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 Post subject: Thinking about the square 1
PostPosted: Fri Apr 01, 2011 6:28 pm 
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Hi all

My square 1 arrived the other day. I don't want algorithms at this stage. Here's my problem: I have been trying to return it to a cube state and have literally zero idea on how to do this. I've searched on the forums and all the solution strategies started with

1) Return to cube (intuitive).

Well it's not remotely intuitive to me. I feel completely stupid.

If anyone could explain to me how to think about returning it to a cube so I can have some idea, I'd be appreciative.

Thanks

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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Apr 01, 2011 6:48 pm 
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I had the same trouble at first. It seems very hard to to return it to a cube. First, you should completely ignore the middle layer.

Most of the intuitive strategies are first to segregate the corners and edges into two groups. That means you want to get 6 corners on the bottom and 2 corners and 8 edges on top. You want to do this in a way that groups the corners next to each other and the 8 edges next to each other.

Just grouping pieces like this can be hard but it is easier to measure your progress and work towards your goal.

Once you have achieved this state it's pretty intuitive to get it back into a cube. Once you group the pieces the trick in going from this state to a cubic state is that you always maintain a mirror symmetry on top and bottom. If you find yourself doing a move that breaks the mirror symmetry then the move is wrong.

Hopefully that will help you get to a cubic state "intuitively".

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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Apr 01, 2011 7:22 pm 
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First, thanks very much for your reply.
Quote:
Once you have achieved this state it's pretty intuitive to get it back into a cube. Once you group the pieces the trick in going from this state to a cubic state is that you always maintain a mirror symmetry on top and bottom. If you find yourself doing a move that breaks the mirror symmetry then the move is wrong.

I managed somehow to get it to the state you said, where the corners and edges were grouped as you described.

Are you able to expand on keeping the mirror symmetry? For instance, does it mean that if I make a move, whatever is on the top must be precisely the same as what's on the bottom?

If so, there's only 1 correct move which I can say (obviously I might have missed some) which is to have on top and bottom 4 corners together and 4 edges together. Is that the correct move?

Thanks

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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Apr 01, 2011 8:12 pm 
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Hi Rline,

bmenrigh gave excellent information, keep turning, you will get it :D . He means the top and bottom remain symetrical shapes.

Burgo.

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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Apr 01, 2011 8:46 pm 
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From that point you basically turn it down the center splitting the group of 8 edges into two groups of 4, and then twist both layers and split those, and so on... you should eventually get a cube (but the middle layer might be misaligned).

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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Apr 01, 2011 8:58 pm 
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Thanks folks, I made a cube!

Believe me, this has made my day.

Now, is there anything special I should be keeping in mind now that I need to go from scrambled cube to solved cube?

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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Apr 01, 2011 9:08 pm 
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rline wrote:
Thanks folks, I made a cube!

Believe me, this has made my day.

Now, is there anything special I should be keeping in mind now that I need to go from scrambled cube to solved cube?
Other than an annoying odd cycle in the edges that requires going out of the cubic shape you can stay in the cube at this point. Always make sure the top or bottom is "misaligned" by one edge width so that when you do a turn the top and bottom faces stay square. You can ignore whatever happens to the center.

Now you should be able to get all of the top corners all showing one color and the bottoms another. This is mostly intuitive. It winds up being pretty easy after a few minutes. Once you get all the top corners on top and bottom ones on bottom you can start thinking about getting the top edges on top, etc. Or you cant start cycling corners to get them all positioned correctly.

Basically you've arrived at the solving phase so unless you just want a step-by-step solution you're mostly on your own.

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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Jun 10, 2011 2:38 pm 
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Attachment:
Cube help.png
Cube help.png [ 232.4 KiB | Viewed 2743 times ]
First Step, get it so it looks like either of the two images here.

The one with all the edges together is refereed to as the Flower, whilst the other I like to call the almost flower.

To solve flower, hold it so that the first slice is as in the picture, and execute this algorithm (I presume you know notation):

/ -2,-4 / -1,-2 / -3,-3 /

To solve almost flower, hold it so the first slice is as in picture, and execute THIS algorithm:

/ -2,2 / 3,4 / -4,3 / 5,4 / 0,-3 /

Attachment:
Cube help.png
Cube help.png [ 232.4 KiB | Viewed 2743 times ]


The puzzle should be a cube now. Let me know if something was wrong :)


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 Post subject: Re: Thinking about the square 1
PostPosted: Sun Jun 12, 2011 8:29 pm 
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So here is something that I've always wondered about the square one. Rather then deal with the possible parity at the end while swapping the centers, just before solving it, is there a way of preventing the parity in the first place while putting it in the cube form? I assume that the parity is a consequence of how the edges are placed during that cube forming process, and I always wondered if that can be noticed and prevented, or if its impossible to tell. Any reason to not commit such a monstrous algorithm to memory is worth exploring!
Let me know what you think.


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 Post subject: Re: Thinking about the square 1
PostPosted: Mon Jun 13, 2011 1:25 am 
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Not impossible to tell at all. Provided that the top and bottom sides both look like they have a 3x3x3 parity or not at all then you won't have parity. However, if one side could have parity, then here will be parity. It's hard to tell considering that the corners and edges are mixed on both layers, hense why it's easier to solve the parity at the end. Sorry if that didn't make sense.

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 Post subject: Re: Thinking about the square 1
PostPosted: Tue Jun 14, 2011 2:50 am 
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Doctor who wrote:
... is there a way of preventing the parity in the first place while putting it in the cube form? I assume that the parity is a consequence of how the edges are placed during that cube forming process, and I always wondered if that can be noticed and prevented, or if its impossible to tell.

You are correct. Parity is really only defined for the cube-shaped positions, and if you don't leave the cube shape the parity will not change. So it would indeed be nice if you could check whether you have the wrong parity the moment you first arrive at the cube shape before investing into solving the pieces.
For the square-1 solving programs I wrote ages ago I defined a parity in some arbitrary way for all positions (which matched the definition of parity on the cube-shape positions), so that the shape and the parity are solved simultaneously.

Luke wrote:
Not impossible to tell at all. Provided that the top and bottom sides both look like they have a 3x3x3 parity or not at all then you won't have parity. However, if one side could have parity, then here will be parity. It's hard to tell considering that the corners and edges are mixed on both layers, hense why it's easier to solve the parity at the end. Sorry if that didn't make sense.


This doesn't work until you have separated the pieces into their correct layers (i.e. the top and bottom faces are already one colour). Not only did you forget to take account of the number of swaps between the layers, you haven't explained how you can consider the pieces in a layer to be a 3x3x3 PLL if they are a mix of pieces from both layers. For example, if both orange edges are in the top layer, that doesn't correspond to a PLL at all.
Once the pieces are separated into their layers, then you can check the parity of each layer PLL separately, and add them together to get the parity of the position as a whole.

It is possible to work out the parity without layer separation, but time consuming. Simply decompose the position into cycles, just like most blindfold cubers do when memorising a cube position. Then count the number cycles of even length (swaps, 4-cycles, 6-cycles, 8-cycles), and if that total is odd, then you have odd parity and need to fix it.

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 Post subject: Re: Thinking about the square 1
PostPosted: Tue Jun 14, 2011 5:34 am 
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Hi Friends,

What helped me to `remember` the square 1 algs was the `shape patterns` that are created. The `numbers` mean nothing to me, other than telling me how to make the patterns. If that makes sense.

I have played around with the first few sequences and found some alternatives, but that is all I have achieved for the square 1 (by myself). In some ways I wish I had persued it myself more (instead of looking up a method). I like the idea of finding the parity before. I have already learned the parity alg though.

But it is the only alg in cubing that I do not understand `how` it is working (I know `what` it does)! Perhaps someone who knows its intricacies can break it into steps and explain what the parts do? Maybe a series of smaller steps, instead of one big alg, would be easier to remember and add to the understanding?

Cheers,
Burgo.

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 Post subject: Re: Thinking about the square 1
PostPosted: Wed Jun 15, 2011 6:55 pm 
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I thank you gentlemen for the input. Luke and Jaap, I do believe that you're likely correct about not being able to avoid the parity before getting it into the cube position without a strategy thats even more monstrous then learning the algorithm itself! Burgo, I also agree that it is far more satisfying to learn why an algorithm works in addition to the "how," but I think you are on to something when suggesting to use heuristic techniques of shape recognition rather then number recall. Maybe its time to use the right side of my brain in addition to the left :) I guess if quantum physicists are satisfied with learning "how" quantum particles behave and not "why," perhaps so can I with the cube :)
Allons-y!
The Doctor


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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Jun 17, 2011 6:28 pm 
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Hello Burgo and all other,

trying to think about the funktionaly of parity sequence.

I`m not a gentleman but I try to answer how the parity sequence works.
I do not often solve the square-1 , so i use only an easy parity sequence
The idea behind the parity sequence i think:

1)
/( -3,-3)/(-2.-1)/(-2,-2)
Turn each side until the next cutting line arises.
Now you have one side with only big corners. A star shape.


then turn the star ( -2,0) this causes an odd number of corner permutations. ( Splitting the rotation in single permutations)
The parity ist equvalent with a odd number of bigcorner permutation.
then

restore the cube shape with the reverse sequence.

Than restore the rest of square-1.

Or this in compact better form:
/( -3,-3)/(-2.-1)/(-2,-2)

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

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

This easy parity sequence does this.
It permutes the little edges in down layer.

Cheers, Andrea


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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Jun 17, 2011 10:10 pm 
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Hi Andrea,

Thankyou for the advice. Your method is 1 shorter than the sequence I know. Ironically, the way that I `remember` performing it is, `how it works`.

Make Star shape: /(3,3)/(1,0)/(-2,-2)/
Critical move > changing permutations: (2,0)
Rebuild square (reverse of 1st part): /(2,2)/(-1,0)/(-3,-3)/ (NB: You could just do up to here and re-solve: if you want to shorten your `remembered` sequence by 6).
Reallocate U&D layer pieces:(-2,0)/(3,3)/(3,0)/(-1,-1)/(-3,0)/(1,1)/

I think the most `realistic` place to pick this up is to count your edge permutations after solving your corners: to see if they match on the top and bottom layers (NB: with all edges in the correct layer also at this stage).

Cheers,
Burgo.

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PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
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 Post subject: Re: Thinking about the square 1
PostPosted: Sat Jun 18, 2011 12:00 am 
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Thanks Andrea and, as always, Burgo for the insights. And understand, being a gentleman is neither a necessary nor sufficient criteria for being brilliant at solving, so thanks for correcting that :) Tell me what you think of this strategy for parity fixing, as it not only takes care of the parity and lands the corners in the right position, but it also puts all the edges in the right layer, leaving you to just swap the edges within the same layer, but without parity:
A) Rearrange edges to wipe out the parity:
/ (-3,-3) / (-2,-1) / (-2,-2) / (-2,0) /
B) Reestablish the cube with:
(2,2) / (-1,-2) / (-3,-3) /
C) Place corners back in place and edges back to the right level with:
(-2,0) / (2,2) / (0,1)
So this becomes in it's entirety:
/ (-3,-3) / (-2,-1) / (-2,-2) / (-2,0) /
(2,2) / (-1,-2) / (-3,-3) / (-2,0) / (2,2) / (0,1)
These are very similar to what you two presented, but a with a little variability which is why I presented it.
Thanks!


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 Post subject: Re: Thinking about the square 1
PostPosted: Sat Jun 18, 2011 2:25 am 
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Hi Doctor,
Burgo wrote:
it not only takes care of the parity and lands the corners in the right position, but it also puts all the edges in the right layer, leaving you to just swap the edges within the same layer, but without parity:
Both of our sequences do this also. Mine switches 2 edges F&B in the top layer, Andrea's rotates 4 in the lower layer. Yours rotates 4 in the lower layer also.

It reduces the parity alg to 10 moves, nice. I wouldn't commit the last move to memory, it leaves sides misaligned anyway, so I think you have it to 9 moves.

But for ease of memory: 3 moves, replace the corner, reverse the 3 moves, is very good, I think. I think more people might like the cube if they know this!

I am going to commit your alg to memory, it doesn't mirror, but it is much shorter.
Cheers,
Burgo.

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PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
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 Post subject: Re: Thinking about the square 1
PostPosted: Sat Jun 18, 2011 4:14 am 
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Hello together

Doctor who:
thank you for your posting.

Doctor who wrote:
A) Rearrange edges to wipe out the parity:
/ (-3,-3) / (-2,-1) / (-2,-2) / (-2,0) /
B) Reestablish the cube with:
(2,2) / (-1,-2) / (-3,-3) /
C) Place corners back in place and edges back to the right level with:
(-2,0) / (2,2) / (0,1)
So this becomes in it's entirety:
/ (-3,-3) / (-2,-1) / (-2,-2) / (-2,0) /
(2,2) / (-1,-2) / (-3,-3) / (-2,0) / (2,2) / (0,1)
These are very similar to what you two presented, but a with a little variability which is why I presented it.
Thanks!


Nice Solution. This is exactly the consequent way of my first idea. Perhaps this is better to understand the square-1 and solve it with more intuition.

Permutation corners and permutation edges are connected. E.G If all edges are permuted cyclic you turn 90 degrees (0,3) then the corners are solved and the edges are permuted cyclic. The trick is to separate the corners in one layer.

I use my second posted way of parity solution, because habituation.
Because one edge and one corner together are an angle of 90 degrees. All other solutions without destroying the cube shape must be commutators from 2x2x2 rubik's-cube sequences.

To make the cubic shape I make the tulip-flower shape intuitively and after this i make the cube. Like the pictures in the postings above.


cheers,
Andrea


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 Post subject: Re: Thinking about the square 1
PostPosted: Sat Jun 18, 2011 4:51 am 
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Hi Friends,

I think I will post my full solution because I hinted at it earlier (said I had adapted a few things in the earlier sequences). I just cut down memorizing a bit by adapting common Domino algs, so I will break into and out of Domino algs intentionally to demonstrate that, you will follow my logic:

Make cube shape.
Allocate pieces to U&D layers:
1 Swap right front corners (top & bottom): (1,0) R U R U’ R (where U= 90*)
1b Centre fix: (1,0) (R U2)X3
2 Swap RHS edges (top & bottom): (1,0)/(0,-3)/(0,-3)(-1,-1)/(1,4)/(0,3)/
Solve relative pieces:
3 Swap horizontal corners RF & RB: (1,0) (R U R U’ R) U’ D (R U’ R U R) [NB for the D layer, invert the cube and use (-1,0)]
4 Count edge permutations to check for parity.
5 Parity fix if needed: [/(-3,-3)/(-2,-1)/(-2,-2)/] critical move:(-2,0) [/(2,2)/(-1,-2)/(-3,-3)/] [(-2,0)/(2,2)/] (13 moves down to 9 :D)
6 Swap edge pieces UR & UB + DR & DB: (0,2)/(0,-3)/(1,1)/(-1,2)

I have left the algs completed only to the essential parts (you may need to realign faces after completion of the alg).
The other 2 algs are unchanged `as I learned them`.
And I have included The Doctor's Parity Alg.
Cheers,
Burgo.
PS For step 3 above^^, if you are familiar with them, and creative with their application, you can utilize the methods in `part 2 of my 2x2x2 pocket cube tutorial` on my you tube chanel. This is for exchanging corners very efficiently. You might need to use the step 1b centre fix with them after application, and do some E layer turns to suit the algs :wink: . But it works very well. (4:25 > 8:15)

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PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)


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 Post subject: Re: Thinking about the square 1
PostPosted: Sat Jun 18, 2011 8:57 pm 
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Thanks, ladies and gentlemen for the perspectives on navigating through the parity problem. Burgo, you did a very job of summing up the best approach, combined with Andrea's insightful perspective in simplifying the process. With your domino technique, this renders the square one solve with some of the same techniques applied to known cubes, while using a very novel strategy that can rely more on shape recognition to navigate through the parity, which is quite unique among cubes, and adds a layer of entertainment. I also appreciate what Burgo was saying about the importance of discussing and posting these strategies so that others may use them to help in their solves. That way more people will buy the products, which will help support the builders and modders to continue to provide us with more twisty puzzles!
I'm going to be unable to transmit for the next 7 days as I have found out that the Sontarans and the Rutons are having a skermish at the outer rims of the constellation of Cassiopeia, and that a rogue squadron of Daleks are massing by the outskirts of andromeda, so I'm going to check it out. Happy solving until then!
The Doctor


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 Post subject: Re: Thinking about the square 1
PostPosted: Fri Jun 24, 2011 7:00 am 
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Burgo wrote:
Hi Friends,

I think I will post my full solution because I hinted at it earlier (said I had adapted a few things in the earlier sequences). I just cut down memorizing a bit by adapting common Domino algs, so I will break into and out of Domino algs intentionally to demonstrate that, you will follow my logic:

Make cube shape.
Allocate pieces to U&D layers:
1 Swap right front corners (top & bottom): (1,0) R U R U’ R (where U= 90*)
1b Centre fix: (1,0) (R U2)X3
2 Swap RHS edges (top & bottom): (1,0)/(0,-3)/(0,-3)(-1,-1)/(1,4)/(0,3)/
Solve relative pieces:
3 Swap horizontal corners RF & RB: (1,0) (R U R U’ R) U’ D (R U’ R U R) [NB for the D layer, invert the cube and use (-1,0)]
4 Count edge permutations to check for parity.
5 Parity fix if needed: [/(-3,-3)/(-2,-1)/(-2,-2)/] critical move:(-2,0) [/(2,2)/(-1,-2)/(-3,-3)/] [(-2,0)/(2,2)/] (13 moves down to 9 :D)
6 Swap edge pieces UR & UB + DR & DB: (0,2)/(0,-3)/(1,1)/(-1,2)

I have left the algs completed only to the essential parts (you may need to realign faces after completion of the alg).
The other 2 algs are unchanged `as I learned them`.
And I have included The Doctor's Parity Alg.
Cheers,
Burgo.
PS For step 3 above^^, if you are familiar with them, and creative with their application, you can utilize the methods in `part 2 of my 2x2x2 pocket cube tutorial` on my you tube chanel. This is for exchanging corners very efficiently. You might need to use the step 1b centre fix with them after application, and do some E layer turns to suit the algs :wink: . But it works very well. (4:25 > 8:15)


Well, burgo, I think a miracle has occurred. I managed to solve my square 1 today using your method. So thankyou. And now I have some questions

1) How did you work this out?
2) Are the domino algs you speak of the R U R U' R etc?
3) Is there an easy way you remember what to do, or do you just memorise what to do by osmosis (as in, after the first few hundred times it's easy)

I'm asking these questions because while I'm now confident I can use your method to get it back, I don't really understand it as such, which I'd like to fix if I can.

One other thing (for anyone). Does someone know if there's an easy way to place some particular edge piece next to some particular corner piece? Eg. imagine that all I care about is pairing up an edge piece with (one of) its matching corner pieces. Is this easy to do? If so, how?

Thanks

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 Post subject: Re: Thinking about the square 1
PostPosted: Sat Jun 25, 2011 4:51 am 
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Hi Rline,

I am glad to hear you have solved it :D .

I really enjoyed this small conversation because of the way we all collaborated. Andrea was very helpful with the way she showed how the Parity alg worked, and The Doctor did a great job shortening it. In the end did you decide to use the parity alg as The Doctor shortened it, or its more basic form (more similar to the way Ultimate solves parity)?
Like:
Make Star shape: /(3,3)/(1,0)/(-2,-2)/
Critical move > changing permutations: (2,0)
Rebuild square (reverse of 1st part): /(2,2)/(-1,0)/(-3,-3)/ (and re-solve).

A lot of those algs are not mine, that’s just the way I learned it, the 2 Domino `corner` applications are my own adaption, but I think anyone might think that up. If you do them on a 2x2x2 or 3x3x2 you will see how they work. The longer one interestingly uses the `irrelevantly moved pieces` of the first one to switch new pieces, I like the way it `works for 2 things` (you use the leftovers :wink: ). The square 1 has a lot of similarities to the Domino cube and the 2x2x3 Tower. The 2 edge algs are obvious to watch on the square 1, then it becomes harder to see on the SSQ1.

Cheers,
Burgo.

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1st 3x3 solve Oct 2010 (Even though I lived through the 80s).
PB 3x3 55sec Jan 2011 (When I was a kid 1:30 was speedcubing so I'm stoked).
1st 3x3 Earth (nemesis) solve Jan 2011 My You Tube (Now has ALLCrazy 3X3 Planets with Reduction)


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