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Advanced Square-1 Methods/Theory

DavidWoner

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Edit: this is a work in progress, I failed and hit submit thread before I meant to. Bear with me.

So a few months back I started solving the square-1. Unlike the cubic twisty puzzles, I became very interested in the theory behind this puzzle. However, as I tried to learn/discover as much as I could, I soon found that the theory behind this puzzle remains vastly unexplored (or at least very hard to find). Quite luckily I had some very valuable resources available to me in the forms of some very experienced sq1 solvers like Andrew Nelson, Tomas Kristiansson, and Dan Cohen. Through talks with them and some good old-fashioned experimentation I was able to learn quite a bit, and I would like to share it here.

Some things I will talk about:

1. Beyond Vandenbergh: a few modifications of the Vandenbergh method that are undeveloped or rarely used. (You LBL people are on your own)

2. Turning Styles: I've noticed almost every solver uses one of 3 distinct turning styles, and I'd like to explain them and give examples. I know its not technically theory, but I think its very interesting and didn't want to make another thread.

3. Parity: An explanation of what parity is, how it works on the square-1, and why the parity algorithms work.


1. Beyond Vandenbergh
1.1 Parity CP
Parity CP is method where parity is determined along with corner permutation (instead of during EP), and then parity and corner permutation are solved at the same time. The main advantage of this method is that it only requires 8 additional CP algs, while eliminating the 50 parity EPs. Andrew Nelson is the only person known to use this, and I'd say he's been rather successful with it. His explanation and the algs he generated can be found here.

1.2 EOCP
EOCP does exactly what it sounds like- orients edges and permutes corners at the same time.
1.2.a Precursor
The idea that serves as a sort of precursor to this and eventually led to me thinking of this method is looking ahead to CP. A lot of good sq1 solvers do this, and I recently switched all of my EO algs to ones the preserve CP in order to make this easier. The algs themselves are actually faster than the ones I was using before, and having the extra lookahead helps a lot. But this got me thinking "If certain EO algs switch corners, and other don't, then certainly you should be able to solve EO and CP at the same time."

1.2.b EOCP
Now that I had this idea I had to determine whether or not it was actually practical. However there are lot of possible cases:

6*6 CP in each layer *6 orientation cases = 216 cases

However I soon realized that every case except for adjacent-adjacent can either be reduced to a single edge swap or solve EO with a simple M2 ((1,0)/(-1,-1)/). So with and M2 preface you would only need to learn 72 algs plus the 8 regular CPs.

1.2.c The Algs
Unfortunately most of the algs for this do not exist. A few of them do, in the form of the various EO algs floating around the internet, but the rest would need to be generated. The problem is that Jaap's solver has no was to ignore stickers, so to find the best algs 72 cases* 100 EPs = 7200 cases would have to be tested. Someone proposed writing a program that would determine and generate the inputs for these cases, but I have no idea how to do that. If an experienced programmer who is familiar with Jaap's Solver thinks they can help it would be much appreciated.


2. Turning Style
The only person who really falls outside these 3 groups his Kazuhito Iimura. His turning style is unique and rather difficult to emulate, so I won't really go into it here.
2.1 Beginner Style
This is characterized by lots of wrist turning of the D layer, and turning the U and D layers separately. Most notably / moves are always sliced clockwise with a lot of wrist movement. Most people will begin to properly fingertrick the U and D layers as they get faster, but the wristy / can stay with them well into the low 20 averages. I'm not going to bother finding an example, youtube is filled with videos of this. Chances are very good that you do this now or did so in the past.

2.2 Polish Style
This style is not exclusive to Polish square-1 solvers, but it originated from and is extremely popular there. The U and D layers are fingertricked normally, but whats notable is the / turn. All turns are still clockwise, but the wrist isn't used at all (it often doesn't move at all!) Rather, its turned with some combination of the thumb, ring, and pinky fingers that I have yet to fully figure out. The advantage of this is that there is very little regrip time, the right hand stays in its place allowing you to turn the U/D layers without a delay. Some good examples are Grzegorz Prusak and Piotr Padlewski of Poland and Lee-Seung Woo of Korea.

2.3 Western Style
This is most common among Americans and some western Europeans. It is characterized by alternating clockwise and counterclockwise / turns, with varying sets of U/D fingertricks based on whether a cw or ccw / turn has just occurred. There is zero regrip time when you are alternating, which means some algs can be very fast. However, regrips are sometimes forced, like when a clockwise / is followed by (4,0) or something similar. In cases like these its common for only clockwise / turns to be used, but they can still be quite fast. Some good examples of this style are Dan Cohen, Stephanie Chow, all of my CPs, and Dan's Adj Parity.

3. Parity
3.1 What parity is
(most of what I'm about to say is rather incorrect, I should have known better than to talk about parity on 4x4) Parity occurs when there is an odd number of total swaps on the puzzle. Most people are familiar with parity as it occurs in the reduction method on 4x4. Oll parity is simply 2 edges that need to be swapped, and occurs when there has been an odd number of slice moves perform on the cube, hence the name parity.

3.2 Parity on Square-1
Parity on square-1 is the same thing- an odd number of swaps. However, the way it occurs is different from on cubic puzzles. You may have noticed that solving parity is the only step that requires you to leave cubeshape. This is because while in cubeshape, the square-1 more or less follows the laws of the 3x3 (i.e. no single swaps allowed!) However, when not in cubeshape, there is a lot more freedom and "The Laws of The Cube" do not apply (because its not a cube any more). I'll show you an example using one of the simpler parity algorithms, one of Andrews parity CPs:

/(3,3)/(1,2)/(4,-2)/(-4,2)/(-1,-2)/(-3,-3)/

The bolded red / is where the parity of the puzzle is changed, lets evaluate why. The first thing you may notice is that the shape of the puzzle does not change. This is usually a very good indicator of the point where parity changes. Now if you look closely, you can see that you are actually swapping three corners in the U layer with 3 corners in the D layer- which is three(an odd number!) of 2-swaps. More specifically you are swapping UFR-DBR, UR-DR, UBR-DFR. Then you solve cubeshape by undoing the setup moves you performed earlier. Once its back in cubeshape we can clearly see the results: 1 diagonal corner swap in the U layer and 2 diagonal corner swaps in the D layer- 3 swaps! Almost every parity algorithm generally follows ABA', where A leaves cubeshape, B changes parity without changing cubeshape, and A' solves cubeshape again. Sometimes orientation is disturbed, so I guess you could say they follow ABA'C, where C solves orientation if needed. Here are a few more examples, broken down into their components:

Opp-H: /(-3,-3)/(3,0)/(-3,-3)/(2,0)/( -4,2)/(4,-2)/(1,0)/(-3,-3)/
A: /(-3,-3)/(3,0)/(-3,-3)/(2,0)/( -4,2)
B: /
A': (4,-2)/(1,0)/(-3,-3)/

You probably noticed that A and A' are not inverses of one another. This is because we have taken the alg from earlier, and inserted and Nperm into A. As a result, some moves have been cancelled and A is inherently different from A' (it would be counter-productive to insert the same N perm back into A')

Here is another example, where B is face turns instead of a / move:

ccw O: /(3,3)/(1,0)/(-2,-2)/(2,0)/(2,2)/(-1,0)/(-3,-3)/(0,2)/(-2,-2)/

A: /(3,3)/(1,0)/(-2,-2)/
B: (2,0)
A': /(2,2)/(-1,0)/(-3,-3)/
C: (0,2)/(-2,-2)/

In this case, B is a clockwise 6-cycle, which is 5 swaps(think of BLD solving with Classic Pochmann, solving a 6 cycle uses 5 swaps) When back in cubeshape, you can see that 5 swaps have been performed: cw O (3 swaps) and H perm (2 swaps) resulting in a ccw O perm.

This was the first parity algorithm I learned, and I used it almost exclusively for a very long time. I noticed that this alg followed the ABA' formula and decided that the (2,0) in the middle of the alg had to be the secret behind parity. On a whim, I decided to do (-2,0) instead and discovered my very own clockwise O-perm without having to mirror the whole alg! (note that C changes to (1,0)/(2,2)/) At the time, I had no idea why it worked, but at least I figured out how parity algorithms seemed to work. When I brought this up with Tomas, he pointed out that an odd number of swaps was wholly possible while out of cubeshape, and the final piece fell into place.

I am still not sure how some algs like adjacent parity work, I think there is an insertion somewhere that cancels with B. I'll look into it some more later.
 
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Neutrals01

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nice guide...the methods looks interesting...but the method I use don have relation with lars vandenbergh's...so I am not used to the concept flow of methods related to lars vandenbergh's...

I thought of a method before...now I am still searching algs for the method...only 5 parity algs needed...most cases have easy recognition and the method makes it such that can look ahead at many parts..total of 43 cases needed included the parity cases(only need to learn 41 algs because 2 of them can be done intuitively)...and many parts can be done intuitively so no need algs for them.. the advantage of this method is able to look ahead and also only have to learn 5 long algos(parity cases), other algs are short ones.. this method is for ppl that don like to remember lots of algs and prefer to do it more intuitively..(I think it can go down to 20 secs on avg...but I won't post this method up till I make sure it works well first, now still on experiment..)....

the fingertricks part..I think I am categorized as beginner style....I can't seems to get used to other methods of turning..

I avg around 32~35 now..played about 1 and a half month so far..
 
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blade740

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2.1 Beginner Style
This is characterized by lots of wrist turning of the D layer, and turning the U and D layers separately. Most notably / moves are always sliced clockwise with a lot of wrist movement. Most people will begin to properly fingertrick the U and D layers as they get faster, but the wristy / can stay with them well into the low 20 averages. I'm not going to bother finding an example, youtube is filled with videos of this. Chances are very good that you do this now or did so in the past.

Lars Vandenbergh solves with this style (or he did last I saw).

Also, another thing I notice a lot is ALL turns being done with the right hand. The left hand pretty much stays right where it's at and the right hand does all the work.
 

Gabriel

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Isn't better to do CO + EO in one step and afterthat CP with the parity?

I think wrist movement mustn't be necessarily from amateurs, a good example is Kazhuito, yes, you've said his style is unique, but there are some down-layer's movements which aren't fast with the finger, I do finger D-face with turns of 1,2,3 (Vandenbergh's notation), but 4,5 and 6 I must perform them with a wrist turn.
 

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I don't think CO+EO can be done completely in one step algorithmically, but if you build half of the D layer intuitively (this is quick, a few moves at most) it should be very possible to do the rest with an algorithm. After the half of D there are (I think) 2+2+2+16+12+2+12+10 = 58 non-solved algorithms. So actually it's pretty doable. Then CP+parity and EP would be a cinch.
 
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Pedro

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DavidWoner

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I think one of the best method is Lars + about 50 bonus EP.With this method you can solve edge without repring or doing /6,6/ before Ep."Polish style" I like it :D

By Lars method + 50 do you mean all of the algorithms on his site plus the 50 mirrors?


WTF was that 11.16 solve?! The cube shape was done using just D layer moves, it seems o_O

if I make a video, can you tell me which style I use?
I'm averaging sub-30 normally, best RA is 23.93, but I still get some high-30 or even 40 times...I think I may be able to turn "better" and improve...

Yes there are quite a few 2-gen cubeshapes(there would have to be, otherwise bandaged sq1 would be impossible!), one of the better known ones is Opp-fists. The shape in the vid is passed through while solving opp-fists.
That one is /(0,-4)/(0,-1)/(0,-3)/


Ok, I am going to start writing the parity explanation now.

Edit: Done, let me know if things need to be clarified.
 
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dougbenham

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Excellent parity explanation. Thanks :D

and where can i find "Andrews parity CPs"? Does he have a website?
 

Stefan

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The parity coverage doesn't exactly impress me.

Oll parity is simply 2 edges that need to be swapped, and occurs when there has been an odd number of slice moves perform on the cube, hence the name parity.
Actually parity refers to the permutation of all pieces (of the same kind, in the same orbit, whatever you want to call it). So not just when you're left with two to swap. And it also doesn't refer to a number of moves. There is that direct connection between 4x4 edge permutation parity and the number of inner slice moves, but such a connection doesn't need to exist and the parity refers to the pieces, not the moves. And parity isn't just there when something is odd. When it's not odd, it's even. It's still there. It's not on or off. It's just cubers abusing the word, using it in a very reduced way.

You do say it the right way elsewhere, but you do link to that mathematical definition inside the sentence with the reduced version, so I felt like pointing that out.

solving parity is the only step that requires you to leave cubeshape. This is because while in cubeshape, the square-1 more or less follows the laws of the 3x3 (i.e. no single two-swaps allowed!)
Why not explain why? It's simple. Staying in cube shape (or rather double-square shape) through a / turn means doing four swaps (two edge swaps, two corner swaps) which is even. That's why the parity doesn't change. Done.

/(3,3)/(1,2)/(4,-2)/(-4,2)/(-1,-2)/(-3,-3)/

The bolded red / is where the parity of the puzzle is changed
I disagree. When you do that /, the puzzle is far from cube shape. How do you even define parity there? How do you define parity for non-cube shapes?

I'd say the alg as a whole changes parity, thanks to the three swaps done with that /, in turn made possible with the proper setup. That / is indeed the whole idea behind the alg, but it doesn't by itself change the parity (unless you give me a reasonable definition of parity for non-cube shapes).

Almost every parity algorithm generally follows ABA', where A leaves cubeshape, B changes parity without changing cubeshape, and A' solves cubeshape again.
Only "almost" every? I'd say every. Because you start and end with A=A'=/.

Also, you say A leaves cubeshape, so B starts out of cubeshape and I don't understand what you mean when you say B doesn't change cubeshape.

You probably noticed that A and A' are not inverses of one another.
Then why do you call them A and A'? That's very misleading.

B is a clockwise 6-cycle, which is 5 swaps
Wrong, it *can* be done with five swaps. I can also do it with let's say 7 or 41.

Finally: In your examples, B is always a single turn that represents an odd number of swaps. I'm quite certain there are parity-changing algs that don't contain any such turn.
 
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qqwref

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/(3,3)/(1,2)/(4,-2)/(-4,2)/(-1,-2)/(-3,-3)/

The bolded red / is where the parity of the puzzle is changed
I disagree. When you do that /, the puzzle is far from cube shape. How do you even define parity there? How do you define parity for non-cube shapes?

I'd say the alg as a whole changes parity, thanks to the three swaps done with that /, in turn made possible with the proper setup. That / is indeed the whole idea behind the alg, but it doesn't by itself change the parity (unless you give me a reasonable definition of parity for non-cube shapes).

This is a very good point. I don't think there really is any reasonable way to define parity in cubeshape, except by maintaining a dictionary of specific optimal solutions for each cubeshape and then saying a shape has parity if the dictionary solution ends up with parity in cubeshape. That doesn't give us a way to easily tell when a turn 'creates' parity, though.

The way I would say ABA' works is that, since B does not change the cubeshape, you can express it as some cycles of the cubeshape pieces. In the short parity algorithm here, B does three corner 2-cycles, and the rest of the alg is to set up that algorithm so that it affects cubeshape only. While in a different shape the concept of parity is effectively meaningless, so we can only really create a useful parity algorithm by setting up the three 2-swaps so that they occur in cubeshape.

Interestingly a lot of parity algorithms do seem to follow the ABA' formula (where B is a very short intuitive sequence which does not affect cubeshape, but does an odd permutation of the pieces), or else are a modification of some ABA' algorithm with a short algorithm or modification inserted at the beginning or end. There are also parity algorithms which do not follow this format, and seem to just do random moves that end up at cubeshape again... I don't claim to understand how those work.
 

DavidWoner

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The parity coverage doesn't exactly impress me.


Actually parity refers to the permutation of all pieces (of the same kind, in the same orbit, whatever you want to call it). So not just when you're left with two to swap. And it also doesn't refer to a number of moves. There is that direct connection between 4x4 edge permutation parity and the number of inner slice moves, but such a connection doesn't need to exist and the parity refers to the pieces, not the moves. And parity isn't just there when something is odd. When it's not odd, it's even. It's still there. It's not on or off. It's just cubers abusing the word, using it in a very reduced way.

You do say it the right way elsewhere, but you do link to that mathematical definition inside the sentence with the reduced version, so I felt like pointing that out.

Yeah I think I made the wrong decision to talk about parity on 4x4, knowing that I don't have a full understanding of it. Sorry.

solving parity is the only step that requires you to leave cubeshape. This is because while in cubeshape, the square-1 more or less follows the laws of the 3x3 (i.e. no single two-swaps allowed!)
Why not explain why? It's simple. Staying in cube shape (or rather double-square shape) through a / turn means doing four swaps (two edge swaps, two corner swaps) which is even. That's why the parity doesn't change. Done.

Didn't think of that, its painfully obvious to me now though.

/(3,3)/(1,2)/(4,-2)/(-4,2)/(-1,-2)/(-3,-3)/

The bolded red / is where the parity of the puzzle is changed
I disagree. When you do that /, the puzzle is far from cube shape. How do you even define parity there? How do you define parity for non-cube shapes?

I'd say the alg as a whole changes parity, thanks to the three swaps done with that /, in turn made possible with the proper setup. That / is indeed the whole idea behind the alg, but it doesn't by itself change the parity (unless you give me a reasonable definition of parity for non-cube shapes).

Almost every parity algorithm generally follows ABA', where A leaves cubeshape, B changes parity without changing cubeshape, and A' solves cubeshape again.
Only "almost" every? I'd say every. Because you start and end with A=A'=/.

Also, you say A leaves cubeshape, so B starts out of cubeshape and I don't understand what you mean when you say B doesn't change cubeshape.

You probably noticed that A and A' are not inverses of one another.
Then why do you call them A and A'? That's very misleading.

Sometimes I am rather poor at articulating my thoughts in a manner that is clear to others. I think you knew what I was trying to say, even though I said it incorrectly. Explanations have never been a strong point of mine.

B is a clockwise 6-cycle, which is 5 swaps
Wrong, it *can* be done with five swaps. I can also do it with let's say 7 or 41.

I fail to see how it is wrong. Perhaps we have differing definitions of "is." I would also say that is it 7 and 41 swaps as well.

Finally: In your examples, B is always a single turn that represents an odd number of swaps. I'm quite certain there are parity-changing algs that don't contain any such turn.

Yes I am aware of this, and acknowledge that when I mention adjacent parity at the end. I think some parity algs may be short ones with with other algs inserted to change the final permutation. If the insertion somehow cancelled with B then it would be rather hard to find.
 

Stefan

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I think you knew what I was trying to say, even though I said it incorrectly.
The statement with B not changing cubeshape is still unclear, and qqwref even used it the same way. Do you guys mean "puzzleshape" (so that the inverse of A can then return the puzzle to cubeshape)?

B is a clockwise 6-cycle, which is 5 swaps
Wrong, it *can* be done with five swaps. I can also do it with let's say 7 or 41.

I fail to see how it is wrong. Perhaps we have differing definitions of "is." I would also say that is it 7 and 41 swaps as well.
Well, it's a 6-cycle, that's what it is. It isn't five swaps and it isn't seven swaps. It's in a way *equivalent* to certain five swaps and to certain seven swaps. Somewhat similar to a Ferrari not being $50000, but maybe being worth $50000. And maybe being worth $70000 to some.

I think some parity algs may be short ones with with other algs inserted to change the final permutation. If the insertion somehow cancelled with B then it would be rather hard to find.
I think that's still not quite what I mean. I'm thinking of something fundamentally different from the setup-changeParity-undoSetup. Something like "leave cube shape one way, wander around, return to cube shape a completely unrelated way". Like instead of leaving the house's front door, doing some magic there, and returning through the front door, instead of that think of leaving the front door and returning through the *back* door and somehow get the magic done on your trip around the house.
 
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Stefan

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Another way to put it:

Let me show you how to do an H perm on the 3x3x3. If you look at it closely, you see that the corners need to stay the same relative to each other, and the edges need to stay the same relative to each other as well. It's just that these two rings of four pieces need to be rotated relative to each other by 180 degrees. How do I do this? Easy as ABA'.

A = M2 U2 M2. This setup reverses the order of the edges.

B = U. This rotates the corners clockwise by 90 degrees, and the edges *counterclockwise* by 90 degrees (thanks to A), for a total of 180 degrees rotation of the two rings relative to each other.

A' = M2 U2 M2. Simply undoing the setup.

But... while this is one way to do the H perm, I think you'll agree that not every H perm has the form ABA'. You can do it very different ways. For example by randomly rescrambling the cube and starting the solve all over, hoping that this time you'll get a PLL that you know how to solve.

So you're describing *one* technique to come up or break down certain Square-1 parity algs, but probably not all.
 

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I think you knew what I was trying to say, even though I said it incorrectly.
The statement with B not changing cubeshape is still unclear, and qqwref even used it the same way. Do you guys mean "puzzleshape" (so that the inverse of A can then return the puzzle to cubeshape)?

Yeah. All parity algs need to start and end in cubeshape because that's the only way parity makes sense, and the easiest way to think of doing that is the ABA' format, which also seems to lead to the shortest possible parity algs. I don't think every parity alg can be written this way, but I wouldn't be surprised if most optimal or near-optimal parity algs can be thought of as C ABA' D (for non-parity algs C and D that maintain cubeshape).

As with your H-perm example, the reason we have to do this stuff for Square-1 is that it is a bandaged puzzle with pieces of different sizes, so there is no way to do parity directly - you have to go into a different shape. With a 3x3 it's pretty easy to do a 2-2-swap (M2 U2 M2 U2) which you can then set up into an H-perm, but to make a parity alg on Square-1 the only quick way to affect parity is to do a single move (with an odd permutation effect) while in a shape that permits that. So we have to set it up somehow in order to create intuitive parity algs (in the sense that you can understand what's going on), and possibly in order to create efficient ones too.

It's in a way *equivalent* to certain five swaps and to certain seven swaps.

I'm pretty sure that is what he meant by "is".

I think that's still not quite what I mean. I'm thinking of something fundamentally different from the setup-changeParity-undoSetup. Something like "leave cube shape one way, wander around, return to cube shape a completely unrelated way". Like instead of leaving the house's front door, doing some magic there, and returning through the front door, instead of that think of leaving the front door and returning through the *back* door and somehow get the magic done on your trip around the house.

Clearly there exist parity algs of this form, similar to 3x3 PLL algs that are the inverse of a move that starts at that PLL, scrambles, and re-solves. But, again, I have a hunch that the optimal alg for any specific in-cubeshape position with odd parity will be something of the form ABA' (perhaps with non-parity-affecting algs on one or both sides).


Can we have some more discussion about the EO+CO idea >_>
 

DavidWoner

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I think you knew what I was trying to say, even though I said it incorrectly.
The statement with B not changing cubeshape is still unclear, and qqwref even used it the same way. Do you guys mean "puzzleshape" (so that the inverse of A can then return the puzzle to cubeshape)?

B is a clockwise 6-cycle, which is 5 swaps
Wrong, it *can* be done with five swaps. I can also do it with let's say 7 or 41.

I fail to see how it is wrong. Perhaps we have differing definitions of "is." I would also say that is it 7 and 41 swaps as well.
Well, it's a 6-cycle, that's what it is. It isn't five swaps and it isn't seven swaps. It's in a way *equivalent* to certain five swaps and to certain seven swaps. Somewhat similar to a Ferrari not being $50000, but maybe being worth $50000. And maybe being worth $70000 to some.

I think some parity algs may be short ones with with other algs inserted to change the final permutation. If the insertion somehow cancelled with B then it would be rather hard to find.
I think that's still not quite what I mean. I'm thinking of something fundamentally different from the setup-changeParity-undoSetup. Something like "leave cube shape one way, wander around, return to cube shape a completely unrelated way". Like instead of leaving the house's front door, doing some magic there, and returning through the front door, instead of that think of leaving the front door and returning through the *back* door and somehow get the magic done on your trip around the house.

Yes puzzleshape would be a better word. I've gotten into the habit of saying cubeshape instead of "cubeshape case," or rather, the moves need to return the puzzle to cubeshape(describing the current shape of the puzzle).

And I see what you mean now about 6 cycle.

I knew thats what you meant are ABA' not working for all algs, and I would have to agree. I was being a bit stubborn earlier but now I've eaten and am in a much better mood. There is some sort of ABC where A leaves cubeshape, B cycles an odd number and changes cubeshape (not necessarily one move), C solves cubeshape and any orientation errors. In this case B will be rather tricky to discern given how hard it can be to track cycles over multiple changes in puzzleshape, but I'll continue playing around with it.

Edit: And I agree with Michael about every parity case following ABA' with inserted C and D, not every alg though.

I would also like to add that you are rather good at making analogies.
 
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DavidWoner

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Here, look at the algorithm for W perm:

(0,-1) / (1,-2) / (-4,0) / (0,3) / (1,0) / (3,-2) / (-4,0) / (-4,0) / (-2,2) / (-1,0) / (0,-3) / (-3,0)

First running through it I couldn't find any obvious ABA' format. Then I discovered a cancellation (like what I mentioned earlier) that makes it fit.

Let's break it down.

(0,-1) / (1,-2) / (-4,0) is an orientation (and permutation) fix, just put at the beginning


/ (0,3) / (1,0) / (2,-2) is a setup that leaves cubeshape

note that I broke (3,-2) into (2,-2)(1,0) with no /. These are the types of cancellations I am talking about.

(1,0)/ (-4,0) / (-4,0) / I believe this is B in this case. Here is its effect on the pieces:

I have lettered the corners A-G, starting with A at UBL and going clockwise around the U layer, and E-G going from front to back on the D layer. It performs the cycle:

A->F->D->C->E->G->B->A (equivalent to 6 swaps, doesn't affect parity)

And the 4 U-layer edges are all rotated counter-clockwise (O-perm, equivalent to 3 swaps, creates parity when puzzle is restored to cubeshape)

Note that as I have written it, B does not affect the shape.

(-2,2) / (-1,0) / (0,-3) / restores cubeshape.


I think this is the simplest example that has both a hidden cancellation and a B that is multiple moves.
 
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