)
Alg Rw U' Rw U2 R' U R U Rw' U' r U2 Rw U2 Rw U R' U Rw/(22,19)
Alg Rw U' R' U2 R U Rw R U2 r U2 Rw R U R' U2 R U' Rw/(23,19)
Alg Rw U' Rw U2 R' U R U Rw' U' Rw U2 r U2 Rw U R2 U Rw/(23,19)
Alg Rw U' R' U2 R U' R' U2 r U2 Rw U2 r U R' U2 R U' Rw/(24,19)
Alg Rw U' R U2 R' U' R' U2 r U2 Rw U2 r U R U2 R' U' Rw/(24,19)
Alg Rw U' R' U2 R U r U2 Rw U2 Rw U R U2 R' U R' U2 Rw/(24,19)
Alg Rw U R' U2 R' U2 R U Rw R U2 Rw U2 Rw U R U2 R' U' Rw/(25,20)


OLL Parity + PLL Parity (Double Parity)

Complete 3x3x3 Scrambles

Alg_1 r F2 U2 l F U' R U' r2 B2 r' B2 r2/(19,13)(SiGN)
Alg_2 r F2 U2 l F' U L' U r2 B2 r' B2 r2/(19,13)(SiGN)
Alg r F2 U2 l F U' (L' R) U' r2 B2 r' B2 r2/(19,13)(SiGN)
Alg r F2 U2 l F' U (L' R) U r2 B2 r' B2 r2/(19,13)(SiGN)
reParity r U2 l' U2 x' (r' U2 l U2)2 l'/(19,13)
Alg Uw R' b E' d' L' b L u' b2 Uw M' B' Uw/(15,14)
Alg Uw S L B' R u d f' u d L B' R S Uw/(15,15)
Alg Uw S L' F R' u d f' u d L' F R' S Uw/(15,15)
Alg Rw' U2 r U2 Rw' x' U2 r' U' R' U' Rw' U2 Rw U R U' Rw x/(21,17)


Affect M Layer Only

Alg r U2 r' U2 r' D2 r D2 r' B2 r B2 r'/(19,13)
Alg r U2 r' U2 M' l' U2 r U2 l' U2 r U2 l'/(20,14)
reParity(v2) r U2 l' U2 l r2 U2 l U2 r' U2 l U2 l r2/(23,15)


FR F3L Slot Destroyers

Alg Rw U2 Rw U2 Rw U Rw U2 R' U Rw U Rw2 R U Rw U' Rw/(21,18)


Petrus Parities (Destroy 2 Adjacent Faces)

Alg_1(v2) l' U2 F2 l' U' F R' F l2 U2 r U2 l2 U' x'/(20,14)(SiGN)
Alg_1(v3) l' U2 F2 l' U' F 3l' U l2 x' U2 r U2 l2 U' x'/(20,14)(SiGN)
Alg_2(v2) r U2 F2 r U' F R' F r2 U2 l' U2 r2 U' x'/(20,14)(SiGN)
Alg_2(v3) r U2 F2 r U' F 3l' U r2 x' U2 l' U2 r2 U' x'/(20,14)(SiGN)
Alg Rw' U Rw' U' Rw2 R' U' Rw' U' R U2 Rw' U' Rw' U2 Rw' U2 Rw'/(21,18)
Alg Rw U2 Rw U2 Rw' U' R' Rw2 U2 r U' R' U' Rw U2 r2 U Rw2/(24,18)
Pochmann Rw' U R U (Rw' U2)3 Rw2 U R' U' Rw2 U' R' U Rw'/(24,19)



OLL Parity Algorithms Which Don't Preserve F3L or the Colors of the Centers (WCA Notation)

  • These algorithms are least practical when it comes to use of parity algorithms, and therefore they are mentioned here for theoretical purposes only. They have fewer moves than any other OLL Parity algorithm forms.

OLL Parity

Alg Rw2 U r' U' B2 U r U r' B2 Rw2/(15,11)
Alg Rw2 U' r' U B2 U' r U' r' B2 Rw2/(15,11)
Alg Rw2 U f U' B2 U f' U l B2 Rw2/(15,11)


OLL Parity + PLL Parity (Double Parity)

Alg Uw L U B U' d Fw2 u' Fw2 L' Uw2 y/(14,11)
Alg Uw L U B d Fw2 u' Fw2 U' L' Uw2 y/(14,11)
Alg Rw2 U r' U' B2 U r U r B2 Rw2/(15,11)
Alg Rw2 U' r' U B2 U' r U' r B2 Rw2/(15,11)
Alg Rw2 U f U' B2 U f' U l' B2 Rw2/(15,11)
Alg Rw U2 Rw U2 Rw U2 Rw2 U' R U Rw2 U R U' Rw2/(21,15)
Alg Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U' Rw2 U' R2 U' Rw'/(21,15)


Either

  • The following list of algorithms can be considered either OLL Parity or Double Parity. Just exchange the last move Rw' with Lw', Rw with Lw, etc.
Alg Rw' U' R' U' Rw F' R' F' Rw'/(9,9)
Alg Rw' U' R U' Rw F' R F' Rw'/(9,9)
Alg Rw U' R U' r' D' R D' Rw/(9,9)
Alg Rw' U R' U Rw U F2 U Rw'/(10,9)
Alg Rw U' R U' Rw' D' R2 D' Lw/(10,9)
Alg Rw' U' F2 U' r B' R B' Lw'/(10,9)
Alg Rw U' L U' F2 Lw U' R U' Rw'/(11,10)
Alg Rw U' L' U' F2 l U L U Rw'/(11,10)
Alg Rw' U' L' U' Rw F2 U L U Rw/(11,10)
Alg Rw' U' L' U' B2 Rw' B R B Rw/(11,10)
Alg Rw' U R2 U Rw F2 U L' U Rw/(12,10)
Alg Rw' U F' U l U2 F' U2 F Rw'/(12,10)
Alg Rw U' R' U' Rw' D' L D L' D' Lw/(11,11)
Alg Uw F R B' U f' E U M' U Fw/(11,11)
Alg Rw U R' U R r' U2 B' L' B' Rw'/(12,11)
Alg l' B' u B' D2 B u' B' l D2 l'/(13,11)
Alg Rw F U' R U Rw' F U R F' U Rw/(12,12)
Alg Rw' U F' L F Rw F R B' E y B Rw'/(12,12)
Alg Uw S B D B' R f' R M E R Fw/(12,12)
Alg Rw' U' F R' F' Rw F' U' F2 R U Rw'/(13,12)


  • Although the following list of algorithms is not "OLL Parity", where dedge preservation is important, they are the briefest 2-cycle algorithms yet to be found. They do not preserve F3L or the centers.
Alg r' U' B2 U r U r'/(8,7)
Alg r' U B2 U' r U' r'/(8,7)
Alg r' U L2 U' r D' r'/(8,7)
Alg r' U' R2 U r D r'/(8,7)


Cage Method/Sandwich Method Algorithms (WCA Notation)

  • These algorithms are for fixing the wing edges, but they do not preserve the centers.

Single Dedge Flip

Alg Rw2 U r' U' B2 U r U r' U2 B2 Rw2/(17,12)
Alg Rw2 U' r' U B2 U' r U' r' U2 B2 Rw2/(17,12)
Alg Rw2 (Dw u') l U' R2 U l' U y r U2 F2 Rw2/(17,12)
Alg Lw2 (Dw' u) r' U L2 U' r D' r' U D L2 y' Lw2/(17,13)
Alg Rw2 U' r' U B2 U' r U B2 D2 l D2 Rw2/(19,13)
Alg Rw2 (Dw u') l U' R2 U l' U' D2 y l D2 F2 Rw2/(19,13)
Alg Rw2 U2 r U2 l' U2 l U2 r' F2 r F2 Rw2/(21,13)
Alg Rw2 U2 r U2 l' U2 l U2 r' B2 l B2 Rw2/(21,13)
Alg l U2 l' U2 Lw2 F2 r' F2 Rw2 U2 l' U2 r x2/(21,13)
Alg r2 Uw Bw' Uw' r Uw Rw Bw r' Bw' Rw' Bw Uw' r/(15,14)
Alg x r R' U' Rw U' r U Rw' r' U R U' r U r' x'/(15,15)


OLL Parity + PLL Parity (Double Parity)

  • Most of these algorithms are exactly the same as the single dedge flip ones except that the extra quarter turn is inverted.
Alg Rw' F' u F' D2 F u' F' l D2 F2 Rw/(15,12)
Alg Rw2 U r' U' B2 U r U r U2 B2 Rw2/(17,12)
Alg Rw2 U' r' U B2 U' r U' r U2 B2 Rw2/(17,12)
Alg Rw2 (Dw u') l U' R2 U l' U y r' U2 F2 Rw2/(17,12)
Alg Lw2 (Dw' u) r' U L2 U' r D' r U D L2 y' Lw2/(17,13)
Alg Rw2 U' r' U B2 U' r U B2 D2 l' D2 Rw2/(19,13)
Alg Rw2 (Dw u') l U' R2 U l' U' D2 y l' D2 F2 Rw2/(19,13)
Alg Rw2 U2 r U2 l' U2 l U2 r' F2 r' F2 Rw2/(21,13)


Adjacent Double Parity

Alg [r2 U Rw' U R U: r'/(15,13)]
Alg [r2 F' U2 F r U2: r'/(19,13)]
Alg Rw F' Rw' F' U2 F r F' U2 r F R F Rw'/(16,14)
Alg l2 U' Lw U r2 U2 r U2 r U2 r U Lw' U l2/(21,15)


In Two Adjacent Edges (in the M ring)

Adjacent 2-Swap

Alg r' U r U F2 U' r' U F2 U2/(13,10)
Alg r' U' r U' F2 U r' U' F2 U2/(13,10)
Alg y' U2 R2 U l U' R2 U l' U y r/(13,10)
Alg F2 U2 l U2 r' U2 r U2 l' F2 r /(17,11)
Alg r U' Rw U' R' U r U' R U r' Rw' U/(13,13)


Opposite/Diagonal 2-Swap (The majority are nearly the same as the one dedge flip algorithms.)

Alg Rw U r' U' B2 U r U r' U2 B2 Rw'/(15,12)
Alg Rw U' r' U B2 U' r U' r' U2 B2 Rw'/(15,12)
Alg Rw (Dw u') l U' R2 U l' U y r U2 F2 Rw'/(17,12)
Alg x' U Rw' U r' U' Rw r U' R' U r' U' R x/(13,13)
Alg x' R U Rw' U r' U' Rw r U' R' U r' U' x/(13,13)
Alg r' Uw Bw' Uw' r Uw Rw Bw r' Bw' Rw' Bw Uw'/(13,13)
Alg Lw' (Dw' u) r' U L2 U' r D' r' U D L2 y' Lw/(15,13)
Alg Rw U' r' U B2 U' r U B2 D2 l D2 Rw'/(17,13)
Alg Rw (Dw u') l U' R2 U l' U' D2 y l D2 F2 Rw'/(17,13)
Alg Rw U2 r U2 l' U2 l U2 r' F2 r F2 Rw'/(19,13)
Alg Rw U2 r U2 l' U2 l U2 r' B2 l B2 Rw'/(21,13)


In Two Opposite Edges (in the M Ring)

Adjacent 2-Swap

Alg y Lw' Uw' r Fw r Fw' r2 Uw Lw Uw' r Dw/(13,12)


Opposite/Diagonal 2-Swap (These are the same as the single dedge flips but without first and last moves.)

Alg U r' U' B2 U r U r' U2 B2/(13,10)
Alg U' r' U B2 U' r U' r' U2 B2/(13,10)
Alg (Dw' u) r' U L2 U' r D' r' U D L2 y'/(13,11)
Alg U' r' U B2 U' r U B2 D2 l D2/(15,11)
Alg (Dw u') l U' R2 U l' U' D2 y l D2 F2/(15,11)
Alg U2 l U2 r' U2 r U2 l' F2 r F2/(17,11)
Alg U2 r U2 l' U2 l U2 r' F2 r F2/(17,11)
Alg U2 r U2 l' U2 l U2 r' B2 l B2/(17,11)


4-Cycles in Adjacent Edges (in the M ring)

Checkboard Pattern 4-cycle (Adj. 2-swap 2-Swap + Two Flip)

Alg Rw2 U2 r U2 Rw2/(9,5)


Bowtie/Hourglass (Adj. 2-swap + Pll Parity)

Alg r U2 r' U2 x' U2 r U2 r2 U2 r' U2 r' x/(19,12)


Supercube/Blindfold Solving Algorithms (WCA Notation)

Supercube Safe PLL Parity

In Two Opposite Edges ("Oriented)

Alg r2 U2 B2 l r U2 M' U2 r2 B2 U2 /(19,11)
Alg r2 U2 r U2 r2 U2 r2 U2 r U2 r2 U2/(22,12)
Alg r' F U' R F' U l r U' F R' U F' l' /(14,14)


In Two Adjacent Edges ("Oriented")

Alg R' F l E F2 E' l' r' E F2 E' r F' R/(16,14)
Alg R' F l E' F2 E l' r' E' F2 E r F' R/(16,14)


Supercube Safe OLL Parity

Single Dedge Flip

Alg r' U2 r' U' r U r U r' D' f2 r f2 r' D U2 r' U r U r U r/(27,23)
Alg Rw2 U2 r' U2 r' U' l r u2 l' r' U l r u2 x U2 l' U2 r U2 r' U2 Rw2 x'/(33,23)


2-Cycles in Two Opposite Edges

Adjacent 2-swap

Alg r' U' r' U r U r' U' r' U' r D b2 r' b2 r U2 D' r U'/(23,20)
Alg U2 l' U2 r U2 r' U2 x' U2 r' U2 r' U' l r u2 l' r' U l r u2 x/(29,21)


Opposite/diagonal 2-swap

Alg x' U2 r' U' r U r U r' D' f2 r f2 r' D U2 r' U r U r U x/(25,21)
Alg Rw U2 r' U2 r' U' l r u2 l' r' U l r u2 x U2 l' U2 r U2 r' U2 Rw' x'/(31,23)


2-Cycles in Two Adjacent Edges

Case 1 (Close "Unoriented" Case)

Alg r' U' r' U' r U r U r' D' f2 r f2 r' D U2 r' U r U r2/(25,21)


Case 2 (Far Adjacent "Unoriented" Case)

Alg r' U r' U r U' r U' r' D b2 r b2 r' D' U2 r' U' r U' r2/(25,21)


Case 3 ("Oriented" Case)

Alg r' U' r U' r' D b2 r b2 r' D' U2 r' U' r U' r U r' U r2/(25,21)


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.
Alg Rw U' Lw Uw2 Rw' U Rw Uw2 x Rw2 U/(13,10)
Alg Rw U Rw' U' Rw' F Rw2 U' Rw' U' Rw U Rw' F'/(15,14)
Alg Rw U Lw' U' Lw U Rw' U' Lw' U Lw Rw U' Rw' U'/(15,15)
Alg Rw' U' Lw U Lw' U' Rw U Lw U' Lw' Rw' U Rw U/(15,15)


2-Gen Algorithms (WCA Notation)

  • These algorithms are longer than optimal solutions, but they touch a smaller portion of the cube during execution.
  • The solutions listed are believed to be optimal for this intense move restriction.

One Dedge Flip

(U,r)

Alg r U2 r' U2 r U2 r U r U r U2 r2 U2 r U2 r U r' U' r2 U2/(31,22)
Alg r U2 r' U2 r U2 r U' r U' r U2 r2 U2 r U2 r U' r' U r2 U2/(31,22)
Alg r2 U r' U' r U2 r U2 r2 U2 r U' r U' r U2 r U2 r' U2 r U2/(31,22)
Alg r2 U' r' U r U2 r U2 r2 U2 r U r U r U2 r U2 r' U2 r U2/(31,22)


(U, Rw)

Alg Rw' U' Rw U' Rw U Rw2 U' Rw2 U2 Rw U2 Rw' U2 Rw U2 Rw U' Rw U Rw' U' Rw U2 Rw' U2/(34,26)
Alg Rw' U' Rw U' Rw U' Rw2 U2 Rw2 U' Rw U2 Rw' U2 Rw U2 Rw U Rw U Rw' U' Rw U2 Rw' U2/(34,26)


(U, Rw, r)

  • Although the following algorithm is not 2-gen and is one move longer than the optimal (U, Rw) solutions, it can be applied to larger cube sizes by

1) Letting Rw be the right half of the even cube and the right half + the central slice on odd cubes.

2) Applying the r' and r2 only on this slices which are to be affected.

Alg Rw U2 Rw U Rw U2 Rw U2 Rw' U2 Rw' U r' U' Rw U2 Rw U2 Rw' U2 Rw' U r2 U2 Rw' U2 Rw'/(37,27)

(Similar to other algorithms in this page, the (U, Rw) solutions cannot be applied to the 4x4x4, but the above algorithm can be applied to all big cubes.)


External Links

OLL Parity Algorithms

OLL Parity Algorithms which don't preserve F3L

PLL Parity Algorithms

SuperCube Parity Algorithms

Miscellaneous

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