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)

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)

  • Algorithms 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)
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 (SiGN Notation)

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) (SiGN Notation)

  • 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)


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)

Other PLL Parity Forms

Adjacent PLL Parity

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)


Unoriented PLL Parity

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)


Unoriented Adjacent PLL Parity

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)


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 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)


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 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 Uw' B2 z2 Rw d' Rw' Fw2 Rw2 d' Rw' d Rw' b2 Uw Fw' /(19,15)
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)


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 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)



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 (except for the (15,15) solutions).
  • 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 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)



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 edge 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

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

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)


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)


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)


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)


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