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 in each category.

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)


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 (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' 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)


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)


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


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)
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)
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 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 M' U2 r2 U2 r' U2 r /(26,17)
Alg r' U2 r U2 r' U2 r' U2 l U2 r' U2 r U2 M' U2 r2 /(26,17)


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 Rw2 U x' U2 r U2 r' F2 l F2 U' L' U' l U L U l' x U' Rw2 /(25,19)



Other ELL Cases (WCA Notation)

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)
Alg f2 M2 f2 l2 U2 r S2 r' S2 r' U2 r2 /(21,12)
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 r2 U2 l' U2 l F2 U2 r2 U2 r F2 r' F2 r' F2 /(25,15)
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 (B' R Uw' u' r' f2 Rw2 E) f (E' Rw2 f2 r u Uw R' B) /(21,17)
Alg r U2 l' U2 r U2 r U2 r' U2 r U2 r2 U2 M U2 r' /(26,17)
Alg 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 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 M' U2 l' U2 l U2 r' U2 l2 /(20,13)
Alg r U2 r U2 M' U2 r U2 r' U2 l U2 r2 /(20,13)
Alg z' Lw U' r Uw Lw' Uw' r Fw r Fw' r2 Uw Lw u' Lw' z /(16,15)
Alg l2 D' f' D r D2 r' D' f D' l' F2 r' F2 l' /(19,15)
Alg b u Bw' Rw' f' Rw Bw Rw' f2 Uw' f' Uw f' Rw u' b' /(17,16)
Alg Rw U2 r U2 Rw U2 Rw2 F r F' Rw2 U2 Rw2 F r' F' /(23,16)
Alg Fw' Uw r' Uw' Fw Uw r' Fw' f' L f r' f' L' f Fw r2 Uw' /(19,18)


Opposite/diagonal 2-swap

Alg 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 r2 B2 U2 r' U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14)
Alg r2 B2 U2 r U2 r2 B2 r' U2 r2 U2 B2 r' B2 /(25,14)
Alg Uw Lw' Uw' l' Uw Lw2 Dw' Lw Uw' l' Uw L' Dw Lw' Uw' /(16,15)
Alg Uw Lw' Uw' l' Uw Lw Fw' Lw2 Uw' l' Uw Lw' L' Fw Uw' /(16,15)
Alg Uw Lw' Bw' r' Bw Lw2 Bw' Rw Fw' r' Fw R' Bw Lw' Uw' /(16,15)
Alg Fw' L2 Uw b' Uw' Rw2 Uw B' Uw b' Uw' Bw Uw' l2 Fw z y2 /(18,15)
Alg Fw Uw' B2 Lw u' Lw' Bw2 Rw2 d' Rw' d Rw' b2 Uw Fw' /(19,15)
Alg Uw Rw' F2 Rw' f Rw Bw2 Lw2 u Lw u' Rw u2 Rw Uw' y2 /(19,15)
Alg Rw2 F2 U2 r U2 x U2 Rw2 U2 r' U2 l r U2 l' U2 x' /(25,15)
Alg y' Rw' F U' r Uw2 y Lw2 Dw r Dw' Lw2 Dw r' Uw U F' Rw y /(19,16)
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)
Alg Uw' Lw' Fw R' Fw' d' Fw R Fw' Lw2 Fw2 Rw' u' Rw Fw2 Lw' b Uw /(21,18)
Alg x' l' Uw Rw' Uw' l Uw Fw Uw B' l2 B r' B' l2 B Uw' Fw' Rw Uw' x /(21,19)


2-Cycles in Two Adjacent Edges

Case 1 (Close Adjacent "unoriented")

Alg l2 r' U B2 U' r U B2 Lw U2 r U2 Lw' U' l2 /(21,15)
Alg x' l2 F U2 l U2 r' U2 r U2 F2 r F2 l' F' l2 x /(23,15)
Alg r U2 r U2 F' L F' r F L' F l' U2 l U2 r2 /(21,16)
Alg x' l2 F U2 l U2 r' U2 r U2 (l' Rw) U2 r U2 Rw' F' l2 x /(25,16)
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 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' 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 z' Fw' l' Uw' Bw2 Uw' r Uw Bw2 Uw2 r Uw r' Uw2 l Fw z /(19,15)
Alg x' Fw Rw' Fw' r' Fw Rw Uw' Rw' r' Fw' r' Fw R r' Uw Fw' x /(16,16)
Alg Fw z' 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 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 (Lw r') l U F2 U' l U F2 Rw U2 l U2 Rw' U' l2 (Lw' 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' f' Uw Rw' Fw' r' Fw Rw Uw' f r' Uw Rw' Fw' r' Fw Rw Uw' r2 x /(19,18)
Alg l' Uw Rw' Uw' l Uw Fw Uw B' Rw2 F r' F' Rw2 B Uw' Fw' Rw Uw' /(21,19)


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

  • Since there is disagreement among cubers about whether lowest q moves or lowest btm moves defines the "optimal algorithm", the table below categorizes all 2-cycle and 4-cycle ELL cases on the 4x4x4 by the average of q moves and btm.
  • Algorithms optimal in btm need not be the algorithm with the lowest average of q and btm, and algorithms optimal in q turns need not be the algorithm with the lowest average of q and btm. Therefore, the average for a given case might be from an algorithm optimal in q moves, optimal in btm moves, or optimal in both.
  • Of course, this ranking is based off of required moves. It is not based off of the amount of time it takes to solve a case.
  • These results will change in the future if shorter algorithms are found for any of these cases.
  • The table shows that the "worst case" is Adjacent Double Parity, and that the "easiest case" is the Checkerboard Pattern (in two opposite edges).


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


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.
  • 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)
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 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)
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)
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)



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.

Pure 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.)



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