# New 4x4 parity algs using <R,Rw,U>



## Kare (Jun 24, 2011)

After some serious computations I have found the following parity algs for 4x4 solved using the reduction method. The algs should be the shortest possible using only R, Rw & U while preserving the lower layers.

The move count is higher than the well known parity alg, but this might still be faster. I will leave it to the speed freaks to try out how fast the algs can be executed.

*************************************************************

Flip UB

Depth: 19

Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U R Rw2 U Rw U' Rw 



Depth: 20 

R Rw' U2 R' Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 R Rw2 U' R' U Rw 

Rw2 U' Rw2 U2 Rw' U R U R' Rw' U2 R' Rw2 U R Rw U2 Rw' U2 Rw' 

*************************************************************

Flip UR

Depth: 19 

Rw U' Rw U R Rw2 U Rw U R2 U2 R Rw U Rw U2 Rw U2 Rw 

Rw' U Rw' U' R' Rw2 U' Rw' U' R2 U2 R' Rw' U' Rw' U2 Rw' U2 Rw' 



Depth: 20 

Rw U R' U' R Rw2 U2 Rw2 U Rw U2 Rw' U2 Rw U' R' Rw2 U2 R Rw' 

Rw U2 Rw U2 R' Rw' U' R Rw2 U2 R Rw U' R' U' Rw U2 Rw2 U Rw2 

Rw' U' R U R' Rw2 U2 Rw2 U' Rw' U2 Rw U2 Rw' U R Rw2 U2 R' Rw 

Rw' U2 Rw' U2 R Rw U R' Rw2 U2 R' Rw' U R U Rw' U2 Rw2 U' Rw2 

*************************************************************

Flip UF

Depth: 19

Rw' U2 Rw' U2 Rw' U' R' Rw' U2 R2 U' Rw' U' R' Rw2 U' Rw' U Rw' 



Depth: 20 

R' Rw U2 R Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U2 R' Rw2 U R U' Rw' 

Rw2 U Rw2 U2 Rw U' R' U' R Rw U2 R Rw2 U' R' Rw' U2 Rw U2 Rw 

*************************************************************


I have also been looking for algs for those who like to have one flipping three pieces. The shortest seems to be 22 moves and I have yet to compute all of them. Here is one such alg.


Flip all but UL

Depth: 22

Rw U Rw U2 Rw U2 Rw U Rw' U2 R Rw' U2 Rw' U Rw U2 Rw U2 Rw U Rw 


I have loads of more algs that solves the parities but mess up one or more of the DR corners. If anyone would find that useful I can post those too, including an 18 move alg.


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## masterofthebass (Jun 24, 2011)

Do you know if any of the algs that mess up 1 f2l slot are able to be executed faster? Going through these algs, they just don't seem that fluid, but if the alg only messes up DRF, it can be applied earlier.


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## joey (Jun 24, 2011)

I would like to see the parities that affect the corners.


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## irontwig (Jun 24, 2011)

Is that you, Kåre? How long does it take to run these computations?


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## Kare (Jun 24, 2011)

Here is the complete results, including the ones that mess up f2l corners. Solving the parity before the last pair is an interesting idea. I will have to do quite a few changes to the program but should be able to compute the algs for that too.

Yes, this is Kåre Krig. As usual development takes longer than running the computations. The full 1-flip computations was done in 5-15min while the attached 3-flip list used around 85min, utilizing all four cores on my not to ancient home computer. 

I'm attaching a list of algs for flipping three edges.
View attachment 3FLIP.TXT

*************************************************************
Flip UB

Depth: 18 
DBR Rw U' Rw U R Rw2 U Rw U R' U2 Rw U Rw U2 Rw U2 Rw 
DFR Rw U2 Rw U2 Rw U Rw U2 R' U Rw U R Rw2 U Rw U' Rw 


19 
Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U R Rw2 U Rw U' Rw 
DBR Rw2 U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw' U R U2 R' U' Rw 

20 
DBR R Rw' U2 R' Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 R Rw2 U' R' U Rw 
DBR R' Rw' U2 Rw2 U Rw U2 Rw' U2 Rw U' R' Rw2 U2 R' Rw2 U' R' U Rw 
DBR Rw U2 Rw U2 Rw U R' U2 R U2 Rw U' Rw U R Rw2 U Rw U' Rw 
DFR+DBR Rw' U' R2 U' Rw2 U R2 Rw U2 Rw' U' R2 U R2 U2 Rw U' Rw2 U2 Rw' 
DFR+DBR Rw2 U R' U2 R Rw2 U2 Rw' U R' U Rw' U2 Rw2 U Rw U2 Rw' U2 Rw' 
DBR Rw' U2 R2 Rw2 U Rw U2 Rw' U2 Rw U' R Rw2 U2 R' Rw2 U' R' U Rw 
DFR Rw' U2 R2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw2 U' R2 U' R U2 Rw 
DFR+DBR Rw' U2 Rw2 U' Rw U2 R2 U R2 U' Rw' U2 R2 Rw U Rw2 U' R2 U' Rw' 
DFR+DBR Rw2 U' R' Rw2 U2 R' Rw' U R2 U Rw' U2 R Rw2 U Rw U2 Rw' U2 Rw' 
DFR Rw2 U' R2 Rw2 U2 Rw' U R U R Rw' U2 R Rw2 U Rw U2 Rw' U2 Rw' 
DFR+DBR Rw2 U' Rw2 U2 R2 Rw' U R' U R' Rw' U2 R Rw2 U Rw U2 Rw' U2 Rw' 
Rw2 U' Rw2 U2 Rw' U R U R' Rw' U2 R' Rw2 U R Rw U2 Rw' U2 Rw' 

*************************************************************
Flip UR
Depth: 18 
19 
DFR Rw U' R' U2 R U Rw' U2 Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw2 
Rw U' Rw U R Rw2 U Rw U R2 U2 R Rw U Rw U2 Rw U2 Rw 
DBR Rw' U R U2 R' U' Rw U2 Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U2 Rw2 
Rw' U Rw' U' R' Rw2 U' Rw' U' R2 U2 R' Rw' U' Rw' U2 Rw' U2 Rw' 

20 
DFR Rw U R' U' R Rw2 U2 Rw2 U Rw U2 Rw' U2 Rw U' R' Rw2 U2 R Rw' 
DFR Rw U R' U' R' Rw2 U2 R Rw2 U' Rw U2 Rw' U2 Rw U R2 Rw2 U2 Rw' 
DFR Rw U R' U' R' Rw2 U2 R' Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 R' Rw' 
DFR Rw U' Rw U R Rw2 U Rw U' Rw U2 R U2 R' U Rw U2 Rw U2 Rw 
DBR Rw U2 R U' R2 U' Rw2 U2 Rw2 U' Rw U2 Rw' U2 Rw U R2 Rw2 U2 Rw' 
Rw U2 Rw U2 R' Rw' U' R Rw2 U2 R Rw U' R' U' Rw U2 Rw2 U Rw2 
DBR Rw' U Rw' U' R' Rw2 U' Rw' U Rw' U2 R' U2 R U' Rw' U2 Rw' U2 Rw' 
DBR Rw' U' R U R Rw2 U2 R Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U2 R Rw 
DBR Rw' U' R U R Rw2 U2 R' Rw2 U Rw' U2 Rw U2 Rw' U' R2 Rw2 U2 Rw 
DBR Rw' U' R U R' Rw2 U2 Rw2 U' Rw' U2 Rw U2 Rw' U R Rw2 U2 R' Rw 
DFR Rw' U2 R' U R2 U Rw2 U2 Rw2 U Rw' U2 Rw U2 Rw' U' R2 Rw2 U2 Rw 
Rw' U2 Rw' U2 R Rw U R' Rw2 U2 R' Rw' U R U Rw' U2 Rw2 U' Rw2 

*************************************************************

Flip UF
Depth: 18 
DFR Rw' U Rw' U' R' Rw2 U' Rw' U' R U2 Rw' U' Rw' U2 Rw' U2 Rw' 
DBR Rw' U2 Rw' U2 Rw' U' Rw' U2 R U' Rw' U' R' Rw2 U' Rw' U Rw' 

19 
Rw' U2 Rw' U2 Rw' U' R' Rw' U2 R2 U' Rw' U' R' Rw2 U' Rw' U Rw' 
DFR Rw2 U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw U' R' U2 R U Rw' 

20 
DFR R' Rw U2 R Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U2 R' Rw2 U R U' Rw' 
DFR+DBR Rw U R2 U Rw2 U' R2 Rw' U2 Rw U R2 U' R2 U2 Rw' U Rw2 U2 Rw 
DFR R Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U R Rw2 U2 R Rw2 U R U' Rw' 
DFR Rw U2 R2 Rw2 U' Rw' U2 Rw U2 Rw' U R' Rw2 U2 R Rw2 U R U' Rw' 
DBR Rw U2 R2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U R' U2 Rw' 
DFR+DBR Rw U2 Rw2 U Rw' U2 R2 U' R2 U Rw U2 R2 Rw' U' Rw2 U R2 U Rw 
DFR+DBR Rw2 U R Rw2 U2 R Rw U' R2 U' Rw U2 R' Rw2 U' Rw' U2 Rw U2 Rw 
DBR Rw2 U R2 Rw2 U2 Rw U' R' U' R' Rw U2 R' Rw2 U' Rw' U2 Rw U2 Rw 
DFR+DBR Rw2 U Rw2 U2 R2 Rw U' R U' R Rw U2 R' Rw2 U' Rw' U2 Rw U2 Rw 
Rw2 U Rw2 U2 Rw U' R' U' R Rw U2 R Rw2 U' R' Rw' U2 Rw U2 Rw 
DFR Rw' U2 Rw' U2 Rw' U' R U2 R' U2 Rw' U Rw' U' R' Rw2 U' Rw' U Rw' 
DFR+DBR Rw2 U' R U2 R' Rw2 U2 Rw U' R U' Rw U2 Rw2 U' Rw' U2 Rw U2 Rw 

*************************************************************

Flip UL
Depth: 18 
19 
20 
DFR+DBR Rw U2 Rw U2 Rw' U' R' Rw2 U2 R Rw U' R U' R2 Rw U2 Rw2 U Rw2 
DFR Rw U2 Rw U2 Rw' U' R' Rw2 U2 R' Rw U' R' U' Rw U2 R2 Rw2 U Rw2 
DFR+DBR Rw U2 Rw U2 Rw' U' R' Rw2 U2 Rw U' R2 U' R Rw U2 R Rw2 U Rw2 
DFR+DBR Rw U2 Rw U2 Rw' U' Rw2 U2 Rw U' R U' Rw U2 R' Rw2 U2 R U' Rw2 
DBR Rw' U2 Rw' U2 Rw U R Rw2 U2 R Rw' U R U Rw' U2 R2 Rw2 U' Rw2 
DFR+DBR Rw' U2 Rw' U2 Rw U R Rw2 U2 R' Rw' U R' U R2 Rw' U2 Rw2 U' Rw2 
DFR+DBR Rw' U2 Rw' U2 Rw U R Rw2 U2 Rw' U R2 U R' Rw' U2 R' Rw2 U' Rw2 
DFR+DBR Rw' U2 Rw' U2 Rw U Rw2 U2 Rw' U R' U Rw' U2 R Rw2 U2 R' U Rw2


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## Christopher Mowla (Jun 25, 2011)

Kare said:


> I have also been looking for algs for those who like to have one flipping three pieces. The shortest seems to be 22 moves and I have yet to compute all of them. Here is one such alg.
> 
> 
> Flip all but UL
> ...


I'm happy to see an alg in this move set to do that to be only 22 moves! Wow! Wow! Wow! I made one before to flip three dedges in U, but it's much longer
R' U2 Rw U2 Rw U2 Rw' U R U Rw U2 Rw' U' R' U' Rw R U R U Rw U2 Rw' U' R' U' Rw' U2 Rw' U2 Rw' U2 R (34)

As you probably can tell, I just conjugated this alg I made.
Rw U2 Rw U2 Rw' U R U Rw U2 Rw' U' R' U' Rw R U R U Rw U2 Rw' U' R' U' Rw' U2 Rw' U2 Rw' (30)

And, not that it is exactly relevant to this thread, but I did make an alg before which is pretty close to this move set which also flips 3 dedges in U (a double parity 3 2-cycle instead).
Rw' U2 r U2 Rw' x' U2 r' U' R' U' Rw' U2 Rw U R U' Rw R U2 x (19)

(Just posting it for those interested in a slightly shorter alg that is a little out of the move set focused on in this thread.)

Interesting stuff. Excellent work.


EDIT:

Wait, I have looked at your 3-flip alg (which I quoted above), and it is not in <R, Rw, U>
= Rw U Rw U2 Rw U2 Rw U Rw' U2 *r'* U2 Rw' U Rw U2 Rw U2 Rw U Rw

It makes the alg one move shorter in btm and q, but it still is not in the move set talked about in this thread. Do you have another alg that does this which is in the move set <R, Rw, U>?


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## Kare (Jun 26, 2011)

cmowla said:


> Wait, I have looked at your 3-flip alg (which I quoted above), and it is not in <R, Rw, U>
> = Rw U Rw U2 Rw U2 Rw U Rw' U2 *r'* U2 Rw' U Rw U2 Rw U2 Rw U Rw
> 
> It makes the alg one move shorter in btm and q, but it still is not in the move set talked about in this thread. Do you have another alg that does this which is in the move set <R, Rw, U>?


 
I don't see why the move I posted, using only R Rw U would not be considered to be in the group just because it can be shortended when concidering a different restriction of the 4x4. But I do have a bunch of other algs for the same case:

Rw U' R U2 R' U2 R' U' Rw U2 R Rw U2 Rw U' R' U2 R' U2 R U' Rw 

Rw U' R U2 R' U2 R' U' Rw U2 R Rw U2 Rw U' Rw' U2 R' U2 Rw U' Rw 

Rw U' Rw U2 R' U2 Rw' U' Rw U2 R Rw U2 Rw U' R' U2 R' U2 R U' Rw 

Rw U' Rw U2 R' U2 Rw' U' Rw U2 R Rw U2 Rw U' Rw' U2 R' U2 Rw U' Rw 

Rw' U R' U2 R U2 R U Rw' U2 R' Rw' U2 Rw' U R U2 R U2 R' U Rw' 

Rw' U R' U2 R U2 R U Rw' U2 R' Rw' U2 Rw' U Rw U2 R U2 Rw' U Rw' 

Rw' U Rw' U2 R U2 Rw U Rw' U2 R' Rw' U2 Rw' U R U2 R U2 R' U Rw' 

Rw' U Rw' U2 R U2 Rw U Rw' U2 R' Rw' U2 Rw' U Rw U2 R U2 Rw' U Rw' 

Rw' U' Rw' U2 Rw' U2 Rw' U' Rw U2 R' Rw U2 Rw U' Rw' U2 Rw' U2 Rw' U' Rw'


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## cuBerBruce (Jun 26, 2011)

cmowla said:


> Wait, I have looked at your 3-flip alg (which I quoted above), and it is not in <R, Rw, U>
> = Rw U Rw U2 Rw U2 Rw U Rw' U2 *r'* U2 Rw' U Rw U2 Rw U2 Rw U Rw
> 
> It makes the alg one move shorter in btm and q, but it still is not in the move set talked about in this thread. Do you have another alg that does this which is in the move set <R, Rw, U>?


 


Kare said:


> I don't see why the move I posted, using only R Rw U would not be considered to be in the group just because it can be shortended when concidering a different restriction of the 4x4.


Kare's expression was in fact written as a composition of the generators themselves, so the result of his maneuver is clearly in the group generated by those generators.

In mathematics, whether or not something is a member of a group is not dependent on how it is represented. So either *Rw U Rw U2 Rw U2 Rw U Rw' U2 R Rw U2 Rw' U Rw U2 Rw U2 Rw U Rw* or *Rw U Rw U2 Rw U2 Rw U Rw' U2 r' U2 Rw' U Rw U2 Rw U2 Rw U Rw* in fact represents an element of the group *<U,R,Rw>*.

Mathematicians also talk about "words" in a set of generators. A "word" in this sense is like a maneuver. Many different words can represent the same group element, just as many different maneuvers can represent the same transformation on the cube. So *Rw U Rw U2 Rw U2 Rw U Rw' U2 r' U2 Rw' U Rw U2 Rw U2 Rw U Rw* is _not_ a word in *<U,R,Rw>* because the *r'* is not one of the three generators given, nor a power of any of those generators. However, Kare's version did not use *r'* so Kare's version would certainly be considered a "word" in *<U,R,Rw>*.

Cmowla makes the point that what Kare called a "depth 22" maneuver can really be considered 21 block turns because the *R Rw'* can be considered equivalent to a single block turn ( *r'* ). People count moves in different ways, and I agree with cmowla on this that it may be worth mentioning that the way Kare's program counts turn may not be what others may expect.

Sometimes people infer from the group generator notation what metric should be considered applicable. I consider this somewhat dubious, and generally prefer to be more explicit about how I count moves. But in any case, some people may write, for instance, *<U,R,Rw,r>* with the intention that U, R, Rw, and r are to each count as a single move (and generally their inverses as well), and perhaps U2, R2, Rw2, and r2 as well. Cmowla's version of the maneuver could be seen as a "word" in *<U,R,Rw,r>* of length 21.


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## Kare (Jul 7, 2011)

I have now also searched for sequencens using only Rw U. Sadly those are 25 moves long (21 if one f2l pair is unsolved).
Dumping long lists of algs in the forum seems to be a bit impractical, so Gunnar kindly provided some webspace for a simplistic list of my results.

http://apelgam.se/Rubik/4x4parity/


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## mr. giggums (Jul 8, 2011)

I would be intrested if you did a search with <U,R,Rw> but instead of just the LL unsolved have U and R unsolved.

Just like Rw' U R U Rw' U2 Rw' U2 Rw' U2 Rw2 U R' U' Rw2 U' R' U Rw' but if possible shorter. Also if there isn't one shorter can you post all other algs at similar length so I can see which one flows faster.


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## uberCuber (Jul 8, 2011)

mr. giggums said:


> I would be intrested if you did a search with <U,R,Rw> but instead of just the LL unsolved have U and R unsolved.


 
Ooo, I see where you're going with this; that could be cool

As a similar type of thought, it would be interesting to see <U,R,Rw> algs that could mess with both the U layer and M layer


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## Christopher Mowla (Jul 8, 2011)

Just for fun, here's a 4-cycle double parity that messes up two 1x2 center blocks in M (in addition to the U and R slices).
Rw U2 Rw U2 Rw U2 Rw2 U' R U Rw2 U R U' Rw2 (21,15)

EDIT:

Here's another one which I made from one of Kåre's algs. It destroys the U Layer, two 1x2 center blocks in M, and destroys only one F3L slot.
Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U' Rw2 U' R2 U' Rw' (21,15)
​


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## mr. giggums (Jul 8, 2011)

uberCuber said:


> Ooo, I see where you're going with this; that could be cool
> 
> As a similar type of thought, it would be interesting to see <U,R,Rw> algs that could mess with both the U layer and M layer


 
It would be pointless because you coun't move the DFl and DBl wings so when they get paired back together the other wings the DF and DB edges will be solved so you would end up with only LL scrambled.


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## uberCuber (Jul 8, 2011)

mr. giggums said:


> It would be pointless because you coun't move the DFl and DBl wings so when they get paired back together the other wings the DF and DB edges will be solved so you would end up with only LL scrambled.


 
And there's me failing to think something through again...


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## Christopher Mowla (Jun 14, 2012)

Kare said:


> I have now also searched for sequencens using only Rw U. Sadly those are 25 moves long (21 if one f2l pair is unsolved).
> Dumping long lists of algs in the forum seems to be a bit impractical, so Gunnar kindly provided some webspace for a simplistic list of my results.
> 
> http://apelgam.se/Rubik/4x4parity/


Beautiful collection. Did Gunnar not add all of the algs you listed in this thread (I know some of them are there)? Also, there are algorithms which only mess up the U Layer in categories which claim to mess up more?

Anyway, I just studied the OLL Parity (not double parity) algorithms which destroy the FR F3L slot, and I have took off a move. These solutions mess up the U and R layers (Petrus).

Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw' (25,17)
Rw U2 Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U2 r2 U R2 U' Rw' (25,17)
Rw' U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw2 U' R2 U Rw (25,17)
Rw' U2 Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 r2 U' R2 U Rw (25,17)


mr. giggums,
Are these algs (well, I really just listed 2 algs, and only one is fast) are an example of what you wanted? I wonder what the minimum btm move counts are for <U, Rw, R>, considering that the minimum for all moves allowed is 13.:confused:


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## KingTim96 (Jun 14, 2012)

am i doing something wrong here? im using the very first alg posted, the first 19 move one, and each time it is flipping a different edge, what am i doin wrong?


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## Christopher Mowla (Jun 14, 2012)

KingTim96 said:


> am i doing something wrong here? im using the very first alg posted, the first 19 move one, and each time it is flipping a different edge, what am i doin wrong?


You're doing it correctly. It just flips the edge in the back. You have to add U' at the end of the alg to flip the same edge each time:
Rw U2 Rw U2 Rw U R Rw U2 R2 U Rw U R Rw2 U Rw U' Rw U'


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## Pyjam (Jun 17, 2012)

cmowla said:


> Rw U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U2 Rw2 U R2 U' Rw'


Wonderful. Many thanks.

I do {thumb on F} *(Rw U2' Rw2' U')* {thumb on U} *(Rw' U2 Rw U2' Rw' U) (Rw2' U2' Rw2 U R2' U')* {thumb on U} *Rw'*


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## Christopher Mowla (Dec 2, 2012)

Even though I posted this algorithm here, I wanted to add it to this thread. It's another "Petrus Parity", only this one is shorter than the (25, 17) I derived from Kåre's algorithms, and it's a lot less complicated. I indirectly derived it from cuBerBruce's algorithm.

Rw' U R' U2 R U' Rw' U2 Rw' U2 Rw' U' R U2 R' U Rw' (21, 17) (Not double parity).

If we rewrite my algorithm as the following, you can see that this algorithm is no more than (Rw' U2)4 Rw' with some <U, R> insertions!
(Inserted moves are red.)

Rw' U R' U2 R U' Rw' U2 Rw' U2 Rw' U' R U2 R' U Rw'

It should be very easy to remember, as the red moves are just setup moves to the first and last U2s.
In addition, although this algorithm is not in <U, Rw, R>, if we adjust this algorithm, we can have an inner slice version which just affects the FR F3L slot and the last layer.

r' U R U2 R' U' r' U2 r' U2 r' U' R U2 R' U r' (21, 17) (Not double parity)

Again, this algorithm can be visualized as being (r' U2)4 r' with some move insertions, and therefore it can be easy to remember.

r' U R U2 R' U' r' U2 r' U2 r' U' R U2 R' U r'

The only difference between the inner slice version and the wide turn version is that the first two R moves are inverted in the inner slice version (obviously all slice turns are wide in the wide turn version, and all slice turns are inner turns in the inner slice version).

I can't help but remember Stefan's algorithm when I see my (21,17) wide turn version because he made his double parity algorithm

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

several years ago using an adjustment of (Rw' U2)4 Rw' as well. (Here is his derivation).

Rw' U R U (Rw' U2)3 Rw2 U R' U' Rw2 U' R' U Rw'


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## Christopher Mowla (Dec 11, 2012)

I have gone through all of Kåre's algorithms in his 3-flip text document, web-page, and those he listed in this thread, and I believe I have eliminated all "duplicates". I have separated them into similar categories that Kåre did, but I also have separated the OLL Parity (Only) from the Double Parity algorithms. In addition, I have manually added BQTM and btm move counts after each algorithm. Note that I counted Rw2 R as 3 BQTM instead of 2 BQTM, even though it technically is 2; but I did count every other move in the algorithms in the correct manner.

I tried to put algorithms which started out the same or were very related (written wise) next to each other to hopefully give a better picture of the varasity of this amazing parity algorithm collection.

Lastly, what I mean by "duplicates" is just y2 cube rotations and mirrors of the same algorithm were omitted. These can be gotten back by either inverting all moves or taking the *reverse* of the algorithm.

Speaking of which, I found at least one algorithm that is equal to its reverse, but it is not equal to its inverse!
Rw U' Rw U2 R' U2 Rw' U' Rw U2 Rw R U2 Rw U' Rw' U2 R' U2 Rw U' Rw (28,22)

(Keep in mind that this is easily less than one third of the algorithms Kåre listed everywhere.)



Spoiler: Entire FR F3L Slot Affected



OLL Parity (1 Flip)
Rw U R' U' Rw2 R U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw' (25,18)
Rw U R' U' Rw2 U2 R Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 r' (25,18)
Rw' U2 Rw2 R' U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw2 U' R2 U Rw (26,18)
Rw U R' U' Rw2 U2 R Rw2 U Rw U2 Rw' U2 Rw U' R Rw2 U2 Rw' (26,19)
Rw' U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw2 R U' R U Rw R2 (27,19)
Rw' U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw2 U' R2 U' R2 U2 Rw (29,19)
Rw U' Rw' U2 Rw U2 Rw' U' Rw2 U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw' U2 Rw' (29,21)
Rw U' Rw' U2 Rw U2 Rw' U Rw2 U' Rw2 U2 Rw2 U2 Rw U2 Rw U2 Rw' U Rw' U2 Rw' (33,23)
Rw U' Rw' U2 Rw U2 Rw U2 Rw2 U2 Rw2 U' Rw2 U Rw' U2 Rw U2 Rw' U Rw' U2 Rw' (33,23)
Rw U Rw U2 Rw' U2 Rw U' Rw' U Rw2 U2 Rw U2 Rw' U2 Rw2 U' Rw' U2 Rw2 U2 Rw (33,23)
Rw U2 Rw' U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw' (33,23)
Rw U2 Rw U Rw2 U2 Rw U2 Rw' U2 Rw2 U' Rw U Rw' U2 Rw U2 Rw' U' Rw2 U2 Rw (33,23)

Double Parity (1 Flip)
Rw' U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 R Rw2 U' R U Rw R2 (27,19)
Rw' U2 Rw' U2 Rw U Rw2 U2 Rw' R2 U R U Rw' U2 R Rw2 U' Rw2 (27,19)
Rw' U' Rw2 U Rw2 U' Rw2 U2 Rw U2 Rw U2 Rw' U Rw' U Rw U Rw2 U2 Rw (29,21)
Rw' U2 Rw2 U' Rw' U' Rw U' Rw U2 Rw' U2 Rw U' Rw2 U Rw2 U' Rw2 U2 Rw' (29,21)
Rw' U' Rw2 U Rw2 U' Rw' U2 Rw U2 Rw' U2 Rw' U2 Rw U' Rw' U Rw U Rw2 U2 Rw (31,23)
Rw' U Rw2 U2 Rw2 U' Rw2 U Rw2 U2 Rw' U2 Rw U2 Rw' U Rw' U Rw U Rw2 U2 Rw (33,23)
Rw U Rw U2 Rw' U2 Rw U' Rw' U' Rw2 U2 Rw' U2 Rw' U2 Rw2 U' Rw' U2 Rw2 U2 Rw (33,23)
Rw' U' Rw2 U' Rw2 U2 Rw2 U' Rw2 U2 Rw' U2 Rw U2 Rw' U Rw' U Rw U Rw2 U2 Rw (33,23)
Rw U Rw U2 Rw' U2 Rw U' Rw' U Rw2 U2 Rw U2 Rw' U2 Rw2 U' Rw' U2 Rw2 U2 Rw (33,23)
Rw' U2 Rw2 U' Rw' U' Rw U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw2 U' Rw2 U Rw2 U2 Rw (33,23)
Rw' U2 Rw2 U' Rw' U' Rw U' Rw U2 Rw' U2 Rw U' Rw2 U' Rw2 U2 Rw2 U' Rw2 U2 Rw (33,23)
Rw' U2 Rw2 U' Rw' U2 Rw U2 Rw' U Rw2 U Rw' U2 Rw U2 Rw' U Rw' U2 Rw2 U2 Rw (33,23)





Spoiler: Just DFR Corner Slot Affected



OLL Parity (1 Flip)
Rw U R' U' R Rw2 U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 R' U2 r' (26,19) (Just twists the DFR corner)
Rw U R' U' Rw2 R' U2 R Rw2 U' Rw U2 Rw' U2 Rw U r2 U2 Rw' (26,19)
Rw U' R' U2 R U Rw' U2 Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw2 (27,19)
Rw' U2 R' U R2 U Rw2 U2 Rw2 U Rw' U2 Rw U2 Rw' U' r2 U2 Rw (28,19)
Rw U R' U' Rw2 R' U2 Rw2 R' U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw' R' (27,20)
Rw U R' U' Rw2 R' U2 Rw2 R U' Rw U2 Rw' U2 Rw U Rw2 R2 U2 Rw' (28,20)

OLL Parity (3 Flip)
Rw U2 R U R' U2 R U r U2 Rw U2 Rw U R U2 R' U' Rw (24,19) (Just twists the DFR corner)
Rw U R' U Rw U2 r U2 r U' Rw' U R U R' U2 Rw U' Rw R2 (24,20)
Rw U R' U r U2 r U2 Rw U' Rw' U R U R' U2 Rw U' Rw R2 (24,20)
r U' Rw U2 R' U R U Rw' U' r U2 Rw R2 U2 Rw U R' U Rw (24,20)
Rw U' R' U2 R U Rw U2 Rw U2 Rw U R' U2 R U R' U R U Rw (25,21)
Rw U' Rw U2 R' U R U Rw' U' Rw U2 Rw U2 Rw U R U' R' U2 Rw (25,21)
Rw U' R' U2 R U Rw R2 U2 Rw U2 Rw R2 U R U2 R' U' Rw U2 R' (28,21)
Rw U' R' U2 R U Rw U2 Rw U2 Rw U R2 U2 R2 U R2 U R2 U Rw (29,21) (Just twists the DFR corner)
Rw U2 Rw2 U' R U2 R' U' Rw' U2 Rw2 U2 Rw U Rw2 U2 Rw U2 Rw' U2 Rw2 (32,21)
Rw U' R' U2 R U Rw R U2 Rw U2 Rw U R U2 R' U' R' U' R U Rw (26,22)
R' U2 Rw U R2 U2 R U2 R' U Rw U2 Rw R2 U2 Rw U R U2 R' U' Rw (30,22)
R U2 Rw R2 U' R U2 R' U Rw U2 Rw R2 U2 Rw U R' U2 R U' Rw R2 (30,22)
R U2 Rw U' R2 U2 R2 U Rw U2 Rw U2 Rw U2 R U' R' U2 R U' Rw R2 (31,22)
Rw U' R' U2 R U Rw U2 Rw R2 U2 Rw U R2 U2 R2 U' Rw' U2 R U2 Rw2 (32,22) (This does a pure 3 flip + pure 2 corner twist)

OLL Parity (3 Flip) Only permutes corners/just flips 3 dedges in their own locations
Rw U' R' U2 R U r U2 r U2 r U R' U2 R U' Rw R2 (23,18)
Rw U' R' U2 R U Rw U2 r U2 Rw R2 U R' U2 R U' Rw R2 (25,19)
Rw U' R' U2 R U Rw R2 U2 r U2 Rw U R' U2 R U' Rw R2 (25,19)
Rw U' R' U2 R U Rw R U2 r U2 Rw R U R' U2 R U' Rw R2 (25,20)
Rw U' R' U2 R U' R' U2 r U2 Rw U2 r U R' U2 R U' Rw R2 (26,20)
Rw U' R' U2 R U r U2 Rw U2 r U2 R' U' R' U2 R U' Rw R2 (26,20)
Rw U' R' U R2 U R' U Rw U2 Rw R U2 Rw U R' U2 R U' Rw R2 (26,21)
Rw U2 R U' R U2 R U2 R' U r U2 Rw U2 Rw U R U2 R' U' Rw (27,21)
Rw U2 R U' Rw U2 R U2 Rw' U r U2 Rw U2 Rw U R U2 R' U' Rw (27,21)
Rw U' R' U2 R U r U2 Rw U2 Rw U R U' R2 U2 R2 U R' U' Rw (27,21)
Rw U2 R U R2 U2 R2 U Rw R2 U2 Rw U2 r U R2 U2 R2 U' Rw R (31,21)
Rw U' R' U2 R U' R' U2 Rw R U2 Rw U2 Rw U' R' U2 R' U2 R U' Rw (28,22)
Rw U' R' U2 R U' R' U2 Rw R U2 Rw U2 Rw U' Rw' U2 R' U2 Rw U' Rw (28,22)
Rw R U R U2 R U2 R' U Rw U2 Rw U2 Rw R U R' U2 R U' Rw R2 (28,22)
Rw R U Rw U2 R U2 Rw' U Rw U2 Rw U2 Rw R U R' U2 R U' Rw R2 (28,22)
Rw U' R' U2 R U Rw U2 Rw R U2 Rw U2 R2 U' R' U2 R U2 R2 U Rw (30,22)

Double Parity (1 Flip)
Rw' U Rw' U' Rw2 R' U' Rw' U' R U2 Rw' U' Rw' U2 Rw' U2 Rw' (22,18)
Rw U2 Rw U2 Rw' U' R' Rw2 U2 r U' R' U' Rw U2 r2 U Rw2 (25,18)
Rw U' Rw U R Rw2 U Rw U' Rw U2 R U2 R' U Rw U2 Rw U2 Rw (25,20)
Rw' U' Rw' U Rw U Rw2 U2 Rw' U2 Rw U2 Rw2 U' Rw' U' Rw U' Rw' U2 Rw' U2 Rw2 (31,23)
Rw U2 Rw2 U' Rw' U' Rw U' Rw' U2 Rw' U2 Rw2 U2 Rw' U' Rw' U Rw U Rw2 U2 Rw' (31,23)

Double Parity (3 Flip)
Rw2 U2 Rw U2 Rw U2 Rw2 U r U2 Rw2 R U2 Rw' U R2 U Rw2 U2 Rw (31,20)
Rw R U' R2 U' r2 U2 Rw' U2 Rw U' R2 U' Rw' U2 Rw2 U R U Rw R2 (29,21) (Just twists the DFR corner)
Rw2 U2 Rw U2 Rw U2 Rw2 U' Rw R U2 Rw2 R' U2 r' U R U Rw2 U2 Rw (31,21)
Rw2 U2 Rw R2 U2 Rw U2 Rw2 U r U2 Rw2 U2 Rw' R2 U R U Rw2 U2 Rw (33,21)
Rw2 U2 Rw U2 Rw U2 Rw2 U' R U2 r U2 Rw2 U2 Rw' U R2 U Rw2 U2 Rw (33,21)
Rw2 U' Rw U2 Rw' U2 Rw U Rw U Rw R U2 Rw' R' U' R' U' Rw R2 U' Rw2 (28,22)
Rw U R' U' Rw2 R U' R2 U' Rw2 U Rw U2 Rw' U2 Rw U R' U2 Rw2 U2 Rw' (30,22)
Rw U R' U Rw2 R U2 Rw' U' R' U' Rw U2 Rw' U2 Rw2 U' R2 U R2 U2 Rw (30,22)
Rw2 U2 Rw U2 Rw U2 Rw2 R U' R' U Rw U2 Rw2 U2 Rw' U R U Rw2 U2 Rw (32,22)
Rw2 U2 Rw U2 Rw U2 Rw2 U' Rw U2 Rw2 U R2 U Rw' R2 U R2 U Rw2 U2 Rw (34,22)
Rw U2 Rw2 U R2 U Rw' R2 U2 Rw2 R' U2 Rw U Rw2 U2 Rw U2 Rw R2 U2 Rw2 (35,22)





Spoiler: Just FR Dedge Affected



Double Parity (1 Flip)
Rw' U Rw2 U2 Rw U2 Rw U R2 U' Rw2 U2 Rw2 U Rw' U2 Rw2 U2 Rw' (29,19)





Spoiler: Just DFR and DBR Corners Affected



OLL Parity (3 Flip)
Rw U R' U Rw U2 Rw R U2 r U' Rw' U R U R' U2 Rw U' Rw (23,20)
Rw U R' U Rw U2 Rw U2 r U' Rw' U R U R' U2 Rw U' Rw R (23,20)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R U R' U2 R U R' U2 Rw (25,20)
Rw R2 U' R U2 R' U Rw U2 Rw U2 Rw U R' U2 R U' Rw U2 R2 (27,20)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R2 U R2 U2 R2 U R2 U2 Rw (29,20)
Rw R U R2 U Rw R U2 Rw U2 r U' Rw' U R U R' U2 Rw U' Rw (25,21)
Rw R U' Rw U2 R' U R U Rw' U' Rw U2 r U2 Rw U R2 U Rw R (25,21)
Rw U' R' U2 R U Rw U2 Rw U2 Rw U' R' U R U R' U2 R U' Rw (25,21)
Rw U' R' U2 R U Rw U2 Rw U2 Rw U R U2 R' U R U R' U Rw (25,21)
Rw U' R' U2 R U Rw U2 Rw U2 Rw U R U2 R' U R U R' U Rw (25,21)
Rw U' R' U2 R U R' U R U' Rw U2 Rw U2 Rw U R U2 R' U' Rw (25,21)
Rw U' Rw U2 R' U R U Rw' U' r U2 Rw U2 Rw U R' U Rw U2 R (25,21)
Rw U' Rw U2 R' U R U Rw' U' Rw R U2 r U2 Rw U R U Rw R2 (25,21)
Rw U' Rw U2 R' U R U Rw' U' Rw U2 r U2 Rw R U R U Rw R2 (25,21)
Rw U' Rw U2 R' U R U Rw' U R U2 r U2 Rw U2 Rw U R' U Rw (25,21)
Rw U' R' U2 R U Rw U2 Rw U2 r U R' U2 R U R' U R2 U Rw (26,21)
Rw U' R' U2 R U Rw U2 Rw U2 r U R' U2 R U R' U R2 U Rw (28,21)
Rw U' R' U2 R U Rw U2 Rw U2 r U R2 U2 R2 U R2 U R' U Rw (28,21)
Rw U' R' U2 R U r U2 Rw U2 Rw U' R2 U2 R' U2 R U' Rw U2 R (28,21)
Rw U' R' U2 R U r U2 Rw U2 Rw U R2 U2 R2 U2 R U Rw U2 R (29,21)
Rw U' R' U2 R U r U2 Rw U2 Rw U' R' U2 R2 U2 R2 U' Rw U2 R (29,21)
Rw U' R' U2 R U r U2 Rw R U2 Rw R2 U R2 U2 R2 U R U2 Rw (29,21)
Rw U' R' U2 R U' R' U2 Rw U2 Rw U2 Rw U R2 U2 R2 U' Rw U2 R (29,21)
Rw U' R' U2 R U' R2 U2 Rw U2 Rw U2 Rw U R' U2 R U' Rw U2 R2 (29,21)
Rw U R' U Rw' U2 r' U2 Rw' U R' U Rw' U R2 U2 R2 U2 R2 U Rw2 (29,21)
Rw U R' U Rw' U2 r' U2 Rw' U R' U Rw' U' R2 U2 R2 U2 R2 U' Rw2 (29,21)
Rw U R' U Rw' U2 r' U2 Rw' U R' U Rw' U Rw2 U2 R2 U2 Rw2 U Rw2 (29,21)
Rw U R' U Rw' U2 r' U2 Rw' U R' U Rw' U' Rw2 U2 R2 U2 Rw2 U' Rw2 (29,21)
Rw U R U2 R2 U2 R2 U Rw U2 Rw U2 r U R U2 R' U' Rw U2 R (29,21)
Rw U' R2 U2 R2 U R2 U R2 U' Rw U2 Rw U2 Rw U R U2 R' U' Rw (29,21)
Rw U' R2 U2 R2 U2 R' U' Rw U2 Rw U2 r U R U2 R' U' Rw U2 R (29,21)
r U2 Rw2 U Rw' U2 Rw U R2 U Rw' R2 U' Rw2 R' U2 Rw2 U R U' Rw' (29,21)
Rw U2 Rw2 U' R2 U2 R2 U' Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw' U2 Rw2 (34,21)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R U R U2 R' U R U R2 U Rw (27,22)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R2 U R U2 R' U R U R U Rw (27,22)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R2 U R' U2 R U R' U R' U Rw (27,22)
Rw U R2 U2 R U R U R2 U Rw U2 Rw U2 Rw R U R U2 R' U' Rw (28,22)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R U R2 U2 R2 U R2 U R U Rw (29,22)
Rw U' R' U2 R U Rw R2 U2 Rw U2 Rw U R' U2 R U R2 U2 Rw U2 R' (30,22)
Rw U' R' U2 R U Rw R U2 Rw R U2 Rw U R2 U2 R2 U' Rw R U2 R2 (30,22)
Rw U R2 U2 R U2 R' U Rw U2 Rw R2 U2 Rw U R U2 R' U' Rw U2 R' (30,22)
Rw U2 R2 U R U2 R' U Rw U2 Rw U2 Rw R2 U R U2 R' U' Rw U2 R' (30,22)
Rw R U' R2 U2 R2 U Rw U2 Rw R U2 Rw R U R U2 R' U' Rw U2 R2 (30,22)
Rw R U' R2 U2 R2 U' R U2 Rw U2 Rw U2 Rw U R' U2 R U' Rw U2 R2 (31,22)
Rw U R' U' Rw2 U2 Rw2 U2 Rw U2 Rw' R' U2 Rw' U R U' Rw2 U2 Rw2 U2 Rw' (32,22)
Rw R U' R2 U2 R2 U' R2 U2 Rw U2 Rw U2 Rw U R2 U2 R2 U' Rw U2 R (33,22)

OLL Parity (3 Flip) Only permutes corners/just flips 3 dedges in their own locations
Rw U' R' U2 R U r U2 Rw R U2 r U R U2 R' U' Rw (22,18)
Rw R U' R2 U2 R2 U r U2 Rw U2 r U R U2 R' U' Rw (24,18)
Rw U' R' U2 R U' R U2 r U2 Rw U2 r U R U2 R' U' Rw (24,19)
Rw U' R' U2 R U' R2 U2 r U2 r U2 r U R U2 R' U' Rw (25,19)
Rw R U' R2 U2 R2 U r U2 r U2 r U R2 U2 R2 U' Rw R (27,19)
Rw R U2 R' U R' U2 R U Rw U2 Rw U2 r U R U2 R' U' Rw (25,20)
Rw U' R' U2 R U r U2 Rw R2 U2 r U2 R' U' R U2 R' U' Rw (26,20)
Rw U' R' U2 R U Rw U2 Rw R U2 Rw U R U R U' R' U2 R' U' Rw (26,20)
Rw R2 U' R U2 R' U r U2 Rw R U2 r U R' U2 R U' Rw R2 (26,20)
Rw U' R' U2 R U Rw R2 U2 r U2 Rw U2 R2 U' R U2 R' U' Rw (27,20)
Rw U' R' U2 R U Rw U2 r U2 Rw R2 U2 R2 U' R U2 R' U' Rw (27,20)
Rw R2 U' R U2 R' U Rw U2 r U2 Rw U R' U2 R U R2 U2 Rw (27,20)
Rw R2 U2 R2 U R' U2 R U Rw U2 r U2 Rw U R U2 R' U' Rw (27,20)
Rw R2 U2 R' U R2 U2 R2 U r U2 r U2 Rw U R U2 R' U' Rw (28,20)
Rw R2 U2 R' U R2 U2 R2 U Rw U2 r U2 r U R U2 R' U' Rw (28,20)
Rw R U' R2 U2 R2 U r U2 Rw R2 U2 r U R' U2 R U' Rw R2 (28,20)
Rw R U' R2 U2 R2 U Rw R2 U2 r U2 Rw U R2 U2 R2 U' Rw R (29,20)
Rw U2 R2 U R' U2 R U2 R U' Rw R2 U2 Rw U2 Rw U R U2 R' U' Rw (30,20)
Rw U' R' U2 R U Rw U2 r U2 Rw U R U' R U2 R' U R' U' Rw (25,21)
Rw U' R' U2 R U Rw U2 Rw U2 r U R' U' R' U R' U2 R' U' Rw (25,21)
Rw R2 U' R U2 R' U Rw U2 Rw U2 Rw R U R2 U R' U R' U' Rw (26,21)
Rw U' R' U2 R U r U2 Rw R U2 Rw R U R' U2 R U R2 U2 Rw (27,21)
Rw U' R' U2 R U r U2 Rw U2 Rw R2 U R' U2 R U R U2 Rw R (27,21)
Rw U' R' U2 R U Rw U2 Rw U2 r U R U' R2 U' R U R2 U2 Rw (27,21)
Rw U' R' U2 R U Rw U2 Rw U2 r U R U2 R2 U R' U' R2 U' Rw (27,21)
Rw U' R' U2 R U' R2 U2 Rw R U2 r U2 Rw R U R U2 R' U' Rw (27,21)
Rw U2 R U R2 U2 R2 U Rw R2 U2 Rw R U2 r U R U2 R' U' Rw (29,21)
Rw U2 R U R2 U2 R2 U Rw R U2 r U2 Rw U R' U2 R U' Rw R2 (29,21)
Rw U2 R U R2 U2 R2 U Rw U2 r U2 Rw R U R' U2 R U' Rw R2 (29,21)
Rw U2 R2 U R U2 R' U Rw U2 r U2 Rw U2 R2 U' R U2 R' U' Rw (29,21)
Rw R U' R2 U2 R2 U Rw R2 U2 r U2 Rw U2 R U' R U2 R' U' Rw (29,21)
Rw R U' R2 U2 R2 U Rw U2 r U2 Rw R2 U2 R U' R U2 R' U' Rw (29,21)
Rw R U' R2 U2 R2 U r U2 Rw U2 Rw R U R' U2 R U R2 U2 Rw (29,21)
Rw R U' R2 U2 R2 U Rw R U2 r U2 Rw R U R2 U2 R2 U' Rw R (29,21)
Rw R U' R2 U2 R2 U' R U2 Rw R2 U2 r U2 Rw U R U2 R' U' Rw (29,21)
Rw R U' R2 U2 R2 U' R U2 Rw U2 r U2 Rw R2 U R U2 R' U' Rw (29,21)
Rw U' R' U R' U R2 U Rw U2 Rw R U2 Rw U R2 U R' U R' U' Rw (26,22)
Rw U' R' U2 R U Rw R U2 Rw R2 U2 Rw R2 U R' U R2 U R' U' Rw (28,22)
Rw U' R' U2 R U Rw R U2 Rw U2 Rw U2 R2 U' R2 U R' U R' U' Rw (28,22)
Rw U' R' U2 R U Rw R U2 Rw U2 Rw U R2 U R U R U2 R2 U Rw (28,22)
Rw U' R' U2 R U' R2 U2 Rw U2 Rw U2 Rw R U R2 U R' U R' U' Rw (28,22)
Rw U' R' U2 R U' R2 U2 Rw U2 Rw R U2 Rw U R' U R2 U R' U' Rw (28,22)
Rw U' R' U2 R U Rw U2 Rw R U2 Rw U R U2 R' U' R2 U' R2 U Rw (28,22)
Rw U' R' U2 R2 U' R2 U R U Rw U2 Rw U2 Rw R U R U2 R' U' Rw (28,22)
Rw U' R' U R2 U R' U Rw U2 Rw U2 Rw R U R' U2 R U2 R2 U Rw (28,22)
Rw U' R U2 R' U R2 U R U' Rw R2 U2 Rw U2 Rw U R U2 R' U' Rw (28,22)
Rw U' R U2 R U2 R U2 R U Rw R U2 Rw U2 Rw U R U2 R' U' Rw (28,22)
Rw U2 R' U R U2 R' U Rw U2 Rw R U2 Rw U R' U2 R U R' U2 Rw (28,22)
Rw U2 R' U R U2 R' U Rw U2 Rw U2 Rw U R' U2 R U R' U2 Rw R (28,22)
Rw R2 U' R U2 R' U Rw R U2 Rw U2 Rw U R U R2 U R U' Rw R2 (28,22)
Rw R2 U' R U2 R' U Rw U2 Rw U2 Rw U R' U R' U R2 U R U2 Rw (28,22) (Just twists DFR and DBR Corners)
Rw R U R' U2 R' U2 R U Rw U2 Rw R U2 Rw U R' U2 R U' Rw R2 (28,22)
Rw R U Rw' U2 R' U2 Rw U Rw U2 Rw R U2 Rw U R' U2 R U' Rw R2 (28,22)
Rw R U' R2 U R' U R' U Rw U2 Rw U2 Rw R U R2 U2 R2 U' Rw R (28,22)
Rw U' R' U2 R U Rw R2 U2 Rw R2 U2 Rw R U R' U2 R U2 R2 U Rw (30,22)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R2 U' Rw U2 R U2 Rw' U R2 U2 Rw (30,22)
Rw U' R' U2 R U Rw U2 Rw U2 Rw R2 U' R U2 R U2 R' U R2 U2 Rw (30,22)
Rw U' R' U2 R U Rw R U2 Rw U2 Rw U2 R' U R' U2 R2 U2 R2 U' Rw (30,22)
Rw U' R' U2 R U Rw R U2 Rw U2 Rw U2 R' U' R2 U2 R2 U2 R U Rw (30,22)
Rw U' R' U2 R U Rw R U2 Rw U2 R2 U2 R2 U2 Rw U R U2 R' U' Rw (30,22)
Rw U' R' U2 R U Rw R2 U2 Rw R2 U2 Rw R U R' U2 R U2 R2 U Rw (30,22)
Rw U' R' U2 R U' R' U2 Rw U2 Rw R U2 Rw U R2 U2 R2 U2 R U Rw (30,22)
Rw U' R' U2 R U' R' U2 Rw U2 Rw R U2 Rw U' R' U2 R2 U2 R2 U' Rw (30,22)
Rw U' R' U2 R U' R' U2 Rw U2 Rw R U2 Rw U R2 U2 R2 U2 R U Rw (30,22)
Rw R U' R2 U2 R2 U Rw U2 Rw R U2 Rw U' Rw U2 R U2 Rw' U' Rw R2 (30,22)
Rw R U' R2 U2 R2 U Rw U2 Rw R U2 Rw U' R U2 R U2 R' U' Rw R2 (30,22)
Rw R2 U' R U2 R' U Rw U2 Rw U2 Rw R U R U2 R' U2 R2 U Rw R2 (30,22)
Rw U R' U' Rw2 U2 Rw2 U2 Rw U2 Rw' R' U2 Rw U2 Rw2 U2 Rw2 U' R' U Rw (32,22)
Rw U2 Rw2 U2 Rw2 U R' U' Rw U2 Rw R U2 Rw U' R' U Rw2 U2 Rw2 U2 Rw (32,22)
Rw R U' R2 U2 R2 U Rw R U2 Rw U2 Rw U R2 U2 R2 U2 R' U Rw R2 (32,22)
Rw R U' R2 U2 R2 U Rw R U2 Rw U2 Rw U' R U2 R2 U2 R2 U' Rw R2 (32,22)

Double Parity (1 Flip)
Rw U2 Rw U2 Rw' U' Rw2 U2 Rw U' R U' Rw U2 Rw2 R' U2 R U' Rw2 (28,20)
Rw U2 Rw U2 Rw' U' Rw2 R' U2 Rw U' R2 U' Rw R U2 Rw2 R U Rw2 (28,20)
Rw U2 Rw U2 Rw' U' Rw2 R' U2 Rw R U' R U' Rw R2 U2 Rw2 U Rw2 (28,20)
Rw' U' R2 U' Rw2 U R2 Rw U2 Rw' U' R2 U R2 U2 Rw U' Rw2 U2 Rw' (29,20)

Double Parity (3 Flip)
Rw U R' U Rw2 U2 r' U' R' U' Rw R2 U2 Rw' U2 r2 U' R' U' Rw (26,20)
Rw U' R' U' Rw2 U2 Rw' R2 U2 Rw U' R U' r' U2 r2 U R' U Rw (26,20)
Rw U' R' U' Rw2 U2 Rw' R' U2 r U' R U' Rw' R' U2 Rw2 U R' U Rw (26,21)
Rw U2 Rw2 U R U' Rw U2 Rw U2 Rw U2 Rw U2 Rw U' R' U' Rw2 U2 Rw' (29,21)
r U2 Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U R2 U' R' U2 Rw2 U R U' Rw' (29,21)
Rw U2 r2 U R' U Rw' R' U2 Rw2 R U2 Rw U Rw2 U2 Rw U2 Rw U2 Rw2 (31,21)
Rw U2 Rw2 U R2 U' Rw U2 Rw U2 Rw U2 Rw U2 Rw U' R2 U' Rw2 U2 Rw' (31,21)
Rw R U2 Rw2 R' U R U r' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw U2 Rw2 (31,21)
Rw U2 r2 U R' U Rw' U2 Rw2 U2 Rw R U Rw2 U2 Rw U2 Rw R2 U2 Rw2 (32,21)
Rw U2 r2 U R' U' R' U2 Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw U2 Rw2 (32,21)
Rw U2 Rw2 U R2 U Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw U2 r2 U R2 (33,21)
Rw U2 Rw2 R' U R2 U Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw R2 U2 Rw U2 Rw2 (33,21)
Rw U2 Rw2 U R U Rw' U2 Rw2 U2 r U' Rw2 U2 Rw R2 U2 Rw R2 U2 Rw2 (33,21)
Rw R2 U2 Rw2 U R' U Rw' R2 U2 Rw2 U2 r U' Rw2 U2 Rw U2 Rw U2 Rw2 (33,21)
Rw R2 U2 Rw2 U R' U Rw' U2 Rw2 U2 r U Rw2 U2 Rw R2 U2 Rw U2 Rw2 (33,21)
Rw R2 U2 Rw2 U R2 U Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw R2 U2 Rw2 (34,21)
r U2 Rw2 U R2 U Rw' U2 Rw2 U2 Rw U2 R2 U Rw2 U2 Rw U2 Rw U2 Rw2 (34,21)
Rw U2 Rw' U2 Rw2 U' Rw U2 Rw' R' U' R U' Rw U Rw2 U R2 U2 R2 U' Rw2 (31,22)
Rw U2 Rw2 R U R U' R2 U2 Rw' U2 Rw2 U2 Rw U Rw2 U2 Rw U2 Rw U2 Rw2 (34,22)
Rw U2 Rw2 U2 R U' R U Rw' R2 U2 Rw2 U2 Rw U Rw2 U2 Rw U2 Rw U2 Rw2 (34,22)
Rw U2 Rw2 U2 R2 U' R' U Rw' R' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw U2 Rw2 (34,22)
Rw R2 U2 Rw2 U R' U Rw' U2 Rw2 U2 Rw U Rw2 U2 Rw U2 Rw U2 Rw2 U R2 (34,22)
Rw U2 Rw2 U R2 U Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw R2 U2 Rw2 U R2 (35,22)
Rw U2 Rw2 U R2 U' R U2 Rw' U2 Rw2 U2 Rw U Rw2 U2 Rw R2 U2 Rw U2 Rw2 (35,22)
Rw2 U2 Rw U2 Rw U2 Rw2 U Rw U2 Rw2 U2 Rw' U R' U Rw2 U2 Rw R2 U2 R2 (35,22)
Rw2 U2 Rw R2 U2 Rw U2 Rw2 U Rw U2 Rw2 U2 Rw' U R' U Rw2 U2 Rw U2 R2 (35,22)
Rw U2 Rw2 U R2 U Rw' R2 U2 Rw2 U2 Rw R2 U' Rw2 U2 Rw R2 U2 Rw U2 Rw2 (36,22)





Spoiler: Just Last Layer Affected



OLL Parity (1 Flip)
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' (32,25) (Single Dedge Flip)
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' (32,25) (Single Dedge Flip)
Rw' U' Rw2 U' Rw' U Rw' U' Rw' U2 Rw U Rw U' Rw U Rw2 U Rw2 U2 Rw U2 Rw2 U2 Rw (33,25)
Rw' U2 Rw' U2 Rw U Rw' U Rw' U2 Rw2 U Rw2 U' Rw2 U' Rw' U' Rw U Rw' U2 Rw2 U2 Rw (34,25)
Rw' U2 Rw' U2 Rw U Rw' U Rw U' Rw2 U' Rw2 U2 Rw2 U' Rw U' Rw U Rw' U2 Rw2 U2 Rw (34,25)
Rw' U2 Rw' U2 Rw U Rw' U Rw U Rw2 U2 Rw2 U' Rw2 U Rw U' Rw U Rw' U2 Rw2 U2 Rw (34,25)
Rw' U Rw' U' Rw' U2 Rw2 U2 Rw2 U' Rw U2 Rw U2 Rw' U2 Rw' U' Rw2 U Rw2 U Rw' U2 Rw (35,25)
Rw' U Rw' U' Rw' U2 Rw2 U2 Rw2 U Rw U2 Rw U2 Rw' U2 Rw' U Rw2 U Rw2 U Rw' U2 Rw (35,25)
Rw' U Rw' U' Rw' U2 Rw2 U2 Rw2 U Rw U2 Rw U2 Rw' U2 Rw U' Rw2 U' Rw2 U' Rw U2 Rw (35,25)
Rw' U Rw' U' Rw U' Rw2 U2 Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw' U' Rw2 U Rw2 U Rw' U2 Rw (35,25)
Rw' U Rw' U' Rw U Rw2 U2 Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw' U Rw2 U Rw2 U Rw' U2 Rw (35,25)
Rw' U Rw' U' Rw U Rw2 U2 Rw2 U2 Rw' U2 Rw U2 Rw' U2 Rw U' Rw2 U' Rw2 U' Rw U2 Rw (35,25)

OLL Parity (3 Flip) (These just twist and flip)
Rw U Rw U2 Rw U2 Rw U Rw' U2 r' U2 Rw' U Rw U2 Rw U2 Rw U Rw (27,21) (Pure 3 Flip)
Rw U' Rw U2 R' U2 Rw' U' Rw U2 Rw R U2 Rw U' Rw' U2 R' U2 Rw U' Rw (28,22)***This alg equals its reverse!
Rw U' R U2 R' U2 R' U' Rw U2 Rw R U2 Rw U' Rw' U2 R' U2 Rw U' Rw (28,22)
Rw U' R U2 R' U2 R' U' Rw U2 Rw R U2 Rw U' R' U2 R' U2 R U' Rw (28,22) 

Double Parity (1 Flip)
Rw' U Rw' U' Rw2 R' U' Rw' U' R2 U2 Rw' R' U' Rw' U2 Rw' U2 Rw' (24,19)
Rw' U2 Rw' U2 Rw R U Rw2 R' U2 Rw' R' U R U Rw' U2 Rw2 U' Rw2 (27,20)
Rw' U' Rw' U' Rw U Rw U2 Rw2 U2 Rw U2 Rw U Rw' U Rw U Rw2 U2 Rw' U2 Rw U2 Rw' (33,25)
Rw' U Rw2 U Rw2 U2 Rw U2 Rw' U2 Rw U Rw U Rw U2 Rw' U2 Rw' U2 Rw2 U' Rw2 U Rw' (35,25)

Double Parity (3 Flip)
Rw U' Rw U' Rw2 U Rw' U2 Rw U2 Rw' U Rw' U' Rw U2 Rw U2 Rw U2 Rw U' Rw' U2 Rw' U2 Rw (35,27)





Spoiler: Petrus (Scrambles U and R faces)



(All of these were in the 3 flip text document)
OLL Parity
r U' R2 U2 R2 U r U2 Rw U2 r U R U2 R' U' Rw (23,17)
r U' Rw U2 R' U R U Rw' U' r U2 Rw U2 Rw U R' U Rw (22,19)
r U2 R' U R' U2 R U Rw U2 Rw U2 r U R U2 R' U' Rw (24,19)
r U R2 U Rw R U2 Rw U2 r U' Rw' U R U R' U2 Rw U' Rw (24,20)
r U2 R U R U2 R' U Rw R2 U2 Rw U2 r U R U2 R' U' Rw (26,20)
r U' R2 U2 R2 U r U2 Rw U2 Rw R U R' U2 R U R2 U2 Rw (28,20)
r U' R2 U2 R2 U r U2 Rw U2 Rw R2 U R2 U2 R2 U R U2 Rw (30,20)
r U' R2 U2 R2 U Rw R2 U2 r U2 Rw U2 R U' R U2 R' U' Rw (28,20)
r U' R2 U2 R2 U Rw U2 r U2 Rw R2 U2 R U' R U2 R' U' Rw (28,20)
r U' R2 U2 R2 U' R U2 Rw U2 r U2 Rw R2 U R U2 R' U' Rw (28,20)
r U' R2 U2 R2 U' R U2 Rw R2 U2 r U2 Rw U R U2 R' U' Rw (28,20)
r U2 R' U R U2 R' U Rw U2 Rw U2 Rw U R' U2 R U R' U2 Rw (27,21)
Rw R U2 Rw2 U Rw' U2 Rw U R2 U Rw' R2 U' Rw2 R' U2 Rw2 U R U' Rw' (30,22)

Double Parity
Rw R U2 Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U R2 U' R' U2 Rw2 U R U' Rw' (30,22)





Spoiler: Complete Petrus Set



Here is an additional set of petrus algorithms which include all previous algorithms listed (including the Petrus section above) which:
1) Began or ended with an inner layer slice turn--I made those wide turns
and/or
2) Began or ended with outer layer slice turns--I omitted all of them

OLL Parity
Rw U' R' U2 R U r U2 r U2 r U R' U2 R U' Rw (21,17)
Rw U' R U2 R' U Rw U2 Rw U2 Rw U R' U2 R U' Rw (21,17) (This is the same alg I found and presented in my last post)
Rw U' R2 U2 R2 U r U2 Rw U2 r U R U2 R' U' Rw (23,17)
Rw U' R2 U2 R2 U r U2 r U2 r U R2 U2 R2 U' Rw (25,17)
Rw U' R U2 R' U r U2 Rw R U2 r U R' U2 R U' Rw (22,18)
Rw U' R' U2 R U Rw U2 r U2 Rw R2 U R' U2 R U' Rw (23,18)
Rw U' R U2 R' U Rw U2 r U2 Rw R2 U R U2 R' U' Rw (23,18)
Rw U' R U2 R' U Rw U2 Rw R2 U2 Rw U R' U2 R U' Rw (23,18)
Rw U' R2 U2 R2 U r U2 Rw R2 U2 r U R' U2 R U' Rw (25,18)
Rw U R' U' Rw2 U2 R Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 Rw' (25,18)
Rw U R U' Rw2 R U2 Rw2 U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw' (25,18)
Rw U' R2 U2 R2 U Rw R2 U2 r U2 Rw U R2 U2 R2 U' Rw (27,18)
Rw U R' U Rw U2 r U2 r U' Rw' U R U R' U2 Rw U' Rw (22,19)
Rw U R' U r U2 r U2 Rw U' Rw' U R U R' U2 Rw U' Rw (22,19)
Rw U' Rw U2 R' U R U Rw' U' r U2 Rw U2 Rw U R' U Rw (22,19)
Rw U' R' U2 R U Rw R U2 r U2 Rw R U R' U2 R U' Rw (23,19)
Rw U' Rw U2 R' U R U Rw' U' Rw U2 r U2 Rw U R2 U Rw (23,19)
Rw U' R' U2 R U' R' U2 r U2 Rw U2 r U R' U2 R U' Rw (24,19)
Rw U' R U2 R' U' R' U2 r U2 Rw U2 r U R U2 R' U' Rw (24,19)
Rw U' R' U2 R U r U2 Rw U2 Rw U R U2 R' U R' U2 Rw (24,19)
Rw U' R' U2 R U Rw U2 r U2 Rw U R U2 R' U R2 U2 Rw (25,19)
Rw U' R' U2 R U Rw R2 U2 Rw U2 Rw R2 U R U2 R' U' Rw (25,19)
Rw U' R' U2 R U Rw R U2 Rw R U2 Rw U R2 U2 R2 U' Rw (25,19)
Rw U' R' U2 R U' R2 U2 Rw U2 Rw U2 Rw U R' U2 R U' Rw (25,19)
Rw U' R U2 R' U Rw U2 r U2 Rw U R' U2 R U R2 U2 Rw (25,19)
Rw U' R U2 R' U2 R2 U' Rw U2 Rw U2 r U R U2 R' U' Rw (25,19)
Rw U' R' U2 R U r U2 Rw U2 Rw U R2 U2 R2 U2 R U Rw (26,19)
Rw U' R' U2 R U r U2 Rw U2 Rw U' R' U2 R2 U2 R2 U' Rw (26,19)
Rw U' R' U2 R U r U2 r U2 Rw U R2 U2 R2 U R' U2 Rw (26,19)
Rw U' R' U2 R U Rw U2 r U2 r U R2 U2 R2 U R' U2 Rw (26,19)
Rw U' R2 U2 R2 U' R U2 Rw U2 Rw U2 Rw U R' U2 R U' Rw (26,19)
Rw U' R2 U2 R2 U Rw U2 Rw U2 Rw U2 R U' R' U2 R U' Rw (26,19)
Rw U' R2 U2 R2 U Rw U2 Rw U2 Rw U2 R' U' R U2 R' U' Rw (26,19)
Rw U R' U' R Rw2 U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 R' U2 Rw' (26,19)
Rw U R' U' Rw2 R' U2 Rw2 R' U' Rw U2 Rw' U2 Rw U Rw2 U2 Rw' (26,19)
Rw U' R2 U2 R2 U2 R' U' Rw U2 Rw U2 r U R U2 R' U' Rw (26,19)
Rw U' R2 U2 R2 U Rw R U2 r U2 Rw R U R2 U2 R2 U' Rw (27,19)
Rw U' R2 U2 R2 U' R2 U2 Rw U2 Rw U2 Rw U R2 U2 R2 U' Rw (29,19)
Rw U' Rw U2 R' U R U Rw' U' Rw U2 r U2 Rw R U R U Rw (23,20)
Rw U' Rw U2 R' U R U Rw' U' Rw R U2 r U2 Rw U R U Rw (23,20)
Rw U' R U2 R' U Rw R U2 Rw U2 Rw U R U R2 U R U' Rw (24,20)
Rw U' R U2 R' U Rw U2 Rw U2 Rw R U R2 U R' U R' U' Rw (24,20)
Rw U' R U2 R' U Rw U2 Rw R U2 Rw U R' U R2 U R' U' Rw (24,20)
Rw U R2 U Rw R U2 Rw U2 r U' Rw' U R U R' U2 Rw U' Rw (24,20)
Rw U' Rw U2 R' U R U Rw' U' r U2 Rw R2 U2 Rw U R' U Rw (24,20)
Rw U R U2 R U2 R' U Rw U2 Rw U2 Rw R U R' U2 R U' Rw (25,20)
Rw U R' U2 R' U2 R U Rw U2 Rw R U2 Rw U R' U2 R U' Rw (25,20)
Rw U Rw U2 R U2 Rw' U Rw U2 Rw U2 Rw R U R' U2 R U' Rw (25,20)
Rw U Rw' U2 R' U2 Rw U Rw U2 Rw R U2 Rw U R' U2 R U' Rw (25,20)
Rw U' R U2 R' U Rw U2 Rw U2 Rw R U R U2 R' U2 R2 U Rw (26,20)
Rw U' R' U2 R U r U2 Rw U2 Rw R2 U R' U2 R U R U2 Rw (26,20)
Rw U' R2 U R' U R' U Rw U2 Rw U2 Rw R U R2 U2 R2 U' Rw (26,20)
Rw U' R U2 R' U Rw R U2 r U2 Rw U R2 U2 R2 U R U2 Rw (27,20)
Rw U' R U2 R' U Rw U2 r U2 Rw R U R2 U2 R2 U R U2 Rw (27,20)
Rw U' R' U2 R U Rw R2 U2 Rw U2 Rw U R' U2 R U R2 U2 Rw (27,20)
Rw U' R2 U2 R2 U Rw U2 Rw R U2 Rw U' R U2 R U2 R' U' Rw (27,20)
Rw U R2 U2 R U2 R' U Rw U2 Rw R2 U2 Rw U R U2 R' U' Rw (27,20)
Rw U' R2 U2 R2 U Rw U2 Rw R U2 Rw U' Rw U2 R U2 Rw' U' Rw (27,20)
Rw U' R2 U2 R2 U r U2 Rw U2 Rw R U R' U2 R U R2 U2 Rw (28,20)
Rw U' R2 U2 R2 U Rw R2 U2 r U2 Rw U2 R U' R U2 R' U' Rw (28,20)
Rw U' R2 U2 R2 U Rw U2 r U2 Rw R2 U2 R U' R U2 R' U' Rw (28,20)
Rw U' R2 U2 R2 U' R U2 Rw R2 U2 r U2 Rw U R U2 R' U' Rw (28,20)
Rw U' R2 U2 R2 U' R U2 Rw U2 r U2 Rw R2 U R U2 R' U' Rw (28,20)
Rw U' R2 U2 R2 U Rw R U2 Rw U2 Rw U R2 U2 R2 U2 R' U Rw (29,20)
Rw U' R2 U2 R2 U Rw R U2 Rw U2 Rw U' R U2 R2 U2 R2 U' Rw (29,20)
Rw U' R2 U2 R2 U r U2 Rw U2 Rw R2 U R2 U2 R2 U R U2 Rw (30,20)
Rw U' R U2 R' U Rw U2 Rw U2 Rw U R' U R' U R2 U R U2 Rw (26,21)
Rw U2 R' U R U2 R' U Rw U2 Rw U2 Rw U R' U2 R U R' U2 Rw (27,21)
Rw U2 Rw2 U Rw' U2 Rw U R2 U Rw' R2 U' Rw2 R' U2 Rw2 U R U' Rw' (29,21)

Double Parity
Rw' U2 Rw2 U Rw U2 Rw' U2 Rw U' Rw2 U2 R Rw2 U' R U Rw (25,18)
Rw U' R2 U' r2 U2 Rw' U2 Rw U' R2 U' Rw' U2 Rw2 U R U Rw (26,19)
Rw U2 Rw2 U R' U Rw' U2 Rw2 U2 Rw U Rw2 U2 Rw U2 Rw U2 Rw2 (29,19)
Rw U2 Rw2 U R2 U Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw U2 Rw2 (30,19)
Rw U2 Rw2 R' U R U r' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw U2 Rw2 (30,20)
Rw U2 Rw2 U R' U Rw' U2 Rw2 U2 r U Rw2 U2 Rw R2 U2 Rw U2 Rw2 (31,20)
Rw U2 Rw2 U R' U Rw' U2 Rw2 U2 Rw U Rw2 U2 Rw U2 Rw R2 U2 Rw2 (31,20)
Rw U2 Rw2 U R' U Rw' R2 U2 Rw2 U2 r U' Rw2 U2 Rw U2 Rw U2 Rw2 (31,20)
Rw U2 Rw2 U R2 U Rw' U2 Rw2 U2 Rw U' Rw2 U2 Rw U2 Rw R2 U2 Rw2 (32,20)
Rw U2 Rw2 U Rw' U2 Rw U2 Rw' U' Rw2 U R2 U' R' U2 Rw2 U R U' Rw' (29,21)
Rw U2 Rw2 U R2 U Rw' U2 Rw2 U2 Rw U2 R2 U Rw2 U2 Rw U2 Rw U2 Rw2 (34,21)


I couldn't find Stefan's (24,19) Petrus Double Parity algorithm. Therefore Kåre's algorithms are only a subset of 22 ftm and shorter OLL Parity and Double Parity algorithms in <U, Rw, R>.

I am going to add all Petrus solutions which are below 25q and below 20 btm to the 4x4x4 parity algorithms page in the wiki. I'll be sure to provide a link to this page to give the complete set of algorithms.


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## qqwref (Dec 11, 2012)

cmowla said:


> Rw' U R' U2 R U' Rw' U2 Rw' U2 Rw' U' R U2 R' U Rw' (21, 17) (Not double parity).


This alg is EXTREMELY good for OBLBL.


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## Christopher Mowla (Dec 12, 2012)

qqwref said:


> This alg is EXTREMELY good for OBLBL.


I'm glad I finally found an alg that someone likes for speed! However, before the fact that I started looking at Kåre's algorithms in detail, I didn't realize that he actually had that algorithm in his 3 Flip text document. It was just surrounded by beginning and ending outer layer turns. 

However, at the same time that I found that alg, I tested alternate but related paths. Here's the best one I found that uses L turns as well and could possibly be even faster than that alg.

Rw' U' L' U2 L U Rw' U2 Rw' U2 Rw' U L' U2 L U' Rw' (21,17)


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