Solving the whole cube intuitively

[mrCage]

Twisting 2 corners can be done with 2 so-called sunes like so:

R' D' R D' R' D2 R - L D L' D L D2 L' (2 mirrored sunes)

or commutator style like so:

R D R' F D F' - U - F D' F' R D' R' - U' (for instance)

Yet another 2-twist:

(R' D2 R B' U2 B)*2

"Final example":
(L B' D2 B L' - U2)*2 (yet another commutator).

See how the 2 last ones are essentially the same algorithm!!

One can do a 4-twist by a slight variation of the "final example" like so:

L B' D2 L' B2 D B' - U2 - B D' B2 L D2 B L' - U2 D)

Understanding commutators is not exactly intuitive, but with some practice and experience they become intuitive In the sense that making up new ones can be done easily using basic principles. (Check 2nd algorithm above!)

Bonus corner 3-twist:

(R' D' R L - z)*4

 

[badmephisto]

you're right. I phrased the question wrong by putting in those terms, and I was just hoping that people would try to avoid commutators, but obviously I could have seen it coming that people would instead just attack my assumption that commutators are not intuitive

well to be perfectly honest I want to be able to teach this completely intuitive method to people, in the end. And explaining how commutators work is extremely hard. If I were to explain to a really smart person (albeit a beginner) about how you can use commutators to solve 2 corners at a time, I would almost surely fail, miserably.

So, I was looking for a method that a very bright beginner can pick up, and i didnt want it to involve the idea of commutators because they are too hard to explain. And I dont mean just the idea... but more of the idea of how to use them, effectively, in a directed fashion to your immediate advantage.

(Of course... Now im just waiting for someone to swoop in and tell me that is not so...)


EDIT:
I would consider the R' D2 R thing and M D2 M' thing as an intuitive way to finish up PLL step. So the problem then becomes just how to orient the corners.
And I never tried the corner first approach, I should play around with that.

 

[cmhardw]

So, I was looking for a method that a very bright beginner can pick up, and i didnt want it to involve the idea of commutators because they are too hard to explain.

Completely agree, 100%. Teaching commutators and how they work is extremely difficult, and every time I have tried I succeed in only frustrating the learner to the point of not wanting to cube.

I do, however, still teach commutators. I have only drastically changed my approach to how I teach them.

For cycling corners I teach the same commutator method that Joël van Noort teaches on his site. The actual 3 cycle, in full notation, is:
R' D2 R U R' D2 R U R' D2 R U2 R' D2 R

As a commutator this is: [R' D2 R U : R' D2 R, U]

or

S = Setup Move = R' D2 R U
A = R' D2 R
B = U

and the algorithm is: S A B A' B' S'

Although this is a commutator, and a complicated one at that, I teach it as follows:

R' D2 R I call the "baseball move". In a visual sense it is like someone is throwing a low sidearm throw. The R' is the dropping of the arm, the D2 turn "throws" the corner at DRF to DLB. Then the R is the motion of the arm coming back up. The U turns I call "pinwheel" turns. I tell students that it is like one of those pinwheels on a plastic stick. You blow into the pinwheel and it spins, like a windmill. The U layer turns are like that.

So a student remembers the algorithm
(R' D2 R U) (R' D2 R) (U) (R' D2 R) (U') (U' R' D2 R)

like this:

(baseball move) (pinwheel the next corner around) (baseball move) (pinwheel the next corner around) (baseball move) (pinwheel the original corner around) (baseball move).

They are doing commutators, and complicated ones at that, but don't even realize it!

------------

I also teach twisting corners as commutators.

R' D2 R F D2 F' U F D2 F' R' D2 R U' is a commutator I teach. In notation it is written:
[R' D2 R F D2 F', U]

I also teach this as baseball moves and pinwheels. The R' D2 R I call "the right baseball glove move" and F D2 F' I call "the front baseball glove move"

Students remember:
(R' D2 R F D2 F') (U) (F D2 F' R' D2 R) (U')

as

(right baseball glove move) (front baseball glove move) (pinwheel the next twisted corner over) (front baseball glove move) (right baseball glove move) (pinwheel the corner back to its original spot)

Easy as pie. Trust me, if 6 and 7 year old kids can learn this, then any adult can. The failing is not in the learner but in the teacher. I don't mean this as an insult to you, but rather to get your attention. I have learned so much in terms of how to teach cubing through my many failures in teaching kids how to cube. I have tried teaching commutators as commutators, and just like you said it was a miserable failure. Just find a way to present the topic to the student that makes more sense to the student. You'd be surprised how complicated of a thing they can learn by doing this. Don't ever teach them the term "commutator" or talk about As and Bs and (A')s and (B')s, just phrase it in a very visual sense.

Again, I'm not trying to come across as insulting, I am just relaying things I have learned by teaching cubing to young kids. The failure of the student to learn is really in not teaching it a good way to learn for that particular student. If you can find a good method for that student then they will comprehend it just as well as you do, they might only think of the concept in completely different terms than you do.

Hope this helps, and yes I am an ally to what you are doing, even though it might not sound like it!

Chris

--*******EDIT*******--
To preempt the question of how to twist 3 corners, here is how I teach kids to do this:

(R' D2 R F D2 F') (R' D2 R F D2 F') U (F D2 F' R' D2 R) U (F D2 F' R' D2 R) U2

As a commutator, this is the best I can come up with as to how to represent how that algorithm works:
Let
A = R' D2 R F D2 F'
B = U

The commutator is actually not a single commutator, but rather a cyclic shift of the concatentation of a commutator with a conjugated commutator. Here is what I mean by that.

Using the above definitions of A and B we will call the "conjugated commutator": (A'A') A B A' B' (AA)
The "commutator" we will call: (A'A') B' (AA) B

The concatenation of the two will do the commutator first, followed by the conjugated commutator. This concatenation is:
(A'A') B' . (AA) B (A'A') A B A' B' (AA)

The cyclic shift starts at the period and wraps around after reaching the end. This makes the algorithm become:
(AA) B (A'A') A B A' B' (AA) (A'A') B'

Including the cancellations this becomes: A A B A' B A' B' B' which is the structure of the above alg that twists 3 corners.

---

Anyway, the way I teach kids to do this is to cycle the first corner "the wrong way" the very first time. This means they do R' D2 R F D2 F' to twist the corner in the opposite direction that they should. After this, they will cycle all corners the "correct" way. They finish the first corner with R' D2 R F D2 F', which is now the correct way to twist the corner. Then they "pinwheel" the next corner over, and continue similar to above.

In the student's mind they are doing:
(Read the first corner incorrectly the first time around) (Read the first corner correctly the second time around) (pinwheel the next twisted corner over) (twist the next corner correctly) (pinwheel the next corner over) (twist the final corner correctly) (bring the U layer back into the original start position)

In actuality I have them skip the last U turn, as the corners all show the correct sticker on top, which makes this step already completed by that point.

[/ridiculously long post]

 

[fanwuq]

Very insightful post Chris!
I think the corners are the hardest to solve intuitively. For edges, solving middle layer than using M2 is pretty easy. I've taught one beginner to solve edges that way and it's not too hard to learn.

 

[Am1n-]

Maybe you can use something like the human thistlewaite, there, you orient the edges first, make a cross on D and U (while you tread oposite colours as the same), then you insert the corners on D and U while preserving orientation. This can all be done easily without any algs. Then permute the corners, this has 1 alg, but a very simple one, so you could teach this as intuitive. Next you permute the edges, wich is also is intuitive.

some tutorials you can find here:
writen: http://www.ryanheise.com/cube/human_thistlethwaite_algorithm.html
youtube: http://www.youtube.com/watch?v=xw5luzfkO48

most of this can be done intuitive, but its quite hard in the beginning.
hope this helps

mvg

 

[badmephisto]

Thank you chris. Naturally, I did not take any offense, and I appreciate your insight. It is certainly a good strategy to relate new concepts to the old ones already acquired, in general. The trouble with that of course is that that way is not intuitive at all-- It's just a memory trick. My aim here is to try to find a way to solve the cube that a bright beginner could easily pick up, given enough fiddling-around time. ie no memory work at all. I think we can agree that for that purpose then, commutators are not exactly applicable.

 

[cmhardw]

My aim here is to try to find a way to solve the cube that a bright beginner could easily pick up, given enough fiddling-around time. ie no memory work at all. I think we can agree that for that purpose then, commutators are not exactly applicable.

Yes I can see that commutators would not be immediately considered intuitive by a beginner, and would start out as a memory trick - understanding would have to come later.

All that comes to mind right now is Am1n-'s suggestion of human thistlewaite. Explain the concept of "reduction" to a simpler sub-group of the cube as making the cube a "simpler puzzle" and just continue from there. I'm not sure if block building techniques work because of the LL. Also, if the method has to start out intuitive with no memory work whatsoever then commutators would have to be explained in the pure sense, which would most often result in frustrating the learner to the point of not learning :-(

I'd also vote for human thistlewaite, or some other form of reducing the cube to simpler sub-groups. That is if we're restricting the beginner to 100% intuitive with no memory work.

Also, I think some memory followed by understanding is ok for a beginner. To explain an entire solution to the point of 100% understanding might delay the learner from learning at the pace they want to. Remember that most beginners just "want to solve the damn thing". So to explain to them in a pure sense why every single turn they use works might take longer than using a mix of intuition and memory. If you use the right kind of memory tricks they will, hopefully, come to understand how those parts of the solve work later.

Just an idea. Perhaps we're asking the wrong question. Why should we teach beginners methods that are 100% intuitive right from the start? If there is a very simple way to do this, then of course that approach makes sense. But it seems we are down to the very theoretical options like Human Thistlewaite to accomplish this restriction.

Chris