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 Post subject: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 11:34 am 
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Hi. I'm back.

What is the general approach used to solve twisty puzzles? Commutators (and conjugates), right? Well, how do you figure out how to come up with and use them? I don't know, but I want to.

I've read Joël van Noort's page on commutators (the original website isn't around now) and Jaap's Useful Mathematics page (up until "Metrics"). Also, I've gone through these videos: Solving Twisty Puzzles With Commutators by Nan Ma, Commutators and Conjugates - The Ultimate Instructional Video, and Rubik's Cube Theory: Commutators.

I understand how basic commutators work. I can follow what is happening when I see one performed.

I don't know how to
-come up with/find commutators
-use commutators to solve a scrambled puzzle

Any suggestions or ideas for how I can learn how to solve puzzles?

I had the idea to go through a puzzle solving thread (such as "Solving the Mixup Plus") with the puzzle and seeing if I can figure out the discussion. Is that a good idea? If so, I can give a list of the puzzles I own so that thread(s) could be suggested.

Thanks.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 11:52 am 
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Sorry for the self-promotion but I doubt I have anything more to say that I haven't already have said in my videos:


Finding commutators is all about finding a way to isolate a piece or set of pieces you want to cycle. The specific way to isolate the pieces is puzzle-dependent and requires some intuition. The sequences that will work nearly always follow a specific form of construction which I have discussed many other times in many other places. I'm slowly working towards formalizing my ideas and putting together a thesis-of-sorts. This is not coming "soon" though since I have a lot of related work to do before it can be formalized.

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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 1:40 pm 
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Zeotor wrote:
[...] I don't know how to
-come up with/find commutators
-use commutators to solve a scrambled puzzle
[...]

I don't know about others' approach, but for me, I started with this commutator:

R'U'LU RU'L'U

It's kinda hard to see from the standard notation, but there is a very easy mnemonic for this algo: if you look at the way slices move from your point of view looking at the front of the cube, it's (left up)(top right) (right up) (top left) ; (left down) (top right) (right down) (top left). It makes a sort of inverted U shape where the first half of the algo moves R and L up (from the POV of the front of the cube), and the second half is the same as the first half except R and L move down.

But anyway. The effect of this commutator is that it cycles 3 corners: LD -> LU -> RD -> LD. There are, of course, many formulations of the 3-cycle commutator for corners, but this one is particularly easy to remember intuitively: the first move you make (R') moves the LD corner to LU, where the orientation of that corner is what it will be after the algo is finished. That is, it always moves the corner such that the bottom sticker comes up to the upper left corner of the front face, and the front sticker to the top face. This is the first point.

The second point is that the RD (lower right) corner gets moved to the LD corner such that its sticker on the front face remains on the front face, that is, it's as if you turned the front face clockwise.

I described these specific movements in detail, because, as you'll see, other commutators derived from the same pattern exhibits analogous movements of the cycled pieces, which makes it very easy to intuitively derive new commutators that have a similar effect, not just on the cubes, but on all kinds of twisty puzzles.

Let's see what happens when we modify this commutator a bit: instead of moving the left slice, substitute all left slice twists with middle slice twists (in the same directions as before):

M'U'LU MU'L'U

This modified commutator cycles three edges. Now take note: the first twist (M') moves the bottom center edge to the top center edge, such that its front sticker gets on the top face, and its bottom sticker gets on the front face. This is, in fact, the orientation it will end up in after the algo is done. Does this remind you of what the original commutator did to the lower left corner? It should. Now look at its effect on the right edge. It gets moved to the bottom, with its front sticker still on the front face afterwards. See how this is analogous to the effect the original algo had on the lower right corner?

Now let's try this on a 4x4x4, where things get interesting. If we apply the algo to the left and right slices of the 4x4x4, of course, we get what's expected: the lower left, upper left, and lower right corners cycle, with the same orientation changes as before. If we substitute one of the middle slices for the left slice turns, then we get a 3-cycle of the (half) edges, with exactly analogous orientation changes as the 3-edge cycle on the 3x3x3. So we already have edge-cycling algos on the 4x4x4 for free.

But it gets better. What if we substitute the left/right slices in the algo with the left/right inner slices of the 4x4x4? Try it out and see. What we end up with is perhaps somewhat unexpected, but very useful: now it's 3 face centers that are cycled.

So you see, we already have all the basic algos to solve the 4x4x4. Parity isn't even an issue; if you end up with two swapped edges, just turn a middle slice 90° then move everything back into place with the same commutators. A tedious, somewhat inefficient approach, granted, but certainly workable.

Now let's go to the 5x5x5. Substituting one of the slice turns with any of the inner slices gives you 3-cycles for various inner/outer edges. Substituting the slice turns with two inner slices gives you 3-cycles for both kinds of face centers. So again, you can solve the entire 5x5x5 with just commutators alone. How's that for a one-algo-solves-all approach?

But there's more(tm). :) So far we proved that it works on the NxNxN cubes. But what about other puzzles?

Let's take the megaminx. For convenience, let's translate the algorithm so that the U face is still the upper face on the megaminx, and the L and R turns are now two faces adjacent to the upper face, but one face apart between themselves. The algo, adapted in this way, causes a 3-corner cycling on the megaminx. Now again, note the way the orientation of the corners change: the first twist moves the lower left corner to the top face, and that's the orientation it will remain in at the end. The lower right corner gets moved to the lower left corner, with the front sticker remaining on the front, just like on the cubes (except that in this case, it is the equivalent of what happens when you turn the front face twice, because of the pentagonal shape).

Unfortunately, there is no middle slice on the megaminx, so this algo doesn't really give you a way to easily permute edges. That's no trouble, though, if you just solve all edges first (which should be relatively easy on the 12-color megaminx; on the 6-color, you may get a pair of swapped edges, in which case you have to exchange one of them with an identical-looking piece elsewhere on the puzzle). Then the commutator we derived lets you solve all the corners without touching the edges.

It doesn't stop here. Let's go one-up to a gigaminx. The 3-corner case remains the same. If you substitute an inner slice for the R turns (respectively L turns), then you get a 3-cycle of the outer edges. Again, the orientation changes on the cycled pieces are analogous, so you don't even have to memorize anything more. You can pretty much invent your own commutator using the above pattern to suit the need at hand. If both R and L are substituted with inner slice turns, then you get a 3-cycle of face centers. So this gives you almost all the tools you need to solve the gigaminx: first solve the edge centers, then the rest can be solved by commutators alone. Inefficient, yes, but it does work. If you just tweak the method a little by "intuitively" solving the face centers first, then you have a pretty good solution method for the gigaminx.

The same thing applies to the teraminx, petaminx, etc.. Substituting a single inner slice for the L or R turns gives you edge-cycling commutators; substituting two inner slices gives you face-cycling commutators. All of the cycled pieces move in analogous ways to the original algo, which makes it almost trivial to know which variation to use for a given situation. What more can you ask for? :)

Dodecahedrons are by no means the only thing you can use this commutator (or should I call it, category of commutators?) for. Let's go back to cubes to illustrate this point. But not the usual cubes; let's take a "weird cube" like the helicopter cube.

Here, there is no direct equivalents of L, R and U turns, but have no fear: just use analogy to derive the commutators, like this: substitute all the L turns with FL turns, the U turns with FU turns, and R turns with FR turns. On the helicoptor cube, because all turns are 180° (unless you jumble, but let's not consider that here), the direction of the twists (clockwise/anticlockwise) don't matter anymore. This does not reduce the effectiveness of the commutator, though. The resulting algo moves the lower left corner to the upper left -- and note -- the final orientation of the corner is exactly the orientation it got into when you made that first FL twist. Interesting, huh? So the mnemonic still works, even though the turns are only loosely analogous to the original algo! Furthermore, it moves the lower right corner to the lower left -- and keeps the front sticker facing the front, exactly like it did in all the above cases!!. Is that cool or what??

Now of course, this only gives us a way to solve half the puzzle, but face centers on the helicopter cube are relatively easy to solve "intuitively". So just do those first, then the above commutator lets you solve all the corners leaving the face centers untouched.

I could go on, but I'll leave you to discover it on your own (it's much more fun that way). The above pattern of commutator is extremely flexible; the L and R turns can be substituted for almost any kind of turns, and as long as they are two non-intersecting slices, its effect can be easily predicted in the same way as above. Using this one commutator, I've been able to derive useful algos on all kinds of puzzles, from the skewb to the skewb ultimate to the master pyraminx to the pyraminx crystal, even the cuboids. If you're into memorizing as few algos as possible, memorize this one. It works for all kinds of puzzles, is easy to adapt to new puzzles, and is very easy to predict the effects of. You can't get much better than that. :)


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 9:22 pm 
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bmenrigh wrote:
I doubt I have anything more to say that I haven't already have said in my videos

I will go through those videos soon. I have seen them on your YouTube channel before, but have never watched them. Now I will.

quickfur wrote:
I don't know about others' approach, but for me, I started with this commutator:

R'U'LU RU'L'U

That's basically just the Corner Piece Series from the Ultimate Solution, right? I am familiar with the method. In fact, before I made this thread, I saw your post here about what you just mentioned. Thank you for the explanation.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 9:44 pm 
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Zeotor wrote:
[...]
quickfur wrote:
I don't know about others' approach, but for me, I started with this commutator:

R'U'LU RU'L'U

That's basically just the Corner Piece Series from the Ultimate Solution, right? I am familiar with the method. In fact, before I made this thread, I saw your post here about what you just mentioned. Thank you for the explanation.

Yep, that's the corner piece series from the Ultimate Solution.

Before I found the Ultimate Solution, I had tried learning other methods: the layer-by-layer method, which I only managed to use once or twice at the most because it required the memorization of too many algorithms; the corners-first method, which is not bad, slightly easier to learn, but is hard to generalize to other puzzles. The layer-by-layer method is the slowest of all, and most difficult to learn (well, for me anyway) because it required memorizing too many algorithms. I suck at memorization. The corners-first method is slightly better; more intuitive, faster, and required fewer algorithms. But I did best with the Ultimate Solution: only two algos to memorize (and the first is so trivial you don't really need to memorize it at all), and best of all, it works for all sorts of things, not just the 3x3x3.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 10:04 pm 
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Zeotor wrote:
quickfur wrote:
I don't know about others' approach, but for me, I started with this commutator:

R'U'LU RU'L'U

That's basically just the Corner Piece Series from the Ultimate Solution, right? I am familiar with the method. In fact, before I made this thread, I saw your post here about what you just mentioned. Thank you for the explanation.
quickfur wrote:
Yep, that's the corner piece series from the Ultimate Solution.

Um, actually, no it's not. The corner piece series from the ultimate solution method is

URU'L' UR'U'L

You can see the whole official ultimate solution method at this site. The only comment I'll make on the site is that the method itself is far simpler than the site.

Phil Marshall chose this commutator (I think, since I've never talked to him) because

1. Its symmetry made for a nice upper-view pattern for explanation, which he used
2. All corners moved were on the up face.

His CPS starts by turning the upper face. Every corner 3-cycle is, I believe, in essence, a variation on the same theme, which is what bmenrigh talked about.

The 1st 3 moves to isolate a piece, the 4th to replace it with another piece, the 5th, 6th and 7th to undo the isolation and the 8th to undo the replacement.

You can see nearly 100 different puzzles solved using the two algorithms of the ultimate solution at my website or youtube channel. Like bmenrigh, I'm not trying to self-promote, rather give an example of how a couple of very simple ideas can be extended (sometimes easily, sometimes with a lot of work) to many many different kinds of puzzles. I have solutions on there for puzzles such as the supercubes, 4x4x4 rhombic dodecahedron, pretty much all the cuboids that have been mass-produced, the constrained cube ultimate, crazy cubes, tetrahedra and pentahedra, bandaged cubes, latch cube, mixup 3 and 4, rex, teraminx and others.

Your original question was
Quote:
I don't know how to
-come up with/find commutators
-use commutators to solve a scrambled puzzle

Any suggestions or ideas for how I can learn how to solve puzzles?

First, I thoroughly recommend watching bmenrigh's video above. It's very clear and helpful.

Essentially, though, how to find commutators is, I think, a matter of trying different moves in order to isolate a piece. Once you've found a set of moves to isolate a piece, you're home. The trick is not in finding a commutator, but in finding a "nice" or "elegant" commutator. Others can correct me if I'm wrong here.

As to how to use them to solve scrambled puzzles, again, see bmenrigh's video. but i think a lot of it is in identifying the different piece types on the puzzle, and seeing what they also move when they move. A simple example of this is the standard rubik's cube.

The edges have their own commutator, but this commutator also moves around corners.
The corners have their own commutator, and this commutator does not move anything else around.

Therefore, do the edges first. And once the edges are done (having moved around the corners, which we don't care about), then move the corners around (which of course won't affect the edges).

This approach works basically the same on more complex puzzles.

One last thing I'd say is this: suffer through the inevitable feelings of self-defeat and uselessness when you can't find how to approach a puzzle straight away. (I speak from my own experience, maybe no-one else feels like this!) The satisfaction and (I'd go so far as to say) joy that comes from having solved a new puzzle by yourself, with your method, is much better than the feeling of seeing someone else's approach first. Once you've come up with something yourself, then by all means go and have a look at what others have done. To me, that's one of the key attractions of this forum. Being able to have a go at something myself, and then go and see how others have done it better :lol:

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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 10:32 pm 
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rline wrote:
Zeotor wrote:
quickfur wrote:
I don't know about others' approach, but for me, I started with this commutator:

R'U'LU RU'L'U

That's basically just the Corner Piece Series from the Ultimate Solution, right? I am familiar with the method. In fact, before I made this thread, I saw your post here about what you just mentioned. Thank you for the explanation.
quickfur wrote:
Yep, that's the corner piece series from the Ultimate Solution.

Um, actually, no it's not. The corner piece series from the ultimate solution method is

URU'L' UR'U'L

You can see the whole official ultimate solution method at this site. The only comment I'll make on the site is that the method itself is far simpler than the site.

Hmm. Then I don't know where I got my commutator from. I'm pretty sure it was Marshall's series. Or maybe it is, just seen from a different angle? In any case, there is a nice mnemonic for the way it moves pieces around, which is what makes it so useful.

Quote:
[...] A simple example of this is the standard rubik's cube.

The edges have their own commutator, but this commutator also moves around corners.

Actually, there is a commutator that only permutes edges without touching corners. It's basically the commutator I posted, with the left slice turns substituted with middle slice turns. It's not as simple as the edge commutator that does permute the corners (4 turns vs. 8 turns), but it's very useful in the larger cubes, where, suitably modified, it lets you move edge pieces around without touching anything else.

Quote:
[...]One last thing I'd say is this: suffer through the inevitable feelings of self-defeat and uselessness when you can't find how to approach a puzzle straight away. (I speak from my own experience, maybe no-one else feels like this!) The satisfaction and (I'd go so far as to say) joy that comes from having solved a new puzzle by yourself, with your method, is much better than the feeling of seeing someone else's approach first. Once you've come up with something yourself, then by all means go and have a look at what others have done. To me, that's one of the key attractions of this forum. Being able to have a go at something myself, and then go and see how others have done it better :lol:

Funny story. When I first got the pyraminx crystal, I just couldn't resist scrambling it immediately just to see if I can solve it on my own. The corners were easy -- you can solve them just like a megaminx's corners. But the edges eluded me. After struggling with it for a good number of days, I managed to come up with a 27-turn algorithm that cycles 7 edge pieces. Using this algo, along with lots of conjugations, I managed to solve the puzzle. Based on what I knew at the time, I thought the best algo for solving the edges must probably be very complicated, even if it wouldn't be the 27-turn algo, which I was pretty proud of at the time. And then I started peeking online for pyraminx crystal solutions, and was surprised to find that edges were considered among the easiest to solve. Imagine my chagrin when I learned that there was a 4-move algo to cycle three edges. I felt really stupid for inventing the 27-turn algo. :P


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 10:49 pm 
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rline wrote:
Zeotor wrote:
quickfur wrote:
[...]
R'U'LU RU'L'U

That's basically just the Corner Piece Series from the Ultimate Solution, right?[...]

Um, actually, no it's not. The corner piece series from the ultimate solution method is

URU'L' UR'U'L
[...]

Fundamentally they are the same sequence.

Goal: Turn [R', U', L, U, R, U', L', U] into [U, R, U', L', U, R', U', L].

Step 1: Mirror([R' U' L U R U' L' U]) -> [L, U, R', U', L', U, R, U']

Step 2: LeftRotate4([L, U, R', U', L', U, R, U']) -> [L', U, R, U', L, U, R', U']

Step 3: Invert([L', U, R, U', L, U, R', U']) -> [U, R, U', L', U, R', U', L]


I think this discussion is missing a broader point which is the [[A:B],C] construction is extremely versatile. Every minimal 8-move commutator is always of this form. In the Gelatinbrain solving thread the shorthand we use for this is [3,1] or when we're feeling extra specific, [[1:1],1] and it is by far the most-used commutator construction across a huge variety of puzzles.

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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Oct 24, 2012 11:04 pm 
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bmenrigh wrote:
[...]
I think this discussion is missing a broader point which is the [[A:B],C] construction is extremely versatile. Every minimal 8-move commutator is always of this form. In the Gelatinbrain solving thread the shorthand we use for this is [3,1] or when we're feeling extra specific, [[1:1],1] and it is by far the most-used commutator construction across a huge variety of puzzles.

Cool. I knew it was a very general construction, but I didn't realize that every 8-move commutator has this form. Out of curiosity, are 8-move commutators good enough for 4D puzzles? Or are there more complex ones (12-move? 16-move?)?

I wanted to add that, over the course of generalizing this construction (or discovering its many facets, depending on how you look at it), I have learned to do it mirror-imaged, forwards, backwards, etc.. It's extremely useful in all sorts of situations. And like bmenrigh says, you can substitute A, B and C with just about anything. Even on the good ole 3x3x3, you can come up with some interesting algos that might be useful in rare occasions; for example, by replacing the left slice turn with a left slice 180° turn, you can cycle corners that are separated across opposite faces of the cube. All sorts of variations are possible, and if you've learned the pattern, you can predict its effects easily, so you can pretty much invent a new variation on the spot, as the need arises.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Thu Oct 25, 2012 12:31 am 
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quickfur wrote:
Cool. I knew it was a very general construction, but I didn't realize that every 8-move commutator has this form. Out of curiosity, are 8-move commutators good enough for 4D puzzles? Or are there more complex ones (12-move? 16-move?)?

Regarding 4D puzzles, the answer is "yes and no" which is really the same for 3D puzzles. There are plenty of puzzles that require longer commutators. There are plenty of pieces that we know require at least a [6,1] commutator to perform a 3-cycle. This is true for 3D and 4D.

There are many cases where a [3,1] commutator will perform a useful operation on a 4D puzzles but there are many 4D puzzles with pieces that need longer commutators.

Actually, in terms of commutator sequences 3D and 4D twisty puzzles are extremely similar.

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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Fri Nov 02, 2012 10:59 am 
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Update.

I have watched those two videos. They were very helpful. I feel like I understand the basic approach to solving that is presented in the second one.

I have two more questions.

1. In the puzzle from the second video, Gelatinbrain's 1.4.14, Brandon said that there were no orbits for all of the pieces, except for one and its mirror. What is an orbit?

2. Is the Master Skewb a good first puzzle to try to solve using commutators and conjugates?


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Fri Nov 02, 2012 11:05 am 
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Zeotor wrote:
Update.

I have watched those two videos. They were very helpful. I feel like I understand the basic approach to solving that is presented in the second one.

I have two more questions.

1. In the puzzle from the second video, Gelatinbrain's 1.4.14, Brandon said that there were no orbits for all of the pieces, except for one and its mirror. What is an orbit?

2. Is the Master Skewb a good first puzzle to try to solve using commutators and conjugates?

An orbit is pretty much like it is in astronomy. The planets have an orbit around the sun. They have a set movement. An example of an orbit is on the helicopter cube. A triangle can never go into the spot of the triangle immediately next to it (without jumbling, of course).

The master skewb is an excellent puzzle to start with. There aren't that many pieces where it is tedious (like the starminx), and there are only 2 or 3 commutators needed. It is good practice with set-up moves, and with commutators that use 2 pieces of the same color.

-Doug

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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Fri Nov 02, 2012 11:13 am 
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Zeotor wrote:
[...] I have two more questions.

1. In the puzzle from the second video, Gelatinbrain's 1.4.14, Brandon said that there were no orbits for all of the pieces, except for one and its mirror. What is an orbit?

An orbit is, in simple terms, the set of all possible positions a given piece can end up when you perform all combinations of a particular set of moves (usually taken to be all possible moves on a puzzle). For example, if you only allow turning a single face on the 3x3x3, then the corners in that face have an orbit of 4 possible positions (the four places where they can end up). If you allow turning two adjacent faces, then the corners have an orbit of 6 possible positions (any one of the 6 corners touched by the two faces).

Of course, on the 3x3x3 things are rather boring, since the orbit just covers all possible positions covered by the moves, so it doesn't tell you anything interesting. In some puzzles, however, things are different. For example, on the helicopter cube, if you don't allow jumbling moves, then you'll discover that the face centers cannot be arbitrarily exchanged with each other; in fact, each face center belong to an orbit of 6 possible positions, and each face center of the same color belongs to a different orbit. Which means that no matter what you do, as long as you don't use jumbling moves, you can never swap two face centers of the same color. Each face center can only ever end up in one of 6 positions -- the 6 positions that lie in its orbit. The other face center positions are unreachable.

Once you allow jumbling moves, though, you can move the face centers "between orbits". Or, to be technically correct, the orbit of a face center under jumbling moves covers all positions on the puzzle, not just 6 positions.

So basically, the concept of an orbit is most useful when it doesn't cover all possible positions; then it tells you which pieces can end up where, and where they can never end up. So it saves you a lot of frustration trying to get a piece somewhere that's impossible to get to. :)

Quote:

2. Is the Master Skewb a good first puzzle to try to solve using commutators and conjugates?

I'll defer this to other posters.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Fri Nov 02, 2012 11:50 am 
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Zeotor wrote:
2. Is the Master Skewb a good first puzzle to try to solve using commutators and conjugates?
In my opinion no. The corners on the Master Skewb (Skewb corners) have some strange properties / restrictions that are not typical of other pieces on other twisty puzzles. I think a Starminx is the best place to start.

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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Fri Nov 02, 2012 11:53 am 
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bmenrigh wrote:
Zeotor wrote:
2. Is the Master Skewb a good first puzzle to try to solve using commutators and conjugates?
In my opinion no. The corners on the Master Skewb (Skewb corners) have some strange properties / restrictions that are not typical of other pieces on other twisty puzzles. I think a Starminx is the best place to start.

That's a good point, but starminx points are really tedious and honestly, kind of boring. It has super easy set up moves, and I think is a little unrealistic if you want to apply those strategies to other puzzles.
Although the centers are nice.

-Doug

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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Fri Nov 02, 2012 9:20 pm 
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Rex Cube or FTO would be a better place to start than Master Skewb, then. :)


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Fri Nov 02, 2012 11:49 pm 
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A good place to start solving with commutators:
I agree with the FTO and Rex Cube, but I'd add the Dayan Gem3 and with a bit more experience the Dayan Gem4. The Pyraminx Crystal is an obvious choice too I think. I presume you've already exploited your 333.

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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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Sun Nov 04, 2012 11:42 am 
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Burgo wrote:
I presume you've already exploited your 333.

Actually, I haven't. I will start with that and do the following.

3x3x3 - The one that started it all.
4x4x4 - More layers, more pieces.
5x5x5 - Even more layers, even more pieces.
FTO - Basically a Master Skewb, but with less pieces.
Rex Cube - A Master Skewb without corners.
Starminx - Something different.
Master Skewb - It's time to take on the corners.

It will take me a while to go through all of those. That's not a bad thing though.

Thanks for all of the suggestions everyone. I will post my successes in the "Accomplishments thread." (Now I have to go and order some of those puzzles.)


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Nov 07, 2012 1:22 am 
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Zeotor wrote:
Burgo wrote:
I presume you've already exploited your 333.

Actually, I haven't.

I will post my successes in the "Accomplishments thread." (Now I have to go and order some of those puzzles.)
Hi Zeotor,

If you haven't solved the 333 with commutators, yes.. definately start there. But try to forget everything you know (especially from a layer by layer point of view). Look at the piece types and try to solve `them` seperately.

Start with your cube in the solved state. Look back to Brandon's videos.. and remember that you are trying to isolate a particular piece on a face with a short series of moves (it doesn't matter what the moves are), there are plenty of ways to do this on a 333. Then give that face a twist, reverse your initial sequence, and undo your twist. Once you've found one way to isolate a piece.. try and see how many different ways you can do it. Try flipping edges and orientating corners, etc.

Also, I think you should post your success and journey here, it would be handy for someone else doing the same thing in the future, and it would just get buried in the accomplishments thread.

Cheers,
Burgo.

_________________
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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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Nov 07, 2012 12:43 pm 
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Burgo wrote:
[...] If you haven't solved the 333 with commutators, yes.. definately start there. But try to forget everything you know (especially from a layer by layer point of view). Look at the piece types and try to solve `them` seperately.

Funny, I've never really liked the layer by layer POV, because it seems unnecessarily complicated to me (it doesn't line up nicely with the structure of the puzzle -- edges and corners). I've always preferred treating the puzzle as 12 edges and 8 corners, vs. 3 layers of various combinations of edges and corners.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Nov 07, 2012 12:51 pm 
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quickfur wrote:
Funny, I've never really liked the layer by layer POV, because it seems unnecessarily complicated to me (it doesn't line up nicely with the structure of the puzzle -- edges and corners). I've always preferred treating the puzzle as 12 edges and 8 corners, vs. 3 layers of various combinations of edges and corners.

The pieces you haven't solved yet don't need to be preserved. If you can group pieces that are solved together then you delay that you maximize the number of solved pieces while still minimizing the amount of the puzzle that is restricted you maintain more solving freedom over the course of the solve.

The layer-by-layer method just captures this idea into a rule.

Having competed in fewest moves solving on many of Gelatinbrain's puzzles, it's obvious to me that the more of a particular region of a puzzle you can solve without relying on set sequences the more efficient you're likely to be overall. Daniel Kwan has demonstrated this to great effect in many of his records.

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Prior to using my real name I posted under the account named bmenrigh.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Nov 07, 2012 1:28 pm 
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bmenrigh wrote:
quickfur wrote:
Funny, I've never really liked the layer by layer POV, because it seems unnecessarily complicated to me (it doesn't line up nicely with the structure of the puzzle -- edges and corners). I've always preferred treating the puzzle as 12 edges and 8 corners, vs. 3 layers of various combinations of edges and corners.

The pieces you haven't solved yet don't need to be preserved. If you can group pieces that are solved together then you delay that you maximize the number of solved pieces while still minimizing the amount of the puzzle that is restricted you maintain more solving freedom over the course of the solve.

The layer-by-layer method just captures this idea into a rule.

Correct, which is why I usually don't even bother with the (1,1) commutator, because I usually order my solve such that pieces that require 8-move (or more) commutators come last, so that at the beginning of the puzzle I'm mostly solving "intuitively".

Quote:
Having competed in fewest moves solving on many of Gelatinbrain's puzzles, it's obvious to me that the more of a particular region of a puzzle you can solve without relying on set sequences the more efficient you're likely to be overall. Daniel Kwan has demonstrated this to great effect in many of his records.

True. Though I'm lazy and bad at memorization (but mostly just lazy) so I usually just arrange my solves to use long commutators last, shorter ones first, and mostly solve "intuitively" at the beginning.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Nov 07, 2012 1:53 pm 
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Case in point: recently I refined my rex cube solution a little more (it's still not perfect yet, but this is what I have so far), thus:

1) Pair up an edge with a face center

2) Solve face centers (by remembering which colors are opposite which -- due to parity this will always result in the correct final permutation of face centers) - this is simple enough to solve intuitively, perhaps with 1 or two applications of the [1,1] commutator.

3) Using the face centers as reference, solve for edges on a single face (the face centers get messed up again, but no matter, the initial solve was just to ensure I don't end up with a mirror image configuration of face colors, which is unsolvable because the face centers will be in odd parity).

4) Use the side colors of this single face to solve for the rest of the edges -- again, this can be done "intuitively", and near the end, a couple of [1,1] commutators.

5) Solve 3 faces by (a) pairing up two corners in a single lune (crescent-shaped slice) without the correct face center, moving those into the matching face with [1,1] commutators, and (b) pairing up another two corners of the same color, but this time also with the correct face center, and using (1,1) commutators to move that into the matching face. At this point, there's enough room on the puzzle to move things around easily, though there's a need for some [3,1] commutators to get the corner pieces to line up. (The [3,1] commutator here is slightly easier to execute -- they cycle 3 corner+face center pairs.)

6) Once there are only 3 unsolved faces left, room gets a bit tight, so at this point I solve the last 3 face centers (trivial application of [1,1]), then use another [3,1] commutator -- the one involving the middle slice, which only cycles 3 corners without touching the face centers -- to solve for the remaining edges. Some amount of conjugation is needed here, because the 3-cycle involves faces of 3 different colors, which makes it trickier to use than the analogous algorithm on the FTO, which cycles 3 face centers between 2 faces. So this last step is where the most involved algorithms are needed. Step (5) is, technically speaking, the same as this step, but I split them up so that I can postpone the complicated conjugations + middle slice turns (which are slow on the rex cube due to the uneven shape of the middle slice) to the very last.


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Nov 28, 2012 2:07 pm 
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Bump/Update

I've solved the 3x3x3 using commutators.

I tried to come up with commutators that I hadn't seen before. For example, I didn't use R U R' U' or any of its variants for the edges. All of the sequences that I used are pure, so edges or corners could be solved first. I have solved the puzzle using both approaches.

Here are the commutators that I used.

Corners

(U R U' R') D (R U R' U') D' = [U R U' R', D] = 1
URF to FRD to FDL

D (U R U' R') D' (R U R' U') = [D, U R U' R'] = 1'
URF to FDL to FDR

(U' B U2) 1 (U2 B' U) 1' = [U B U2, 1]
URF twisted clockwise, FDL twisted counter-clockwise

Edges

(U M' U' M) D (M' U M U') D' = [U M' U' M, D] = 1
UF to FD to LD

D (U M' U' M) D' (M' U M U') = [D, U M' U' M] = 1'
UF to LD to FD

L 1 L' = [L:1]
UF to FD to FL

(U' R B U2) 1 (U2 B' R' U) 1' = [U' R B U2, 1] = 2
flips UF and DL

L 2 L' = [L:2]
flips UF and FL

Other combinations of setup moves and those commutators were used.
I know that there are more efficient ways to do some of those things. However, I tried to come up with things that I hadn't seen before.

I welcome any suggestions and thoughts about my solution.


I did encounter a problem when coming up with the commutators. With the solved cube, I would do a set of moves to see the effect. I would often do something wrong, though. I would have to solve the puzzle using a method that I know and try again. Should I have written down what I was going to do and then tried it? When I get to the FTO, for example, if I make a mistake, I won't know how to solve it. What can I do to prevent accidental scrambling?


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 Post subject: Re: Learning How to Solve Puzzles
PostPosted: Wed Nov 28, 2012 5:08 pm 
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Zeotor wrote:
Bump/Update

I've solved the 3x3x3 using commutators.

I tried to come up with commutators that I hadn't seen before. For example, I didn't use R U R' U' or any of its variants for the edges. All of the sequences that I used are pure, so edges or corners could be solved first. I have solved the puzzle using both approaches.

Nice!

This is the "long" way to solve it, but it is useful in that it teaches you how to deal with corners independently of edges (and vice versa) without touching anything else. Sometimes this can be handy in combination with a usual "dirty" solving method. For example, I usually solve edges first, but on even cubes, sometimes I end up with swapped corners due to flipped edge parity, so I adopt an algo from a corners-first solution method (which messes up the edges), and then use edge commutators to fix things up afterwards.

Knowing how to deal with these cases also lets you solve a not-fully-scrambled puzzle quicker, because you can identify special cases that you can take advantage of to minimize touching the parts of the puzzle that aren't scrambled, and you don't have to resort to a full-blown solution method.

Quote:
[...] I did encounter a problem when coming up with the commutators. With the solved cube, I would do a set of moves to see the effect. I would often do something wrong, though. I would have to solve the puzzle using a method that I know and try again. Should I have written down what I was going to do and then tried it? When I get to the FTO, for example, if I make a mistake, I won't know how to solve it. What can I do to prevent accidental scrambling?

Accidental scrambling happens to me all the time. Esp. on lubed puzzles where accidental turning happens easily, and sometimes unconsciously when I'm replaying a complex algo. Assuming you aren't being pressed for time, the first thing to do is, don't panic. :) Usually, there's a way to reverse the mistake if it's within 2-3 turns of the last known good state.

Usually you can guess, by the parts of the puzzle that are still partially-solved, how to reverse the mistake. Sometimes you can get it exactly, but sometimes it's not exact but if you do it carefully you can still get most of the pieces back, excepting maybe a few that are wrong because you did the turns in the wrong order. You can then use a commutator to fix those.

Writing down what you did, of course, lets you restore the puzzle 100% of the time, but usually if you're familiar with the puzzle it's not really necessary.

OTOH, accidental scrambling can be a wonderful way of learning how to solve a new puzzle from scratch. :) I mentioned somewhere that I have this tendency of scrambling every new puzzle I get without first learning how to solve it. Then as I try to solve it, I discover commutators I can use, and any quirks that I can take advantage of. It's the ultimate test of whether you can truly invent new commutators on-the-fly as opposed to taking the "easy" way out by starting with a solved puzzle. During the process, sometimes you can get lucky and "accidentally" solve the puzzle (or rather, make an educated guess that coincidentally also fixed other parts of the puzzle you didn't know how to deal with yet). But after a few rounds of deliberate scrambling and solving a new puzzle cold-turkey, you kinda get to know how things work and eventually you can refine it into an actual solution method.

I did this for the FTO... it took me quite a few "accidental" solves before I figured out the algo for cycling 3 face centers. For the Rex Cube, I had a lot of trouble figuring out the "endgame" (solving the last 2-3 corner pieces) until I realized its equivalence to the FTO, and so was able to adopt some FTO algorithms to help out.

Anyway, the point is that accidental scrambles happen, but you shouldn't need to be afraid of them. They're a learning process. :) If you learn it this way, then in the future you'll know how to recover from mistakes in real-time, instead of having to restart from scratch each time.


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