New 4x4 parity algs using R, Rw, U

[Kare]

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.

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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'

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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

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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

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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.

[masterofthebass]

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.

[joey]

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

[irontwig]

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

[Kare]

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.
3FLIP.TXT

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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'

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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

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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

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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

[cmowla]

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'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>?

[Kare]

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'

[cuBerBruce]

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.

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.

[Kare]

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/

[tx789]

I need to handle parity better