4x4x4 parity algorithms
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
Alg | [r: [U2, r [r2 U2: r]] /(13,9)] |
In fact, we can do the same 4-cycle of wing edges with just a conjugate alone.
Alg | [r2 d' r2 Uw2 S': r' /(17,11)] |
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.
Contents
- 1 PLL Parity (WCA Notation)
- 2 Impure Dedge Flips (OLL Parity)(Turn the inner and outer layers simultaneously)
- 3 Impure Dedge Flips with Wide Turns/Pure Flips with Inner Layer Turns (SiGN Notation)
- 4 Pure Flips (WCA Notation)
- 5 Other ELL Cases (WCA Notation)
- 6 Summary of 2-Cycle and 4-Cycle ELL Case Movecounts
- 7 OLL Parity Algorithms Which Don't Preserve F3L (WCA Notation)
- 8 OLL Parity Algorithms Which Don't Preserve F3L or the Colors of the Centers (WCA Notation)
- 9 Cage Method/Sandwich Method Algorithms (WCA Notation)
- 10 Supercube/Blindfold Solving Algorithms (WCA Notation)
- 11 2-Gen Algorithms (WCA Notation)
- 12 External Links
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) |
- Algorithms for almost all PLL Parity cases can be found here http://www.math.leidenuniv.nl/~mfung/speedcubing/algs/4x4x4/
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).
- Purpose of the notation. Besides its great application to larger cube sizes, it allows us to write wide turn algorithms more simply and compactly.
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) |
- For algorithms for OLL + OLL Parity and OLL Parity + F3L, see http://www.math.leidenuniv.nl/~mfung/speedcubing/algs/4x4x4/
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_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 | r2 D2 r' D2 l D2 l' D2 B2 l' B2 r' /(19,12) |
Alg_2(v1) | r U2 r U2 x U2 r U2 l' x' U2 l U2 r2 /(19,12) |
Alg_2(v2) | l' U2 l' U2 x U2 l' U2 r x' U2 r' U2 l2 /(19,12) |
Alg_3(v1) | r U2 r U2 M' U2 r U2 r' U2 l U2 r2 /(20,13) |
Alg_3(v2) | l' U2 l' U2 M' U2 l' U2 l U2 r' U2 l2 /(20,13) |
Alg_4 | Rw' U2 r' U2 (Rw' l) U2 r' U2 r U2 l' U2 Rw2 /(20,13) |
Alg_5 | z' Lw U' r Uw Lw' Uw' r Fw r Fw' r2 Uw Lw u' Lw' z /(16,15) |
Alg_6 | l2 D' f' D r D2 r' D' f D' l' F2 r' F2 l' /(19,15) |
Alg_7 | b u Bw' Rw' f' Rw Bw Rw' f2 Uw' f' Uw f' Rw u' b' /(17,16) |
Alg_8 | Rw U2 r U2 Rw U2 Rw2 F r F' Rw2 U2 Rw2 F r' F' /(23,16) |
Alg_9 | 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_2 | r2 B2 U2 r' U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14) |
Alg_3 | r2 B2 U2 r U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14) |
Alg_4 | Uw Lw' Uw' l' Uw Lw2 Dw' Lw Uw' l' Uw L' Dw Lw' Uw' /(16,15) |
Alg_5 | Uw Lw' Uw' l' Uw Lw Fw' Lw2 Uw' l' Uw Lw' L' Fw Uw' /(16,15) |
Alg_6 | Uw Lw' Bw' r' Bw Lw2 Bw' Rw Fw' r' Fw R' Bw Lw' Uw' /(16,15) |
Alg_7 | Fw' L2 Uw b' Uw' Lw2 Uw L' Uw b' Uw' Lw Uw' l2 Fw /(18,15) |
Alg_8 | Fw Uw' B2 Lw u' Lw' Bw2 Rw2 d' Rw' d Rw' b2 Uw Fw' /(19,15) |
Alg_9 | Uw Rw' F2 Rw' f Rw Bw2 Lw2 u Lw u' Rw u2 Rw Uw' y2 /(19,15) |
Alg_10 | Rw2 F2 U2 r U2 x U2 Rw2 U2 r' U2 l r U2 l' U2 x' /(25,15) |
Alg_11 | Fw' L U' f Uw2 Lw2 Dw r Dw' Lw2 Uw f' Uw U L' Fw /(19,16) |
Alg_12(v1) | l' U2 l U2 l U2 r' U2 l U2 l' U2 F2 l2 F2 r /(25,16) |
Alg_12(v2) | r U2 r' U2 r' U2 l U2 r' U2 r U2 F2 r2 F2 l' /(25,16) |
Alg_13 | r F2 D2 l D2 F r2 U2 r2 F r' F' r2 U2 r2 F /(25,16) |
Alg_14 | r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 U2 r' /(26,17) |
Alg_15 | Uw' Lw' Fw R' Fw' d' Fw R Fw' Lw2 Fw2 Rw' u' Rw Fw2 Lw' b Uw /(21,18) |
Alg_16 | 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) |
- More ELL cases for the 4x4x4 cube can be found here: Kirjava's K4 ELL page
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).
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 | r U2 r' U2 r D2 r D2 r' F L' F' U' r' D2 r/(21,16)(SiGN) |
Alg | r U2 r' U2 r D2 r D2 r' F L' F' U' l' F2 l/(21,16)(SiGN) |
Alg | r U2 r' U2 r D2 r D2 l' U L' U' B' l' U2 l x'/(21,16)(SiGN) |
Alg | r U2 r' U2 r D2 r D2 l' U L' U' B' r' F2 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 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 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/(15,15) |
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) |
- To see more detailed information on how most of the optimal algorithms were found, see http://cubezzz.dyndns.org/drupal/?q=node/view/230
- To see Stefan Pochmann's explanation of his Petrus Parity Algorithm, see http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/13712
- To see a complete list of (U,Rw,R) solutions that have been found, see http://www.speedsolving.com/forum/showthread.php?30127-New-4x4-parity-algs-using-R-Rw-U&p=806608#post806608
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 mainly 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) |
Alg | Uw S B D B' R f' R M E R Fw /(12,12) |
OLL Parity + PLL Parity (Double Parity)
Alg | Uw F R B' U f' E U M' U Fw /(11,11) |
Alg | l' B' u B' D2 B u' B' l D2 l' /(13,11) |
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) |
- Although the following isn't specifically "OLL Parity", where dedge preservation is important, it is the briefest 2-cycle algorithm yet to be found. It also doesn't preserve F3L or the centers.
Alg | Lw' F' R2 F2 u' F' Lw /(9,7) |
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.)
- Algorithms in (U,Rw) which affect just the last layer or two intersecting layers can be found at http://apelgam.se/Rubik/4x4parity/
External Links
OLL Parity Algorithms
- http://www.speedsolving.com/forum/showthread.php?11311-4x4x4-OP-DP-algorithms-%28more-finger-friendly%29
- http://www.speedsolving.com/forum/showthread.php?15614-Odd-parity-Algorithms-%28specifically-single-edge-quot-flip-quot-%29
- http://www.speedsolving.com/forum/showthread.php?2727-4x4-Orientation-parity
- http://www.speedsolving.com/forum/showthread.php?12487-4x4-Parity-Algorithms
- http://www.speedsolving.com/forum/showthread.php?30127-New-4x4-parity-algs-using-R-Rw-U
- http://www.twistypuzzles.com/forum/viewtopic.php?f=8&t=8829
- http://www.stefan-pochmann.info/spocc/other_stuff/4x4_5x5_algs/?section=FixOrientationParity
- http://www.stefan-pochmann.info/spocc/other_stuff/4x4_5x5_algs/?section=FixBothParities
- http://www.mementoslangues.fr/CubeDesign/4xCubes/DoubleMidgeFlip.pdf
- http://www.speedsolving.com/forum/showthread.php?11409-4x4-Optimal-Solver-v2
OLL Parity Algorithms which don't preserve F3L
- http://www.speedsolving.com/forum/showthread.php?37962-Fixing-4x4x4-orientation-parity-earlier-than-usual
- http://www.speedsolving.com/forum/showthread.php?26564-4x4x4-edge-parity-is-there-a-shorter-alg-that-doesn-t-preserve-corners
- http://www.speedsolving.com/forum/showthread.php?17839-WANTED-New-Dedge-Flip-Algorithm!
- http://twistypuzzles.com/forum/viewtopic.php?f=8&t=9502
PLL Parity Algorithms
- http://frederickbadie.free.fr/444PLLparity.html
- http://www.speedsolving.com/forum/showthread.php?21725-New-Two-Corner-Swap-Algorithm-Technique-for-Big-Even-Cubes-%28PLL-Parity%29
SuperCube Parity Algorithms
- http://www.speedsolving.com/forum/showthread.php?4659-4x4-algorithm
- http://www.speedsolving.com/forum/showthread.php?37308-Is-a-SuperCube-Safe-Single-Dedge-Flip-Algorithm-Possible-in-lt-U-Rw-gt
- http://www.speedsolving.com/forum/showthread.php?26622-God-s-algorithm-discovered-for-OLL-parity-edge-flip-%28theory%29