# Methods for Forming 2-Cycle Odd Parity Algorithms for Big Cubes

#### Christopher Mowla

The Holy Grail Pure Edge Flip Parity Algorithm Finally Matched!

Today was my annual search day for a possible 18q single dedge flip algorithm. (I still search about once a year.)

I didn't find an 18q, but, after more than 6 years, today I found a single dedge flip algorithm which matches the move count of the Holy Grail algorithm for the nxnxn!

It is the same quantity of block half turns and block quarter turns for all (and the same) cases, but it is actually shorter than the Holy Grail alg for the nxnxn in the:
• single slice turn metric
• plane half turn metric (OBTM)
• plane quarter turn metric (QTM)

Since the Holy Grail alg was (one of) the shortest algorithms in QTM (for all size cubes), we now have an algorithm that is fewer QTM than any other algorithm in history! This algorithm is 22 QTM on the 4x4x4 and 5x5x5, and 24 QTM on individual orbits on the 6x6x6 and 7x7x7! (It's 22 QTM for nxnxn cases like the ones I posted images of in this post.)

In addition to its move count being shorter in the three above move metrics than the Holy Grail (and matching the lengths of the other two move metrics),

• This algorithm is the first single dedge flip algorithm I have ever seen which can be reverted (not just inverted) to achieve the exact same affect.
• This algorithm, unlike the Holy Grail and all 19q algorithms I have ever found, only exchanges center pieces in the top face on the supercube.

May I represent, "Challenger", and its twin, "Reverter"!
"Challenger in its general form,
Rw Uw' L Dw' R Uw' r Uw R' r' Dw LW2 Dw' r Dw LW2 L' Uw Rw'
= [Rw Uw' L Dw': [R Uw': (r)] [r', Dw LW2 Dw'] ]

and here is its move inverted "twin", "Reverter" (the inverse of its reverse),
Rw' Uw L' Dw R' Uw r' Uw' R r Dw' LW2 Dw r' Dw' LW2 L Uw' Rw
= [Rw' Uw L' Dw: [R' Uw: (r')] [r, Dw' LW2 Dw] ]

• LW2 means to turn the left half of the cube on even cubes and the left half of the cube + the central slice on odd cubes.
• The bold r moves indicate the slices that you apply to the orbits of wings you wish to affect.

Now, I will link (Challenger: you can just invert all moves without changing their locations to get Reverter's animations) to alg.cubing.net in SiGN notation for each cube size (with moves simplified specific to each case).

4x4x4
r u' L d' R u' 2R u r' d l2 d' 2R d l 2L u r'

5x5x5
r u' L d' R u' 2R u r' d 3l2 d' 2R d 3l 2-3l u r'

6x6x6 (outer orbit)
3r 3u' L 3d' R 3u' 2R 3u 2r' 3d 3l2 3d' 2R 3d 3l 2-3l 3u 3r'

6x6x6 (inner orbit)
3r 3u' L 3d' R 3u' 3R 3u R' 3R' 3d 3l2 3d' 3R 3d 3l 2-3l 3u 3r'

7x7x7 (outer orbit)
3r 3u' L 3d' R 3u' 2R 3u 2r' 3d 4l2 3d' 2R 3d 4l 2-4l 3u 3r'

7x7x7 (inner orbit)
3r 3u' L 3d' R 3u' 3R 3u R' 3R' 3d 4l2 3d' 3R 3d 4l 2-4l 3u 3r'

Using the same conjugate of the quarter turn idea, I found a (20,17) even cube algorithm.

• This first single edge flip algorithm with this ratio of block quarter turns to block half turns.
• Reverting (in addition to inverting) produces a working algorithm as well (but it flips UB instead of UF).
• This algorithm is only 20q on the 4x4x4 and on the nxnxn even cube when consecutive slices from the corners need to be affected.
General Form:
Lw E Lw2 Dw r' Dw' Lw2 Dw R2 Uw r' Uw' R2 r Dw' E' Lw'
= [Lw E: [Lw2, Dw r' Dw'] [Dw r': [R2 Uw: (r')] ] ]

On the 4x4x4
l e r2 u 2R' u' r2 d R2 u 2R' u' r R d' e' l'

On the 6x6x6 outer orbit
3l e 3r2 3u 2R' 3u' 3r2 3d R2 3u 2R' 3u' 2r R 3d' e' 3l'

And its "twin"'s general form (this flips the back dedge)
Lw' E' Dw' r R2 Uw' r' Uw R2 Dw Lw2 Dw' r' Dw Lw2 E Lw

I will post these (four) new algorithms in the wiki soon.

#### Christopher Mowla

I have contacted Ben, and he was curious about what mental process I used to find Challenger and Reverter, and I just decided to make a brief (well, brief for me) video about this and my comments (which I already have basically written in my previous post).

Enjoy!

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#### Christopher Mowla

18q Single Dedge Flip Algorithm Found for the 4x4x4!

Am I dreaming? I must be...

I just found an 18q single dedge flip algorithm for the 4x4x4 from my original method! (It's ironic that I really don't care about this that much anymore that I finally found it. I thought I--or someone else would have to--do a computer search of sorts to find one, if I ever did.)
b' D' 2U' r' 2R' f' 2R' f 2R' R d b' r' 2B' r b e' b (18,18)

It's only 18q on the 4x4x4 though. It is 19q (proportionally) on all even size cubes (due to the presence of an interior setup move), but it doesn't work at all on odd cubes, despite that where I got it from did originally work on odd cubes.

Story
I perhaps rediscovered an uncommon outer-most setup move conjugation (like I previously used for the 19q l' F' 2B' r' b' r 2U r' b r 2U2 b' 2U b 2U 2B F l posted in the wiki) to a specific base form. I found that the base algorithm I had wasn't ideal when I rediscovered this setup, but I had a hunch, and after looking at my algs for other 2-cycle types on the 4x4x4 parity algorithms wiki page tonight, I spotted this one that I found in the past.

And it worked. The base alg for this 18q is from "[Fw Uw': [Uw Rw' Fw', r'] [Rw' (1-2r)2 Fw': r'] ] (17,17 for large cubes and 16,16 for 4x4x4 and 5x5x5)" in my "A Collection of Algorithms" PDF, which can be downloaded in this post.

6x6x6 (inner orbit)
3b' D' 2-3u' 2-3r R' 3R 3f' 3R' 3f 2-3r' R 3d 3b' 3r' 3B' 3r 3b e' 3b (19,19)

6x6x6 (outer orbit)
3b' D' 2-3u' 2-3r R' 2R 3f' 2R' 3f 2-3r' R 3d 3b' 3r' 2B' 3r 3b e' 3b (19,19)

On the 4x4x4,
= b' D' 2U' 2R R' 2R f' 2R' f 2R' R d b' r' 2B' r b E' b
= b' D' 2U' 2R' R' 2R' f' 2R' f 2R' R d b' r' 2B' r b E' b
= b' D' 2U' r' 2R' f' 2R' f 2R' R d b' r' 2B' r b e' b (18,18)

Decomposition (on the outer orbit of the 6x6x6)
3b' U D'
3u' 2R
3r' 2-3r2 3f' 2R' 3f 2-3r2 3r
3u 3r' 3f' 2R' 3f 3r
D U' 3b

= [3b' U D' 3u' 2R: [3r' 2-3r2 3f': (2R')] [3u 3r' 3f', 2R'] ]

In short, I did all of the hard work years ago, but I didn't put two and two together to get what I was after. Then, several years after the fact, I guess I was "distant enough" from all of the previous chaos (after the dust settled) to be able to spot a destined to be match just by sheer luck (this was the second algorithm I looked at in the wiki: I could have potentially looked at several more before I would have found one that worked).
[HR][/HR]People like Ben and qqwref (and anyone else following my work) can try to find an alg that is 18q on the 4x4x4 and 5x5x5 and 19q on all size cubes (it could exist).

Until either I or someone does this, the Holy Grail, Challenger, and Reverter still claim the number one seats for the BQTM move metric on the nxnxn.

#### TMarshall

##### Member
Congratulations! This is really cool, even though I know practically nothing about cube theory. Maybe this will motivate me to learn more about the cube

#### G2013

##### Member
Christopher, as your cube notation is not the standard one, I usually don't get it :S
Is 2U the same as Uw, and u the same as u? Or how is it? Also, what does "e" mean? I always get a scrambled cube when I try to perform some of your algs :/

But, as I understand, this is a quite impressive discovery, so congratulations!

#### ryanj92

##### Member
Christopher, as your cube notation is not the standard one, I usually don't get it :S
Is 2U the same as Uw, and u the same as u? Or how is it? Also, what does "e" mean? I always get a scrambled cube when I try to perform some of your algs :/

But, as I understand, this is a quite impressive discovery, so congratulations!
Can you not view the alg.cubing.net links?
To answer your questions, 2U is the inner slice only. e is both inner slices in the E direction (I guess it is lower case because E and e would mean different things on 5x5x5 and up)

#### qqwref

##### Member
Wow, congrats, cmowla! Amazing find. It's so weird how the moves are seemingly random, and yet everything but those two pieces comes back together in the end...

#### rokicki

##### Member
Wow, congrats, cmowla! Amazing find. It's so weird how the moves are seemingly random, and yet everything but those two pieces comes back together in the end...
Congratulations; that's really awesome! Maybe this will inspire some more explorations.

#### bcube

##### Member
I don't realize the aspects of your finding, but based on the responses of others, you have pushed the limits of some sort (again). Hats off to you, sir. Well done.

#### Christopher Mowla

Again, thanks everyone for the compliments!

I don't realize the aspects of your finding, but based on the responses of others, you have pushed the limits of some sort (again). Hats off to you, sir. Well done.
History of 4x4x4 Single Dedge Flip Algorithms

Notes

• I found the 18-24 algorithms by hand, and I assume that the 25 was found using a 3x3x3 setup on an optimal 3x3x3 solver. I was born in the late 80s, and thus I do not have a way of knowing when the first brief single dedge flip algorithm was discovered (and by whom).
• To give an accurate account, I listed the original algorithms that I found for every given length, including the exact cube rotations that were present when I derived them.
• I did not post any of the 24-19 move algorithms until after I started my methods thread. (I don't think I ever posted this 23 anywhere because it's hideous in comparison to others that I found).
• It turns out that if we would have used a program such as Cube Explorer, we could technically have found at least two 22s (and A LOT of 23s) if we would know how to translate them to the 4x4x4 properly.
r' U2 B 2R' B' U2 F2 l' B2 2R' B2 l B 2R F2 B' r

r' U2 B 2R' B' U2 F2 r' U2 2R' U2 r B 2R F2 B' r

(It's also possible that Clément Gallet's 4x4x4 solver could have also found these 22s.)
• Also, make no mistake, we could technically use 3x3x3 solvers to find the "base 2-cycle algorithm" for probably every algorithm I ever found using this "wide turn based method".
• After I found my 19q, I found evidence that cubers such as Chris Hardwick (the coolest cuber ever?) were very close (if they did not discover these type of wide-turn based algorithms for these 2 wing edge swaps and didn't notice what they had).
For example, all he had to do is convert the U face turns and the F turn in the algorithm:
(Rr) U (Rr)' U' (Rr)' F (Rr)2 U' (Rr)' U' (Rr) U (Rr)' (Ff)'
to inner slice turns, and he would have had what I have been calling a "wide turn base algorithm".

In fact, my 24, 23, and 22 were actually just a conjugation of this setup (but I created a different algorithm which generated this setup). (The minimum number of required setup block quarter turn moves is 5, and thus 22 is the minimum--that is if you have a move cancellation between the base and the setup moves.)
• I found the 19 and 20 via cyclic shifting (which I discovered/invented on my own--independently of any external sources--at the time when I found the 20: I didn't know what group theory was, nor was I well-versed in past forum posts.)
• Now that I recall, I found the 18q's base by including a face turn as part of the conjugation of the quarter turn (just as I did for Challenger and Reverter).

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#### Christopher Mowla

Hi everyone. (I know for many of the currently active members will probably not know of me or this thread, so this ought to be fun!)

It's been about two and a half years since I updated this thread, but recently I resumed my work on what I define as wide turn based 2-cycle parity algorithms in the Wiki.

Perhaps older members might have noticed that I have added many more brief algorithms of this type (for all last layer 2-cycle cases) to the Wiki this year (the Wiki page seemed to disappear with the upgrade: here is a backup until it comes back up.), and the following 45 minute video explains why. In addition, with the procedure I introduce in the video, I found probably one of the (if not the) largest collection of 4x4x4 2-cycle parity algorithms ever found. (Although most are useless on probably all accounts.) (A download link to this set is included at the bottom of the video description.)

(Also, I apologize about the video quality. Should have been more careful with the screen-recorder region! I should know better. But, oh well. It's still watchable.)

I present a challenge at the end of the video (for those who are interested to participate/try). I will take submissions of 2-cycle seed algorithms either in the video comments or I suppose in replies to this post.

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#### PapaSmurf

##### Member
Hi. Your discoveries for cube theory have been pretty cool tbh and very impressive! How do you gen your algs, and is there a way you could help me gen my own 4x4 parity alg? I'm looking for one that preserves eoline with 11 edges orientated.

#### Christopher Mowla

Hi everyone. Since I managed to complete my Wide Turn Based 2-cycle parity algorithms research project, I figured why not finish my last parity algorithm project: finding humanly comprehensible 2-Gen parity algorithms by hand for the 4x4x4!

• I am no a guru of 2-Gen, but I managed to create reasonably brief and/or relatively easy-to-understand single dedge flip algorithms for the 4x4x4 in all 5 different 2-Gen move sets. I ended up making a 7-part video series entitled "2-Gen 4x4x4 Parity Algorithms".

• Of course, none of these algorithms are practical to actually use in a solve, especially since even the move optimal solutions are not.

• Currently, the length of the move-optimal single dedge flip algorithm in the move set <Rw,3Uw> is unknown. I am not aware that any single dedge flip algorithms have ever been found in that move set, but I do show how to derive a (LONG) supercube safe one in the final video of the series.

• Many thanks to Ben Whitmore for inspiring me to explore all 5 types. His 2-gen collection on his website is very thorough! I also would like to thank Lucas Garron and Bruce Norskog, as they did computer searches for 2-Gen 4x4x4 parity algorithms in the past as well.

• Lastly, I hope those who are interested in the series do enjoy it!
AND, it is important to watch the videos in the order I present them, as the knowledge gained from one type is potentially used in the next. So, don't just skip to the last video because it's short. You "won't get it" unless you have watched all of the videos before watching it.

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#### xyzzy

##### Member
Currently, the length of the move-optimal single dedge flip algorithm in the move set <Rw,3Uw> is unknown. I am not aware that any single dedge flip algorithms have ever been found in that move set, but I do show how to derive a (LONG) supercube safe one in the final video of the series.
I have some semi-crazy idea for finding a sub-optimal but not unreasonably long alg for this. I haven't watched the video, so I don't know how long your "LONG" alg is, but I sure hope my idea will get something shorter.

We'll see in a couple of hours!

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#### xyzzy

##### Member
In the video his long alg is long: 574 moves. That's definitely suboptimal.

The optimal algorithm is at least 48 moves long.

Here's one algorithm that is 115 moves long (there is cancellation on the repetitions).

(r 3u' r 3u2 r' 3u2 r 3u2 r' 3u2 r' 3u2 r2 3u2 r2 3u' r2 3u2 r2 3u2 r' 3u2 r' 3u2 r 3u2 r' 3u2 r 3u' r' 3u2 r 3u2 r 3u2 r 3u2 r2)3
115 moves might be hard to beat with my method, heh. I was thinking of reducing to subgroups generated by longer sequences, e.g. 3u' r 3u, then using the same idea of forcing everything to have short odd cycles. It takes nine moves to reduce pure UF flip to the ⟨3u' r 3u, r⟩ subgroup, so that doesn't leave too much room for anything else.

Is 39 moves optimal for UF flip + (everything else is a 3-cycle)?

#### rokicki

##### Member
Is 39 moves optimal for UF flip + (everything else is a 3-cycle)?
I would guess it is not; certainly the way I determined the result does not enforce that. You clearly see how I came up with the algorithm.

Depth 48 finished; UF single edge flip is at least 49 in this metric. I may or may not let depth 49 finish.

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#### xyzzy

##### Member
I would guess it is not; certainly the way I determined the result does not enforce that. You clearly see how I came up with the algorithm.
I was wondering if the optimal just happened to also have all the edges and corners solved. But indeed it is not—here's a 34f edge flip with some wing 3-cycles, centre 3-cycles, and corner twists: 3u1 r3 3u2 r2 3u2 r1 3u3 r1 3u2 r1 3u2 r3 3u2 r3 3u3 r3 3u2 r1 3u1 r3 3u2 r1 3u3 r2 3u2 r3 3u1 r2 3u1 r2 3u3 r2 3u1 r2. (This might not be optimal either.)

No move cancellations, so repeating it gives 102 moves.

#### rokicki

##### Member
Wow, that's impressive. Gotta ask, how did you find it? Making a pruning table that enforces only three-cycles seems challenging.