4x4x4 parity algorithms

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

NOTE: Alg template (and alg.garron itself) are completely useless for 4x4x4 so I'll just paste algs here.

ALSO: This really needs editing! I just dumped a lot of algorithms here but this could look pretty nice once someone goes over it.

MORE ALSO: Once the lists are made prettier, feel free to give them HTM/BTM counts and names/authors (e.g. lucasparity, cmowlaparity, etc).

A more comprehensive list of ELL algs for the 4x4x4 cube can be found here: Kirjava's K4 ELL page

OLL Parity (WCA notation)

Alg Rw U2 x Rw U2 Rw U2 Rw' U2 Lw U2 Rw' U2 Rw U2 Rw' U2 Rw'
Alg Rw U2 Rw U2 Rw' U2 Rw U2 Lw' U2 Rw U2 Rw' U2 x' Rw' U2 Rw'

Pure edge flips:

  • (For more pure flips, perform the next group of algorithms in WCA notation.)
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)
lucasparity Rw U2 Rw U2 r' U2 r U2 l' U2 l F2 r' F2 Rw' U2 Rw'/ (25,17)
Alg Rw U2 Rw U2 r' U2 Rw U2 l' U2 r U2 r' U2 (Rw' l) Rw' U2 Rw' /(27,18)
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)
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 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 U r' U r' U' r2 U r2 U2 r' U2 r U2 r' U2 r' U r' U' r U r' U2 r U2 /(34,26) (SiGN)


Impure Edge Flips (SiGN notation)

  • If you used WCA notation, these impure edge flips will become pure edge flips (mirrors and inverses excluded).
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)
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)


Algorithms that solve OLL and PLL parity simultaneously:

Alg Rw2 B2 Rw' U2 Rw' U2 x' U2 Rw' U2 Rw U2 Rw' U2 Rw2 U2 (x)
Alg Lw' U2 Lw U2 Lw' U2 Lw' U2 Rw U2 Lw' U2 Lw U2 x' U2 Lw2


PLL Parity (WCA Notation)

Alg Uw2 Rw2 U2 (Rw2 R2') U2 Rw2 Uw2
Alg Uw2 Rw2 U2 r2 U2 Rw2 Uw2
Alg r2 U2 r2 Uw2 r2 u2
Alg r2 U2 r2 Uw2 r2 Uw2
Alg (Rw2 R2') U2 (Rw2 R2') Uw2 (Rw2 R2') Uw2
Alg Rw2 B2 U2 r2 U2 B2 Rw2


Other ELL Cases

  • 4-Cycles in Two Opposite Edges

Double parity

Alg r U2 r' E2 F2 l F2 l' F2 r F2 r' D2 l y2 /(21,14)
Alg r2 B2 r' U2 r' U2 B2 r' B2 r B2 r' B2 r2 B2/ (25,15)
Alg l2 F2 l' U2 l' U2 F2 l' F2 l F2 l' F2 l2 F2/ (25,15)


Checkerboard pattern

Alg (f2 u' r2 Uw2 S') r (S Uw2 r2 u f2) /(17,11)
Alg r' U2 r2 U2 r U2 r' U2 r U2 r2 U2 r' /(21,13)
Alg r U2 r2 U2 r' U2 r U2 r' U2 r2 U2 r /(21,13)
Alg l U2 l2 U2 l' U2 l U2 l' U2 l2 U2 l /(21,13)


Bowtie (adj. 2-swap + Pll Parity)

Alg F2 l2 F2 l F2 l' F2 r U2 l U2 r' U2 l U2 l' /(25,16)
Alg F2 r2 F2 r' F2 r F2 l' U2 r' U2 l U2 r' U2 r /(25,16)
Alg r U2 l' U2 r U2 r U2 r' U2 r U2 r2 U2 l r' U2 r' /(27,18)
Alg r U2 r l' U2 r2 U2 r' U2 r U2 r' U2 r' U2 l U2 r' /(27,18)


  • 2-Cycles in Two Opposite Edges

Adjacent 2-swap

Alg r2 D2 r' D2 l D2 l' D2 B2 l' B2 r' /(19,12)
Alg r U2 r U2 x U2 r U2 l' x' U2 l U2 r2 /(19,12)
Alg l' U2 l' U2 x U2 l' U2 r x' U2 r' U2 l2 /(19,12)
Alg Rw2 U2 l U2 r' U2 r U2 (Rw l') U2 r U2 Rw /(20,13)
Alg l' U2 l' U2 (l' r) U2 l' U2 l U2 r' U2 l2 /(21,14)
Alg r U2 r U2 (r l') U2 r U2 r' U2 l U2 r2 /(21,14)
Alg z' Lw U' r Uw Lw' Uw' r Fw r Fw' r2 Uw Lw u' Lw' z /(16,15)
Alg l2 D' f' D r D2 r' D' f D' l' F2 r' F2 l' /(19,15)


Opposite/diagonal 2-swap

Alg l' S2 U2 l U2 l' U2 r U2 r' F2 l B2 r z2 /(21,14)
Alg r S2 U2 r' U2 r U2 l' U2 l F2 r' B2 l' z2 /(21,14)
Alg l2 B2 U2 l U2 l2 B2 l U2 l2 U2 B2 l B2 /(25,14)
Alg r2 B2 U2 r' U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14)
Alg Uw Lw' Uw' l' Uw Lw2 Dw' Lw Uw' l' Uw L' Dw Lw' Uw'/ (16,15)
Alg Rw2 F2 U2 r U2 x U2 Rw2 U2 r' U2 l r U2 l' U2 x' /(25,15)
Alg l' U2 l U2 l U2 r' U2 l U2 l' U2 F2 l2 F2 r /(25,16)
Alg r U2 r' U2 r' U2 l U2 r' U2 r U2 F2 r2 F2 l' /(25,16)


Impure Edge Flips (For Reduction) (SiGN)

Alg r U2 x r U2 r U2 r' U2 l U2 r' U2 r U2 r' U2 r'
Alg r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 x' r' U2 r'
Alg r U2 r' U2 r U2 r U2 l' U2 r U2 r' U2 x' U2 r2
Alg r2 B2 U2 l U2 r' U2 r U2 F2 r F2 l' B2 r2
Alg l2 B2 l U2 l U2 x' U2 l U2 l' U2 l U2 l2 U2 x U2
Alg r' U2 r U2 r' F2 r2 U2 r U2 r' U2 x U2 r2 U2
Alg r' U2 r U2 r U2 r2 F2 r' U2 r' U2 F2 r2 F2
Alg r2 U2 l' U2 r U2 r' U2 r U2 r' x' U2 r' U2 r2 F2 r
Alg F2 l2 F2 U2 l' U2 l U2 l2 F2 r' F2 r U2 l'
Alg r' U2 l F2 l' F2 r2 U2 r U2 r' U2 F2 r2 F2
Alg r' U2 x l U2 l' U2 x' r2 U2 r U2 r' U2 F2 r2 F2
Alg r U2 r U2 r' U2 r U2 l' U2 r U2 r' U2 l r2 U2 r'


algs that don't preserve F2L

  • u B' L2 B u B2 u R2 u R F2 R' u
  • r2 B2 r' U2 r2 B R' B U' r B2 U2 r
  • r B' U2 B r B2 l B2 r B D2 B' r

dump of more parity algs

  • long list at 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
  • Fw (lr') R B R' Uw d (lr') b' (l'r) Uw u (fb') D Lw (x' y') (WCA notation)
  • Rw B' u (l'r) l' F' u F r' u2 Rw (f'b) U' Rw (z' y) (WCA notation)
  • { Uw, u, d } (lr') R D R' Fw f (ud') l' (u'd) Fw b (u'd) F { Lw, l, r } (WCA notation, choose any one from the brackets)
  • { Uw, u, d } R' b (ud') d' L' b L u' b2 Uw (l'r) B' { Uw, u, d } (WCA notation, choose any one from the brackets)
  • { Uw, u, d } (fb') L B' R u d f' u d L B' R (fb') { Uw, u, d } (WCA notation, choose any one from the brackets)
  • { Uw, u, d } (fb') L' F R' u d f' u d L' F R' (fb') { Uw, u, d } (WCA notation, choose any one from the brackets)
  • r B' U2 B r B2 l B2 r B D2 B' r (WCA notation)
  • r' U2 F U2 F' U2 r' U2 r' U2 r' F U2 F' r' (WCA notation)
  • r' U2 F' U2 F U2 r' U2 r' U2 r' F' U2 F r' (WCA notation)
  • u B L2 B' u B2 d B2 u B' R2 B u (WCA notation)
  • u B' L2 B u B2 d B2 u B R2 B' u (WCA notation)
  • u B L2 B' u B2 u R2 d B R2 B' u (WCA notation)
  • u B' L2 B u B2 u R2 d B' R2 B u (WCA notation)
  • u F2 L2 d F L' U L' u2 B2 u' B2 u2 (WCA notation)
  • u F2 L2 d F' L D' L u2 B2 u' B2 u2 (WCA notation)
  • u F2 L2 d F L' (U D') L' u2 B2 u' B2 u2 (WCA notation)
  • u F2 L2 d F' L (U D') L u2 B2 u' B2 u2 (WCA notation)
  • x' r' U F2 U' l' U2 r' U2 r' F' U2 F r'
  • x' r' U F2 U' l' U2 r' U2 l' U' B2 U r' x'
  • x' r' U F2 U' l' U2 r' 3d2 r' U' F2 U l' x
  • r U F2 U' l F2 2r U2 r U B2 U' r
  • r U' F2 U l F2 2r U2 r U' B2 U r
  • l' U2 F2 l' U' F R' F l2 U2 r U2 l2 U' x'
  • l' U2 F2 l' U' F 3l' U l2 x' U2 r U2 l2 U' x'
  • r U2 F2 r U' F R' F r2 U2 l' U2 r2 U' x'
  • r U2 F2 r U' F 3l' U r2 x' U2 l' U2 r2 U' x'
  • more in this topic? http://www.speedsolving.com/forum/showthread.php?22969-Methods-for-Forming-2-Cycle-Odd-Parity-Algorithms-for-Big-Cubes
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