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.

Algorithms with a lower btm (block half turn) move count are listed first.

PLL Parity (WCA Notation)

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)


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

  • Unless mentioned otherwise, all algorithms in this category are performed in SiGN notation. Lowercase letters mean to move the inner and outer layer at the same time.
  • 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)
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)


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


Impure Edge Flips with Wide Turns/Pure Flips with Inner Layer Turns

OLL Parity

  • If the following group of algorithms is executed using WCA notation (where lowercase = inner layer turns), pure one edge 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)
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)


OLL Parity + PLL Parity (Double Parity)

  • If the following group of algorithms is executed using WCA notation (where lowercase = inner layer turns), pure one edge 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)


Pure Flips (In WCA Notation Unless Mentioned Otherwise.)

OLL Parity: One Dedge Flips

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)
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 U2 Rw U2 r' U2 Rw U2 l' U2 r U2 r' U2 (Rw' l) Rw' U2 Rw' /(27,18)
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)


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)
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)
Alg r' U2 r' U2 l U2 r' U2 r U2 l' r U2 r2 U2 r' U2 r/ (27,18)
Alg r' U2 r U2 r' U2 r' U2 l U2 r' U2 r U2 l' r U2 r2/ (27,18)


Other ELL Cases

4-Cycles in Two Opposite Edges

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/Hourglass (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 Rw' U2 r' U2 (Rw' l) U2 r' U2 r U2 l' U2 Rw2 /(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)



Brief Algorithms Which Don't Preserve F2L

  • 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 algorithms work on all cube sizes.
  • For a list of more variations of these 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), but they won't preserve as much.

OLL Parity

Alg_1(WCA) Rw B U2 B' r B2 l B2 r B' D2 B Rw /(17,13)
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)(WCA)
Alg r' U2 F' U2 F U2 r' U2 r' U2 r' F' U2 F r' /(21,15)(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)
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)
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)


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