Difference between revisions of "4x4x4 parity algorithms"

Revision as of 10:32, 18 August 2012

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.

1 PLL Parity (WCA Notation)
2 Impure Dedge Flips (OLL Parity)(Turn the inner and outer layers simultaneously)
- 2.1 OLL Parity
- 2.2 OLL Parity + PLL Parity (Double Parity)
3 Impure Dedge Flips with Wide Turns/Pure Flips with Inner Layer Turns (SiGN Notation)
- 3.1 OLL Parity
- 3.2 OLL Parity + PLL Parity (Double Parity) (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
- 7.1 OLL Parity
- 7.2 OLL Parity + PLL Parity (Double Parity)
8 OLL Parity Algorithms Which Don't Preserve F3L or the Colors of the Centers (WCA Notation)
- 8.1 OLL Parity
- 8.2 OLL Parity + PLL Parity (Double Parity)
9 Cage Method/Sandwich Method Algorithms (WCA Notation)
10 Supercube/Blindfold Solving Algorithms (WCA Notation)
11 2-Gen Algorithms (WCA Notation)
- 11.1 One Dedge Flip
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

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' 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

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)

r' U2 r' U2 l U2 r' U2 r U2 x U2 r2 U2 r' U2 r /(25,16)

Alg_3(v2)

l U2 l U2 r' U2 l U2 l' U2 x U2 l2 U2 l U2 l' /(25,16)

Alg_4(v1)

r2 U2 x' U2 r' U2 r U2 l' U2 r U2 r U2 r' U2 r /(25,16)

Alg_4(v2)

l2 U2 x' U2 l U2 l' U2 r U2 l' U2 l' U2 l U2 l' /(25,16)

Alg_5(v1)

r U2 r' U2 r2 U2 x U2 r U2 r' U2 l U2 r' U2 r' /(25,16)

Alg_5(v2)

l' U2 l U2 l2 U2 x U2 l' U2 l U2 r' U2 l U2 l /(25,16)

Alg_6(v1)

Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U Rw2 R U Rw U' Rw /(23,19)(WCA)

Alg_6(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.

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 x l U2 l' U2 x' r2 U2 r U2 r' U2 F2 r2 F2 /(25,15)

Alg

r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 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 /(25,15)

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).

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

One advantage of having access to algorithms which don't preserve the first three layers of the 4x4x4 is that they often require less moves than algorithms which preserve the first three layers.
Unlike algorithms which preserve the first three layers, which are at minimum 15 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. 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

All algorithms are in SiGN Notation unless mentioned otherwise, and you may perform the following algorithms without wide turns (except for the (15,15) solutions and the (U, Rw, R) solutions), but they won't preserve as much.

OLL Parity

Alg_1

Rw B U2 B' r B2 l B2 r B' D2 B Rw /(17,13)(WCA)

Alg_2

r B' U2 B r B2 l B2 r B D2 B' r /(17,13)

Alg_2(v2)

x' r' U F2 U' l' U2 r' U2 r' F' U2 F r' /(17,13)

Alg_2(v3)

x' r' U F2 U' l' U2 r' U2 l' U' B2 U r' x' /(17,13)

Alg_2(v4)

x' r' U F2 U' l' U2 r' 3d2 r' U' F2 U l' x /(17,13)

Alg_3

r U F2 U' l F2 2R U2 r U B2 U' r /(17,13)

Alg_4

r U' F2 U l F2 2R U2 r U' B2 U r /(17,13)

Alg

Uw M R D R' Fw f E' l' E Fw b E F Lw /(15,15)(WCA)

Alg

r' U2 F U2 F' U2 r' U2 r' U2 r' F U2 F' r' /(21,15)

Alg

r' U2 F' U2 F U2 r' U2 r' U2 r' F' U2 F r' /(21,15)(WCA)

Alg

l U2 l' F U R U' Lw' D2 r D2 r U2 Rw' U2 Lw F /(22,17)(WCA)

Alg

F' Lw' U2 Rw U2 r' D2 r' D2 Rw B R' B' U' x' l U2 l' /(22,17)(WCA)

Alg

Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F r' U2 r F /(22,17)(WCA)

Alg

Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F Rw' U2 Rw F /(22,17)(WCA)

Alg

Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw' /(25,17)(WCA)

OLL Parity + PLL Parity (Double Parity)

Alg_1

r F2 U2 l F U' R U' r2 B2 r' B2 r2 /(19,13)

Alg_1(v2)

l' U2 F2 l' U' F R' F l2 U2 r U2 l2 U' x' /(20,14)

Alg_1(v3)

l' U2 F2 l' U' F 3l' U l2 x' U2 r U2 l2 U' x' /(20,14)

Alg_2

r F2 U2 l F' U L' U r2 B2 r' B2 r2 /(19,13)

Alg_2(v2)

r U2 F2 r U' F R' F r2 U2 l' U2 r2 U' x' /(20,14)

Alg_2(v3)

r U2 F2 r U' F 3l' U r2 x' U2 l' U2 r2 U' x' /(20,14)

Alg_3

r F2 U2 l F U' (L' R) U' r2 B2 r' B2 r2 /(19,13)

reParity

r U2 l' U2 x' (r' U2 l U2)2 l' /(19,13)

Alg_4

r F2 U2 l F' U (L' R) U r2 B2 r' B2 r2 /(19,13)

Affects_M-Layer_Only

r U2 r' U2 r' D2 r D2 r' B2 r B2 r' /(19,13)(WCA)

Affects_M-Layer_Only(v2)

r U2 r' U2 M' l' U2 r U2 l' U2 r U2 l' /(20,14)(WCA)

Alg

Uw R' b E' d' L' b L u' b2 Uw M' B' Uw /(15,15)(WCA)

Alg

Uw S L B' R u d f' u d L B' R S Uw /(15,15)(WCA)

Alg

Uw S L' F R' u d f' u d L' F R' S Uw /(15,15)(WCA)

Alg

Rw' U2 r U2 Rw' x' U2 r' U' R' U' Rw' U2 Rw U R U' Rw x /(21,17)(WCA)

Alg

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

Pochmann_PetrusParity

Rw' U R U (Rw' U2)3 Rw2 U R' U' Rw2 U' R' U Rw' /(24,19)(WCA)

To see more detailed information on how most of the brief algorithms were found, see http://cubezzz.dyndns.org/drupal/?q=node/view/230

To see Stefan's explanation of his Petrus Parity Algorithm, see http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/13712

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)(WCA)

Alg

Rw2 U' r' U B2 U' r U' r' B2 Rw2 /(15,11)(WCA)

Alg

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

Alg

Uw S B D B' R f' R M E R Fw /(12,12)(WCA)

OLL Parity + PLL Parity (Double Parity)

Alg

Uw F R B' U f' E U M' U Fw /(11,11)(WCA)

Alg

l' B' u B' D2 B u' B' l D2 l' /(13,11)(WCA)

Alg

Uw L U B U' d Fw2 u' Fw2 L' Uw2 y /(14,11)(WCA)

Alg

Uw L U B d Fw2 u' Fw2 U' L' Uw2 y /(14,11)(WCA)

Alg

Rw2 U r' U' B2 U r U r B2 Rw2 /(15,11)(WCA)

Alg

Rw2 U' r' U B2 U' r U' r B2 Rw2 /(15,11)(WCA)

Alg

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

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)(WCA)

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)

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

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 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)

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

OLL Parity Algorithms which don't preserve F3L

PLL Parity Algorithms

http://www.speedsolving.com/forum/showthread.php?21725-New-Two-Corner-Swap-Algorithm-Technique-for-Big-Even-Cubes-%28PLL-Parity%29

@@ Line 1: / Line 1: @@
-'''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 piece]]s 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.
+'''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 piece]]s 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)|cube=4x4x4&notation=WCA&animtype=Generator}}
+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)|cube=4x4x4&notation=WCA&animtype=Generator}}
+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)|cube=4x4x4&notation=WCA&animtype=Generator}}
+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.
-Algorithms with a lower btm (block half turn) move count are listed first in each category.
-NOTE: These algorithms can be applied to the 6x6 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)==
@@ Line 21: / Line 39: @@
 {{Alg| y' R' F l E F2 E' l' r' E F2 E' r F' R y /(16,14) |cube=4x4x4&notation=WCA}}
 {{Alg| y' R' F l E' F2 E l' r' E' F2 E r F' R y /(16,14) |cube=4x4x4&notation=WCA}}
+{{Alg| (Rw' U R U Lw' U2 Rw' U2) r2 (U2 Rw U2 Lw U' R' U' Rw) /(22,17) |cube=4x4x4&notation=WCA}}
 PLL Parity in Two Opposite Edges ("Unoriented")
@@ Line 32: / Line 51: @@
 PLL Parity in Two Adjacent Edges ("Unoriented")
 {{Alg| R B r2 U2 r2 Uw2 r2 u2 B' R' /(16,10) |cube=4x4x4&notation=WCA}}
-{{Alg| (R B Rw2 F2 U2) r2 (U2 F2 Rw2 B' R') /(18,11) |cube=4x4x4&notation=WCA}}
 {{Alg| (R B Rw2 F2 U2) r2 (U2 F2 Rw2 B' R') /(18,11) |cube=4x4x4&notation=WCA}}
 {{Alg| ((Lw r') U Rw2 U2 F2) r2 (F2 U2 Rw2 U' (Lw' r)) /(18,11) |cube=4x4x4&notation=WCA}}
@@ Line 41: / Line 59: @@
 {{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) |cube=4x4x4&notation=WCA}}
 {{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
 {{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) |cube=4x4x4&notation=WCA}}
 {{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) |cube=4x4x4&notation=WCA}}
-Diagonal Two Corner Swap
+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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+Opposite/Diagonal Two Corner Swap
 {{Alg| Rw2 f2 U2 Fw2 U' Rw2 U2 Fw2 U Fw2 R2 U2 F2 Rw2 U /(27,15) |cube=4x4x4&notation=WCA}}
 {{Alg| Rw2 F2 U2 y Rw2 U' Rw2 U D Lw2' U' Lw2' y' r2 U2 F2 Rw2 U /(27,16) |cube=4x4x4&notation=WCA}}
@@ Line 52: / Line 82: @@
 {{Alg| r2 U2 r2 Uw2 r2 Uw2 L' U R' U2 L U' L' R U R' U2 L U' R U /(29,21) |cube=4x4x4&notation=WCA}}
 {{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) |cube=4x4x4&notation=WCA}}
+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) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
 *Algorithms for almost all PLL Parity cases can be found here http://www.math.leidenuniv.nl/~mfung/speedcubing/algs/4x4x4/
@@ Line 70: / Line 106: @@
 ===OLL Parity + PLL Parity (Double Parity)===
-{{Alg| r' U2 r' U2 l U2 r' U2 r U2 x U2 r2 U2 r' U2 r /(25,16)|cube=4x4x4}}
+{{Alg| r' U2 r' U2 l U2 r' U2 r U2 x U2 r2 U2 r' U2 r /(25,16)|Alg_1(v1)|cube=4x4x4}}
-{{Alg| l' U2 l U2 l' U2 l' U2 r U2 l' U2 l U2 x' U2 l2 /(25,16)|cube=4x4x4}}
+{{Alg| l U2 l U2 r' U2 l U2 l' U2 x U2 l2 U2 l U2 l' /(25,16)|Alg_1(v2)|cube=4x4x4}}
-{{Alg| Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U Rw2 R U Rw U' Rw /(23,19)(WCA)|cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r' U2 r U2 r U2 l' U2 r U2 r' U2 x' U2 r2 /(25,16)|Alg_2(v1)|cube=4x4x4}}
-{{Alg| Rw' U2 Rw' U2 Rw' U' R' Rw' U2 R2 U' Rw' U' R' Rw2 U' Rw' U Rw' /(23,19)(WCA)|cube=4x4x4&notation=WCA}}
+{{Alg| l' U2 l U2 l' U2 l' U2 r U2 l' U2 l U2 x' U2 l2 /(25,16)|Alg_2(v2)|cube=4x4x4}}
+{{Alg| r' U2 r' U2 l U2 r' U2 r U2 x U2 r2 U2 r' U2 r /(25,16)|Alg_3(v1)|cube=4x4x4}}
+{{Alg| l U2 l U2 r' U2 l U2 l' U2 x U2 l2 U2 l U2 l' /(25,16)|Alg_3(v2)|cube=4x4x4}}
+{{Alg| r2 U2 x' U2 r' U2 r U2 l' U2 r U2 r U2 r' U2 r /(25,16)|Alg_4(v1)|cube=4x4x4}}
+{{Alg| l2 U2 x' U2 l U2 l' U2 r U2 l' U2 l' U2 l U2 l' /(25,16)|Alg_4(v2)|cube=4x4x4}}
+{{Alg| r U2 r' U2 r2 U2 x U2 r U2 r' U2 l U2 r' U2 r' /(25,16)|Alg_5(v1)|cube=4x4x4}}
+{{Alg| l' U2 l U2 l2 U2 x U2 l' U2 l U2 r' U2 l U2 l /(25,16)|Alg_5(v2)|cube=4x4x4}}
+{{Alg| Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U Rw2 R U Rw U' Rw /(23,19)(WCA)|Alg_6(v1)|cube=4x4x4&notation=WCA}}
+{{Alg| Rw'U2 Rw'U2 Rw'U'R'Rw'U2 R2 U'Rw'U'Rw2 R'U'Rw' U Rw'/(23,19)(WCA)|Alg_6(v2)|cube=4x4x4&notation=WCA}}
 ==Impure Dedge Flips with Wide Turns/Pure Flips with Inner Layer Turns (SiGN Notation)==
@@ Line 92: / Line 136: @@
 *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)|cube=4x4x4}}
+{{Alg| r U2 r' U2 r U2 r U2 r' x U2 r U2 r' U2 x' U2 r2 U2 /(27,17)|cube=4x4x4}}
-==Pure Flips  (In WCA Notation Unless Mentioned Otherwise.)==
+==Pure Flips  (WCA Notation)==
+(Algorithms in SiGN Notation are labeled.)
 ===OLL Parity:  One Dedge Flip===
@@ Line 102: / Line 148: @@
 {{Alg| Rw U2 F2 D r' D' F2 D r D' Rw F2 r' F2 Rw' U2 Rw' /(23,17)|cube=4x4x4&notation=WCA}}
 {{Alg| Rw U2 Rw U2 r' U2 r U2 l' U2 l F2 r' F2 Rw' U2 Rw'/ (25,17)|lucasparity|cube=4x4x4&notation=WCA}}
-{{Alg| z Dw' M D Lw' Uw' r' Uw Lw Uw' Lw2' Bw' r' Bw Rw' R' u y' M' Uw x2 z' /(19,18)|Holy_Grail|cube=4x4x4&notation=WCA}}
+{{Alg| z Dw' M D Lw' Uw' r' Uw Lw Uw' Lw2' Bw' r' Bw Rw' R' u y' M' Uw x2 z' /(19,18)|Holy_Grail|cube=4x4x4&notation=WCA|vidurl=http://www.youtube.com/watch?v=JOiyd3PbXZg}}
 {{Alg| Rw' E Uw2 Fw Uw Rw' r' Fw' r Fw Rw Uw' Fw' Uw r Uw E' Rw /(19,18)|cube=4x4x4&notation=WCA}}
 {{Alg| Lw' S Bw2 Dw r Dw' Bw Dw r' Dw' r' Bw' r Bw r Bw S' Lw /(19,18)|cube=4x4x4&notation=WCA}}
@@ Line 130: / Line 176: @@
 ===One Dedge Flip + PLL Parity (Double Parity)===
-{{Alg| r U2 r' E2 F2 l F2 l' F2 r F2 r' D2 l y2 /(21,14) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r' E2 F2 l F2 l' F2 r F2 r' D2 l y2 /(21,14) |Alg_1|cube=4x4x4&notation=WCA}}
-{{Alg| r2 B2 r' U2 r' U2 B2 r' B2 r B2 r' B2 r2 B2 /(25,15) |cube=4x4x4&notation=WCA}}
+{{Alg| r2 B2 r' U2 r' U2 B2 r' B2 r B2 r' B2 r2 B2 /(25,15) |Alg_2(v1)|cube=4x4x4&notation=WCA}}
-{{Alg| l2 F2 l' U2 l' U2 F2 l' F2 l F2 l' F2 l2 F2 /(25,15) |cube=4x4x4&notation=WCA}}
+{{Alg| r2 F2 r U2 r U2 F2 r F2 r' F2 r F2 r2 F2 /(25,15) |Alg_2(v2)|cube=4x4x4&notation=WCA}}
-{{Alg| r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 U2 r' U2 r /(26,17) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 B2 U2 l r2 U2 r' U2 r U2 F2 r F2 l' B2 Rw2 /(25,15) |Alg_3|cube=4x4x4&notation=WCA}}
-{{Alg| r' U2 r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 /(26,17) |cube=4x4x4&notation=WCA}}
+{{Alg| r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 U2 r' U2 r /(26,17) |Alg_4|cube=4x4x4&notation=WCA}}
+{{Alg| r' U2 r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 /(26,17) |Alg_5|cube=4x4x4&notation=WCA}}
+{{Alg| (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_6|cube=4x4x4&notation=WCA}}
+{{Alg| (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_7||cube=4x4x4&notation=WCA}}
+{{Alg| 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) |Alg_8|cube=4x4x4&notation=WCA}}
 ===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) |cube=4x4x4&notation=WCA}}
+{{Alg| (r2 F Rw' F') U2 r U2 r U2 r U2 r2 (F Rw F' r2) /(23,16) |cube=4x4x4&notation=WCA}}
-{{Alg| Rw' z' L' U r F U2 r U2 r U2 r U2 r2 F' r' U' L z Rw /(23,18) |cube=4x4x4&notation=WCA}}
+{{Alg| (Rw' z' L' U r F) U2 r U2 r U2 r U2 r2 (F' r' U' L z Rw) /(23,18) |cube=4x4x4&notation=WCA}}
-{{Alg| x' r2 U' Rw U M' U2 r' U2 r' U2 r' U2 l r U' Rw' U r2 x /(24,18) |cube=4x4x4&notation=WCA}}
+{{Alg| (x' r2 U' Rw U) M' U2 r' U2 r' U2 r' U2 l r (U' Rw' U r2 x) /(24,18) |cube=4x4x4&notation=WCA}}
 {{Alg| R B r U2 r' E2 F2 l F2 l' F2 r F2 r' D2 l y2 B' R' /(25,18) |cube=4x4x4&notation=WCA}}
-{{Alg| R B r2 B2 r' U2 r' U2 B2 r' B2 r B2 r' B2 r2 B R' /(27,18) |cube=4x4x4&notation=WCA}}
+{{Alg| (R B r2 B2) r' U2 r' U2 B2 r' B2 r B2 r' (B2 r2 B R') /(27,18) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw U2 x U' Rw R U' r U2 r U2 r U2 r2 U' Rw' R' U x' U2 Rw' /(25,19) |cube=4x4x4&notation=WCA}}
 {{Alg| Rw2 U x' U2 r U2 r' F2 l F2 U' L' U' l U L U l' x U' Rw2 /(25,19) |cube=4x4x4&notation=WCA}}
+{{Alg| r2 F' R' F' U2 l F2 D2 r D2 l' F' r' F' U2 F Rw F r2 /(26,19) |cube=4x4x4&notation=WCA}}
+{{Alg| M' U F2 r' B2 r B2 r F2 r' U' r' F R' F r F' R F' l /(24,20) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 F L F U2 l F2 D2 r D2 l' F r' F U2 F' r L' F' Rw2 /(27,20) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
+{{Alg| (Rw2 F' L F r f2 u' r2 Uw2 S') r (S Uw2 r2 u f2 r' F' L' F Rw2) /(29,21) |cube=4x4x4&notation=WCA}}
+{{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) |cube=4x4x4&notation=WCA}}
 *For algorithms for OLL + OLL Parity and OLL Parity + F3L, see http://www.math.leidenuniv.nl/~mfung/speedcubing/algs/4x4x4/
@@ Line 151: / Line 209: @@
 ===4-Cycles in Two Opposite Edges===
 Checkerboard pattern (Adj. 2-swap 2-Swap + Two Flip)
-{{Alg| (f2 u' r2 Uw2 S') r (S Uw2 r2 u f2) /(17,11) |cube=4x4x4&notation=WCA}}
+{{Alg| (f2 u' r2 Uw2 S') r (S Uw2 r2 u f2) /(17,11) |Alg_1|cube=4x4x4&notation=WCA}}
-{{Alg| f2 M2 f2 l2 U2 r S2 r' S2 r' U2 r2 /(21,12) |cube=4x4x4&notation=WCA}}
+{{Alg| f2 M2 f2 l2 U2 r S2 r' S2 r' U2 r2 /(21,12) |Alg_2|cube=4x4x4&notation=WCA}}
-{{Alg| r' U2 r2 U2 r U2 r' U2 r U2 r2 U2 r' /(21,13) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r2 U2 r' U2 r U2 r' U2 r2 U2 r /(21,13) |Alg_3(v1)|cube=4x4x4&notation=WCA}}
-{{Alg| r U2 r2 U2 r' U2 r U2 r' U2 r2 U2 r /(21,13) |cube=4x4x4&notation=WCA}}
+{{Alg| l' U2 l2 U2 l U2 l' U2 l U2 l2 U2 l' /(21,13) |Alg_3(v2)|cube=4x4x4&notation=WCA}}
-{{Alg| l U2 l2 U2 l' U2 l U2 l' U2 l2 U2 l /(21,13) |cube=4x4x4&notation=WCA}}
+{{Alg| r' U2 r2 U2 r U2 r' U2 r U2 r2 U2 r' /(21,13) |Alg_3(v3)|cube=4x4x4&notation=WCA}}
+{{Alg| l U2 l2 U2 l' U2 l U2 l' U2 l2 U2 l /(21,13) |Alg_3(v4)|cube=4x4x4&notation=WCA}}
 Bowtie/Hourglass (Adj. 2-swap + Pll Parity)
-{{Alg| r2 U2 l' U2 l F2 U2 r2 U2 r F2 r' F2 r' F2 /(25,15) |cube=4x4x4&notation=WCA}}
+{{Alg| r2 U2 l' U2 l F2 U2 r2 U2 r F2 r' F2 r' F2 /(25,15) |Alg_1|cube=4x4x4&notation=WCA}}
-{{Alg| F2 l2 F2 l F2 l' F2 r U2 l U2 r' U2 l U2 l' /(25,16) |cube=4x4x4&notation=WCA}}
+{{Alg| r' U2 r U2 l' U2 r U2 l F2 r' F2 r F2 r2 F2 /(25,16) |Alg_2(v1)|cube=4x4x4&notation=WCA}}
-{{Alg| F2 r2 F2 r' F2 r F2 l' U2 r' U2 l U2 r' U2 r /(25,16) |cube=4x4x4&notation=WCA}}
+{{Alg| l U2 l' U2 r U2 l' U2 r' F2 l F2 l' F2 l2 F2 /(25,16) |Alg_2(v2)|cube=4x4x4&notation=WCA}}
-{{Alg| (B' R Uw' u' r' f2 Rw2 E) f (E' Rw2 f2 r u Uw R' B) /(21,17) |cube=4x4x4&notation=WCA}}
+{{Alg| (B' R Uw' u' r' f2 Rw2 E) f (E' Rw2 f2 r u Uw R' B) /(21,17) |Alg_3|cube=4x4x4&notation=WCA}}
-{{Alg| r U2 l' U2 r U2 r U2 r' U2 r U2 r2 U2 M U2 r' /(26,17) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 l' U2 r U2 r U2 r' U2 r U2 r2 U2 M U2 r' /(26,17) |Alg_4|cube=4x4x4&notation=WCA}}
-{{Alg| r U2 M' U2 r2 U2 r' U2 r U2 r' U2 r' U2 l U2 r' /(26,17) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 M' U2 r2 U2 r' U2 r U2 r' U2 r' U2 l U2 r' /(26,17) |Alg_5|cube=4x4x4&notation=WCA}}
 ===4-Cycles in Two Adjacent Edges===
@@ Line 169: / Line 228: @@
 Checkboard Pattern
 {{Alg| (R2 Uw' u' r' f2 Rw2 E) f (E' Rw2 f2 r u Uw R2) /(21,15) |cube=4x4x4&notation=WCA}}
-{{Alg| Rw' F2 M2 F l' U2 l' U2 l2 U2 l' U2 F' M2 F2 Rw /(25,16) |cube=4x4x4&notation=WCA}}
+{{Alg| (Rw' F2 M2 F) l' U2 l' U2 l2 U2 l' U2 (F' M2 F2 Rw) /(25,16) |cube=4x4x4&notation=WCA}}
-{{Alg| Rw U2 M2 U' M U2 r' U2 r2 U2 r' U2 l' U M2 U2 Rw' /(26,17) |cube=4x4x4&notation=WCA}}
+{{Alg| (Rw U2 M2 U') M U2 r' U2 r2 U2 r' U2 l' (U M2 U2 Rw') /(26,17) |cube=4x4x4&notation=WCA}}
 Bowtie/Hourglass
-{{Alg| L' Fw' f' l' u2 Lw2 S' u S Lw2 u2 l f Fw L /(19,15) |cube=4x4x4&notation=WCA}}
+{{Alg| (L' Fw' f' l' u2 Lw2 S') u (S Lw2 u2 l f Fw L) /(19,15) |cube=4x4x4&notation=WCA}}
-{{Alg| Lw' U2 M2 U' x r' U2 r' U2 r2 U2 r' U2 x' U M2 U2 Lw /(25,16) |cube=4x4x4&notation=WCA}}
+{{Alg| (Lw' U2 M2 U' x) r' U2 r' U2 r2 U2 r' U2 (x' U M2 U2 Lw) /(25,16) |cube=4x4x4&notation=WCA}}
 {{Alg| R B r' U2 r2 U2 r U2 r' U2 r U2 r2 U2 r' B' R' /(25,17) |cube=4x4x4&notation=WCA}}
-{{Alg| l' U2 M2 U M' U2 r U2 r2 U2 r U2 l U' M2 U2 l /(26,17) |cube=4x4x4&notation=WCA}}
+{{Alg| (l' U2 M2 U) M' U2 r U2 r2 U2 r U2 l (U' M2 U2 l) /(26,17) |cube=4x4x4&notation=WCA}}
@@ Line 182: / Line 241: @@
 Adjacent 2-swap
-{{Alg| r2 D2 r' D2 l D2 l' D2 B2 l' B2 r' /(19,12) |cube=4x4x4&notation=WCA}}
+{{Alg| r2 D2 r' D2 l D2 l' D2 B2 l' B2 r' /(19,12) |Alg_1|cube=4x4x4&notation=WCA}}
-{{Alg| r U2 r U2 x U2 r U2 l' x' U2 l U2 r2 /(19,12) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r U2 x U2 r U2 l' x' U2 l U2 r2 /(19,12) |Alg_2(v1)|cube=4x4x4&notation=WCA}}
-{{Alg| l' U2 l' U2 x U2 l' U2 r x' U2 r' U2 l2 /(19,12) |cube=4x4x4&notation=WCA}}
+{{Alg| l' U2 l' U2 x U2 l' U2 r x' U2 r' U2 l2 /(19,12) |Alg_2(v2)|cube=4x4x4&notation=WCA}}
-{{Alg| Rw2 U2 l U2 r' U2 r U2 (Rw l') U2 r U2 Rw /(20,13) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r U2 M' U2 r U2 r' U2 l U2 r2 /(20,13) |Alg_3(v1)|cube=4x4x4&notation=WCA}}
-{{Alg| Rw' U2 r' U2 (Rw' l) U2 r' U2 r U2 l' U2 Rw2 /(20,13) |cube=4x4x4&notation=WCA}}
+{{Alg| l' U2 l' U2 M' U2 l' U2 l U2 r' U2 l2 /(20,13) |Alg_3(v2)|cube=4x4x4&notation=WCA}}
-{{Alg| l' U2 l' U2 M' U2 l' U2 l U2 r' U2 l2 /(20,13) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw' U2 r' U2 (Rw' l) U2 r' U2 r U2 l' U2 Rw2 /(20,13) |Alg_4||cube=4x4x4&notation=WCA}}
-{{Alg| r U2 r U2 M' U2 r U2 r' U2 l U2 r2 /(20,13) |cube=4x4x4&notation=WCA}}
+{{Alg| z' Lw U' r Uw Lw' Uw' r Fw r Fw' r2 Uw Lw u' Lw' z /(16,15) |Alg_5|cube=4x4x4&notation=WCA}}
-{{Alg| z' Lw U' r Uw Lw' Uw' r Fw r Fw' r2 Uw Lw u' Lw' z /(16,15) |cube=4x4x4&notation=WCA}}
+{{Alg| l2 D' f' D r D2 r' D' f D' l' F2 r' F2 l' /(19,15) |Alg_6|cube=4x4x4&notation=WCA}}
-{{Alg| l2 D' f' D r D2 r' D' f D' l' F2 r' F2 l' /(19,15) |cube=4x4x4&notation=WCA}}
+{{Alg| b u Bw' Rw' f' Rw Bw Rw' f2 Uw' f' Uw f' Rw u' b' /(17,16) |Alg_7|cube=4x4x4&notation=WCA}}
-{{Alg| b u Bw' Rw' f' Rw Bw Rw' f2 Uw' f' Uw f' Rw u' b' /(17,16) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw U2 r U2 Rw U2 Rw2 F r F' Rw2 U2 Rw2 F r' F' /(23,16) |Alg_8|cube=4x4x4&notation=WCA}}
-{{Alg| Rw U2 r U2 Rw U2 Rw2 F r F' Rw2 U2 Rw2 F r' F' /(23,16) |cube=4x4x4&notation=WCA}}
+{{Alg| Fw' Uw r' Uw' Fw Uw r' Fw' f' L f r' f' L' f Fw r2 Uw' /(19,18) |Alg_9|cube=4x4x4&notation=WCA}}
-{{Alg| Fw' Uw r' Uw' Fw Uw r' Fw' f' L f r' f' L' f Fw r2 Uw' /(19,18) |cube=4x4x4&notation=WCA}}
 Opposite/diagonal 2-swap
-{{Alg| l' S2 U2 l U2 l' U2 r U2 r' F2 l B2 r z2 /(21,14) |cube=4x4x4&notation=WCA}}
+{{Alg| l' S2 U2 l U2 l' U2 r U2 r' F2 l B2 r z2 /(21,14) |Alg_1(v1)|cube=4x4x4&notation=WCA}}
-{{Alg| r S2 U2 r' U2 r U2 l' U2 l F2 r' B2 l' z2 /(21,14) |cube=4x4x4&notation=WCA}}
+{{Alg| r S2 U2 r' U2 r U2 l' U2 l F2 r' B2 l' z2 /(21,14) |Alg_1(v2)|cube=4x4x4&notation=WCA}}
-{{Alg| r2 B2 U2 r' U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14) |cube=4x4x4&notation=WCA}}
+{{Alg| r2 B2 U2 r' U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14) |Alg_2|cube=4x4x4&notation=WCA}}
-{{Alg| r2 B2 U2 r U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14) |cube=4x4x4&notation=WCA}}
+{{Alg| r2 B2 U2 r U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14) |Alg_3|cube=4x4x4&notation=WCA}}
-{{Alg| Uw Lw' Uw' l' Uw Lw2 Dw' Lw Uw' l' Uw L' Dw Lw' Uw' /(16,15) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw Lw' Uw' l' Uw Lw2 Dw' Lw Uw' l' Uw L' Dw Lw' Uw' /(16,15) |Alg_4|cube=4x4x4&notation=WCA}}
-{{Alg| Uw Lw' Uw' l' Uw Lw Fw' Lw2 Uw' l' Uw Lw' L' Fw Uw'  /(16,15) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw Lw' Uw' l' Uw Lw Fw' Lw2 Uw' l' Uw Lw' L' Fw Uw'  /(16,15) |Alg_5|cube=4x4x4&notation=WCA}}
-{{Alg| Uw Lw' Bw' r' Bw Lw2 Bw' Rw Fw' r' Fw R' Bw Lw' Uw' /(16,15) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw Lw' Bw' r' Bw Lw2 Bw' Rw Fw' r' Fw R' Bw Lw' Uw' /(16,15) |Alg_6|cube=4x4x4&notation=WCA}}
-{{Alg| Fw' L2 Uw b' Uw' Rw2 Uw B' Uw b' Uw' Bw Uw' l2 Fw z y2 /(18,15) |cube=4x4x4&notation=WCA}}
+{{Alg| Fw' L2 Uw b' Uw' Lw2 Uw L' Uw b' Uw' Lw Uw' l2 Fw /(18,15) |Alg_7|cube=4x4x4&notation=WCA}}
-{{Alg| Fw Uw' B2 Lw u' Lw' Bw2 Rw2 d' Rw' d Rw' b2 Uw Fw' /(19,15) |cube=4x4x4&notation=WCA}}
+{{Alg| Fw Uw' B2 Lw u' Lw' Bw2 Rw2 d' Rw' d Rw' b2 Uw Fw' /(19,15) |Alg_8|cube=4x4x4&notation=WCA}}
-{{Alg| Uw Rw' F2 Rw' f Rw Bw2 Lw2 u Lw u' Rw u2 Rw Uw' y2 /(19,15) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw Rw' F2 Rw' f Rw Bw2 Lw2 u Lw u' Rw u2 Rw Uw' y2 /(19,15) |Alg_9|cube=4x4x4&notation=WCA}}
-{{Alg| Rw2 F2 U2 r U2 x U2 Rw2 U2 r' U2 l r U2 l' U2 x' /(25,15) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 F2 U2 r U2 x U2 Rw2 U2 r' U2 l r U2 l' U2 x' /(25,15) |Alg_10|cube=4x4x4&notation=WCA}}
-{{Alg| y' Rw' F U' r Uw2 y Lw2 Dw r Dw' Lw2 Dw r' Uw U F' Rw y /(19,16) |cube=4x4x4&notation=WCA}}
+{{Alg| Fw' L U' f Uw2 Lw2 Dw r Dw' Lw2 Uw f' Uw U L' Fw /(19,16) |Alg_11|cube=4x4x4&notation=WCA}}
-{{Alg| l' U2 l U2 l U2 r' U2 l U2 l' U2 F2 l2 F2 r /(25,16) |cube=4x4x4&notation=WCA}}
+{{Alg| l' U2 l U2 l U2 r' U2 l U2 l' U2 F2 l2 F2 r /(25,16) |Alg_12(v1)|cube=4x4x4&notation=WCA}}
-{{Alg| r U2 r' U2 r' U2 l U2 r' U2 r U2 F2 r2 F2 l' /(25,16) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r' U2 r' U2 l U2 r' U2 r U2 F2 r2 F2 l' /(25,16) |Alg_12(v2)|cube=4x4x4&notation=WCA}}
-{{Alg| r F2 D2 l D2 F r2 U2 r2 F r' F' r2 U2 r2 F /(25,16) |cube=4x4x4&notation=WCA}}
+{{Alg| r F2 D2 l D2 F r2 U2 r2 F r' F' r2 U2 r2 F /(25,16) |Alg_13|cube=4x4x4&notation=WCA}}
-{{Alg| r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 U2 r' /(26,17) |cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 U2 r' /(26,17) |Alg_14|cube=4x4x4&notation=WCA}}
-{{Alg| Uw' Lw' Fw R' Fw' d' Fw R Fw' Lw2 Fw2 Rw' u' Rw Fw2 Lw' b Uw /(21,18) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw' Lw' Fw R' Fw' d' Fw R Fw' Lw2 Fw2 Rw' u' Rw Fw2 Lw' b Uw /(21,18) |Alg_15|cube=4x4x4&notation=WCA}}
-{{Alg| x' l' Uw Rw' Uw' l Uw Fw Uw B' l2 B r' B' l2 B Uw' Fw' Rw Uw' x /(21,19) |cube=4x4x4&notation=WCA}}
+{{Alg| x' l' Uw Rw' Uw' l Uw Fw Uw B' l2 B r' B' l2 B Uw' Fw' Rw Uw' x /(21,19) |Alg_16|cube=4x4x4&notation=WCA}}
 ===2-Cycles in Two Adjacent Edges===
@@ Line 247: / Line 305: @@
 {{Alg| Rw2 f Uw' Bw Dw r' Dw' r' Bw' r Bw r Bw' Uw Rw2 /(17,15) |cube=4x4x4&notation=WCA}}
 {{Alg| z' Fw' l' Uw' Bw2 Uw' r Uw Bw2 Uw2 r Uw r' Uw2 l Fw z /(19,15) |cube=4x4x4&notation=WCA}}
-{{Alg| x' Fw Rw' Fw' r' Fw Rw Uw' Rw' r' Fw' r' Fw R r' Uw Fw' x /(16,16) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw Rw' Uw' r' Uw Rw Bw' Rw' r' Uw' r' Uw R r' Bw Uw' /(16,16) |cube=4x4x4&notation=WCA}}
-{{Alg| Fw z' L' Dw' l Dw Lw Bw' Lw L Dw' l Dw Lw2 Fw u' Fw'  /(17,16) |cube=4x4x4&notation=WCA}}
+{{Alg| Bw L' Dw' l Dw Lw Bw' Lw L Dw' l Dw Lw2 Fw u' Fw' /(17,16) |cube=4x4x4&notation=WCA}}
 {{Alg| Uw' Fw l' Fw' Uw Fw l' Uw2 R Uw l' Uw' R' Uw2 l2 Fw' /(19,16) |cube=4x4x4&notation=WCA}}
 {{Alg| r' Uw Lw Uw' r2 Fw2 Lw' Fw r' Fw' Lw Fw2 r' Uw Lw' Uw' /(19,16) |cube=4x4x4&notation=WCA}}
@@ Line 256: / Line 314: @@
 {{Alg| x l2 U2 r U2 r' F2 l F2 U L U l U' L' U' l x' /(21,16) |cube=4x4x4&notation=WCA}}
 {{Alg| l2 r' U F2 U' l U F2 Rw U2 l U2 Rw' U' l r /(21,16) |cube=4x4x4&notation=WCA}}
-{{Alg| (Lw r') l U F2 U' l U F2 Rw U2 l U2 Rw' U' l2 (Lw' r) /(21,16) |cube=4x4x4&notation=WCA}}
 {{Alg| R B r U2 r U2 x U2 r U2 l' x' U2 l U2 r2 B' R' /(23,16) |cube=4x4x4&notation=WCA}}
 {{Alg| x l' U L U l' U' L' U' x U2 l' U2 l U2 r' U2 l r x2 /(21,17) |cube=4x4x4&notation=WCA}}
@@ Line 266: / Line 323: @@
 *More ELL cases for the 4x4x4 cube can be found here: [http://snk.digibase.ca/k4/7.htm Kirjava's K4 ELL page]
 ==Summary of 2-Cycle and 4-Cycle ELL Case Movecounts==
@@ Line 307: / Line 366: @@
-==Algorithms Which Don't Preserve F3L==
+==OLL Parity Algorithms Which Don't Preserve F3L==
-*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.
+*One advantage of having access to algorithms which don't preserve the first three layers of the 4x4x4 is that they often require less moves than algorithms which preserve the first three layers.
-*For a list of more variations of the 13 btm algorithms (but definitely not all possible variations), see
+*Unlike algorithms which preserve the first three layers, which are at minimum 15 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.  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
@@ Line 316: / Line 376: @@
 ===OLL Parity===
-{{Alg| Rw B U2 B' r B2 l B2 r B' D2 B Rw /(17,13) |Alg_1|cube=4x4x4&notation=WCA}}
+{{Alg| Rw B U2 B' r B2 l B2 r B' D2 B Rw /(17,13)(WCA) |Alg_1|cube=4x4x4&notation=WCA}}
 {{Alg| r B' U2 B r B2 l B2 r B D2 B' r /(17,13)|Alg_2|cube=4x4x4&animtype=Generator}}
 {{Alg| x' r' U F2 U' l' U2 r' U2 r' F' U2 F r' /(17,13)|Alg_2(v2)|cube=4x4x4&animtype=Generator}}
@@ Line 326: / Line 386: @@
 {{Alg| r' U2 F U2 F' U2 r' U2 r' U2 r' F U2 F' r' /(21,15) |cube=4x4x4}}
 {{Alg| r' U2 F' U2 F U2 r' U2 r' U2 r' F' U2 F r' /(21,15)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| l U2 l' F U R U' Lw' D2 r D2 r U2 Rw' U2 Lw F /(22,17)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| F' Lw' U2 Rw U2 r' D2 r' D2 Rw B R' B' U' x' l U2 l' /(22,17)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F r' U2 r F /(22,17)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw U2 Lw' U2 l D2 l D2 Rw' U' L U F Rw' U2 Rw F /(22,17)(WCA) |cube=4x4x4&notation=WCA}}
 {{Alg| Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw' /(25,17)(WCA) |cube=4x4x4&notation=WCA}}
 ===OLL Parity + PLL Parity (Double Parity)===
@@ Line 337: / Line 400: @@
 {{Alg| r U2 F2 r U' F 3l' U r2 x' U2 l' U2 r2 U' x' /(20,14) |Alg_2(v3)|cube=4x4x4&animtype=Generator}}
 {{Alg| r F2 U2 l F U' (L' R) U' r2 B2 r' B2 r2 /(19,13) |Alg_3|cube=4x4x4&animtype=Generator}}
+{{Alg| r U2 l' U2 x' (r' U2 l U2)2 l' /(19,13) |reParity|cube=4x4x4&animtype=Generator}}
 {{Alg| r F2 U2 l F' U (L' R) U r2 B2 r' B2 r2 /(19,13) |Alg_4|cube=4x4x4&animtype=Generator}}
 {{Alg| r U2 r' U2 r' D2 r D2 r' B2 r B2 r' /(19,13)(WCA)|Affects_M-Layer_Only|cube=4x4x4&notation=WCA}}
+{{Alg| r U2 r' U2 M' l' U2 r U2 l' U2 r U2 l' /(20,14)(WCA)|Affects_M-Layer_Only(v2)|cube=4x4x4&notation=WCA}}
 {{Alg| Uw R' b E' d' L' b L u' b2 Uw M' B' Uw /(15,15)(WCA) |cube=4x4x4&animtype=Generator&notation=WCA}}
 {{Alg| Uw S L B' R u d f' u d L B' R S Uw /(15,15)(WCA) |cube=4x4x4&animtype=Generator&notation=WCA}}
@@ Line 348: / Line 413: @@
 *To see more detailed information on how most of the brief algorithms were found, see http://cubezzz.dyndns.org/drupal/?q=node/view/230
+*To see Stefan's explanation of his Petrus Parity Algorithm, see http://games.groups.yahoo.com/group/speedsolvingrubikscube/message/13712
+==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)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 U' r' U B2 U' r U' r' B2 Rw2 /(15,11)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 U f U' B2 U f' U l B2 Rw2 /(15,11)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw S B D B' R f' R M E R Fw /(12,12)(WCA) |cube=4x4x4&notation=WCA&animtype=Generator}}
+===OLL Parity + PLL Parity (Double Parity)===
+{{Alg| Uw F R B' U f' E U M' U Fw /(11,11)(WCA) |cube=4x4x4&notation=WCA&animtype=Generator}}
+{{Alg| l' B' u B' D2 B u' B' l D2 l' /(13,11)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw L U B U' d Fw2 u' Fw2 L' Uw2 y /(14,11)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Uw L U B d Fw2 u' Fw2 U' L' Uw2 y /(14,11)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 U r' U' B2 U r U r B2 Rw2 /(15,11)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 U' r' U B2 U' r U' r B2 Rw2 /(15,11)(WCA) |cube=4x4x4&notation=WCA}}
+{{Alg| Rw2 U f U' B2 U f' U l' B2 Rw2 /(15,11)(WCA) |cube=4x4x4&notation=WCA}}
+*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)(WCA) |cube=4x4x4&notation=WCA}}
@@ Line 496: / Line 586: @@
-==Links==
+==External Links==
-*For more on this topic, see http://www.speedsolving.com/forum/showthread.php?22969-Methods-for-Forming-2-Cycle-Odd-Parity-Algorithms-for-Big-Cubes
+===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
+===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://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
+===Miscellaneous===
+*http://www.speedsolving.com/forum/showthread.php?22969-Methods-for-Forming-2-Cycle-Odd-Parity-Algorithms-for-Big-Cubes
+*http://www.speedsolving.com/forum/showthread.php?26477-Contructing-Algorithms-%28not-a-how-to-sorry%29

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Difference between revisions of "4x4x4 parity algorithms"

Revision as of 10:32, 18 August 2012

Contents

PLL Parity (WCA Notation)

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

OLL Parity

OLL Parity + PLL Parity (Double Parity)

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

OLL Parity

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

Pure Flips (WCA Notation)

OLL Parity: One Dedge Flip

One Dedge Flip + PLL Parity (Double Parity)

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

Other ELL Cases (WCA Notation)

4-Cycles in Two Opposite Edges

4-Cycles in Two Adjacent Edges

2-Cycles in Two Opposite Edges

2-Cycles in Two Adjacent Edges

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

OLL Parity Algorithms Which Don't Preserve F3L

OLL Parity

OLL Parity + PLL Parity (Double Parity)

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

OLL Parity

OLL Parity + PLL Parity (Double Parity)

Cage Method/Sandwich Method Algorithms (WCA Notation)

Single Dedge Flip

OLL Parity + PLL Parity (Double Parity)

In Two Adjacent Edges (in the M ring)

In Two Opposite Edges (in the M Ring)

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

Supercube/Blindfold Solving Algorithms (WCA Notation)

Supercube Safe PLL Parity

Supercube Safe OLL Parity

Two X-Center Piece Swap Algorithms

2-Gen Algorithms (WCA Notation)

One Dedge Flip

External Links

OLL Parity Algorithms

OLL Parity Algorithms which don't preserve F3L

PLL Parity Algorithms

SuperCube Parity Algorithms

Miscellaneous

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