4x4 Parity Algorithms

[reThinking the Cube]

Not all odd permutations are parity, and not all parity are an odd number of odd permutations.

 

[Hershey]

Not all odd permutations are parity, and not all parity are an odd number of odd permutations.

Then what is parity?

 

[ianography]

A good PLL parity algorithm is Uw2 r2 U2 r2 U2 r2 Uw2. I use it sometimes

 

[reThinking the Cube]

it's an swap of two pieces.

OLL parity is parity in direct solve methods, or K4 ELL, but PLL parity is not, as its two two swaps

Technically, (lol) there can be no parity within the context of a direct-solve (unless the cube is in an illegal state requiring disassembly to solve correctly). Sure, you may get and recognize a direct solve case as being identical to some OLL/PLL reduction state parity case, but that does NOT make it a parity in the context of a direct-solve. When direct-solving, that reduction state OLL parity is simply just another 2-cycle of edges, with or without some usually unnoticed center piece cycles.

 

[Christopher Mowla]

rU2r'U2r'U2lU2r'U2rU2F2r2F2l' is a direct solve parity.

Here's an entirely different algorithm for that case.
Uw' Lw Uw l Uw' Rw' Uw Rw2 Bw l Bw' Lw L Uw' Rw z (16q, 15 btm)

Here are other conventional algorithms.
l2 F2 U2 l' U2 l2 F2 l' U2 l2 U2 F2 l' F2 (25q, 14 btm)
Lw U2 l' U2 l' U2 r U2 l' U2 l U2 r' l U2 l2 U2 Lw' (27q, 18 btm)

Not sure if this is faster, but you might like it:
Rw' U R U Lw' U2 Rw' U2
r2
U2 Rw U2 Lw U' R' U' Rw

OMG alg.

Does this mean you like it?

 

[Jorghi]

Technically, (lol) there can be no parity within the context of a direct-solve (unless the cube is in an illegal state requiring disassembly to solve correctly). Sure, you may get and recognize a direct solve case as being identical to some OLL/PLL reduction state parity case, but that does NOT make it a parity in the context of a direct-solve. When direct-solving, that reduction state OLL parity is simply just another 2-cycle of edges, with or without some usually unnoticed center piece cycles.

Yeah this makes sense. People act like Parity is some legendary thing when its just part of the solve.

 

[Hershey]

Here's an entirely different algorithm for that case.
Uw' Lw Uw l Uw' Rw' Uw Rw2 Bw l Bw' Lw L Uw' Rw z (16q, 15 btm)

Here are other conventional algorithms.
l2 F2 U2 l' U2 l2 F2 l' U2 l2 U2 F2 l' F2 (25q, 14 btm)
Lw U2 l' U2 l' U2 r U2 l' U2 l U2 r' l U2 l2 U2 Lw' (27q, 18 btm)



Does this mean you like it?

Yes. Also, how do you find all of these 4x4 algorithms?

 

[Christopher Mowla]

Yes. Also, how do you find all of these 4x4 algorithms?

The first one is a one move conjugation of the base for "The Holy Grail of Pure Edge Flip Algorithms." So that one was by hand. The second algorithm (25q/14 btm) I found with cube explorer using the center facelet twist option (the set-up was an N-Perm). The last algorithm I also found by hand, using the same techniques I have shown in my methods parity thread in puzzle theory.

EDIT:
Also, for the adjacent PLL parity algorithm, I found it by hand just using the technique of conjugating the move r2. As I have posted here in the K4 thread, here is a link to a large collection of 2 2-cycles using this technique.
View attachment 2_2-Cycles.txt
That technique has obviously been known a while, due to the prior existence of the algorithms [r2 F2 U2: r2] and [Uw2 r2 U2: r2]. I just looked at those algorithms in a different way to see the bigger picture. Under special circumstances, we can stretch this technique even to handle 4 2-cycles by conjugating (l2 r2) (or M2 on the 4x4x4). For example, [F2 r2 F2 U' M2 U: M2].

Also,
Here are possible optimal algorithms (in quarter turns, at least) for all of the other LL 2-cycle cases. (All found by hand, of course).

Opposite [Prior publicly known minimum until this post, 19q]
y r u Rw' Uw' r' Uw Lw Uw' r2 Fw' r' Fw r' Uw f' r' x y' (17q, 16 btm) (Optimal Algorithm in btm is 12).

Unoriented Adjacent 1 (minor modification of the HG Edge Flip Alg) [Prior publicly known minimum until this post, 21q] (Optimal alg in btm is 15)
z Dw' (M R') D Lw' Uw' r' Uw Lw Uw' Lw'2 Bw' r' Bw Rw' R' u y' (M' R) Uw x2 z' (19q, 18 btm)

Unoriented Adjacent 2 [Prior publicly known minimum until this post, 21q] (optimal alg in btm is 15).
Rw b' Dw r2 x' Uw' r' Uw x' r' Uw Lw' Uw' r' Uw Lw Uw2 x' d Rw' x' (19q, 17 btm)

Oriented Adjacent [Prior publicly known minimum until this post, 21q]
x' Uw l Uw' Rw2 x Uw' l' Uw Lw' L' x Uw Rw' Uw' l' Uw (Rw l') Uw' x (17q, 16 btm)

 

[Nico1]

I don't want to take credit for this extremely easy and fast algorithm because it is the creation of my friend, but here it is:

Lw U2 Lw U2 Lwi U2 Lw U2 Rwi U2 Lw U2 Lwi U2 Lw2 Rw U2 Lwi

This alg simply turns the UB dedge "in its socket."

 

[Rpotts]

Most people would use y2 r U2 x r U2 r U2 r' U2 l U2 r' U2 r U2 r' U2 r'

aka Lucas Parity. Your friend may have stumbled upon that, but it is already known.

Also, whoah 3 year necro.

 

[cubernya]

Most people would use y2 r U2 x r U2 r U2 r' U2 l U2 r' U2 r U2 r' U2 r'

aka Lucas Parity. Your friend may have stumbled upon that, but it is already known.

Also, whoah 3 year necro.

I prefer r' U2 r U2 r' F2 r2 U2 r U2 r' U2 x U2 r2 U2 x'
Or F2 r2 F2

 

[Nico1]

Most people would use y2 r U2 x r U2 r U2 r' U2 l U2 r' U2 r U2 r' U2 r'

aka Lucas Parity. Your friend may have stumbled upon that, but it is already known.

Also, whoah 3 year necro.

Okay thanks! Didn't know that had already been discovered (although i figured.) I prefer this way though because my sexy move on 3x3 M2i U M2i U Mi U2 M2i U2 Mi U2. I perform this with my left ring finger. Therefore, an Lwi is much easier for me than an Rw, also for some reason (I'm not left handed) Lw is easier with the left index finger than Rwi with the right index finger (probably because of the U2 not U2i that follows.)

Also, this might seem like a dumb question, but what is "3 year necro."

 

[vcuber13]

i assume its referring to the bump

 

[Rpotts]

3 year necro (necro bump) means you replied to a thread that hadn't been posted to in 3 years, you bumped a 3 year old thread to the front page.

 

[BlackStahli]

I personally like Rw' U2 Lw F2 Lw' F2 Rw2 U2 Rw U2 Rw' U2 F2 Rw2 F2, but there are probably better ones.

I use that one too^
However, these days I see all the fast 4x4 solvers like Mats and Feliks use the Lucas parity. Is that because of preference or just because it has a faster execution? I tried both but they're both equally fast for me, but the Lucas parity algorithm just doesn't feel as natural as the other one...

 

[CubezUBR]

4x4 pll. parity not seen before (help)

So im new to 4x4 and found this parity/pll on the last layer. the whole cube is solved but the 2 edges are switched on 2 faces next to each other. here is a pciture
http://imgur.com/B2QJhFW
can you please tell me a algorithm to solve this or a link

 

[JasonK]

R B (r U2 r U2 F2 r F2 l' U2 l U2 r2) B' R'
View at alg.garron.us

 

[Cubenovice]

Makes me wonder which method you used?
Or did you just forgot / did not know how to pair up the last two edges?

 

[CubezUBR]

i use reduction method for the first 3 layers but then use oll/pll to do the last layer. it works usually when i dont get this case and some others that i work out how to solve

 

[scottishcuber]

i use reduction method for the first 3 layers but then use oll/pll to do the last layer. it works usually when i dont get this case and some others that i work out how to solve

I'm curious, what do you average with that?