The Limits of Solving the Cube

[Dane man]

After looking into many methods, and even making a few, I have been thinking about the limits of methods when it comes to the ratios of move count vs number of algorithms in use. Not only that, but the "human" limits when it comes to intuition, recognition, and memorization.

It appears that each method has it's own pros and cons, each is a balancing act.

The general tendency is to increase the number of algorithms in order to decrease the number of moves. If you want to decrease the number of algorithms, you generally have to increase the number of moves or even the number of substeps.

If you want to increase the chance of quick recognition and recall(memorization success), you generally have to reduce the number of algs. If you want to reduce the number of moves and substeps, you generally have to increase the amount recognition and recall required of the cuber.

And there are different issues based on your goal. For example, for speedcubing, sometimes a larger algorithm needs to be chosen to achieve faster fingertricks and thus execution. While in FMC, it's all about the fewest moves, and so shorter algs can be chosen regardless of fingertrick ability.

When it comes to methods, speedcubing tends to use methods with swift recognition and a logical step by step progression at the sacrifice of the number of moves in order to benefit from the speed of execution. While in FMC, some methods require more complex thinking and abstract concepts for the benefit of reducing the number of moves, at the sacrifice of swift recognition and execution time. And yet in both, every benefit counts, even the time used in FMC.

These appear to be a constant battle, and I've been wondering recently, what are the limits of these balances? How can we go about finding an optimal balance? What sorts of undiscovered methods are out there that might be much more efficient than what we are using right now? What sorts of extremes can be explored? For example, at the expense of alg count, what kind of methods can be found that solve the cube logically with as few steps as possible? (the most extreme being 43 quintillion algorithms for 1-Look solved from all orientations.) OR at the sacrifice of steps, using as few algorithms as possible, how intuitive can a method be made? What would be the most intuitive?

From everything I've seen, there's always a trade off, and the trade off isn't always equal (For example: If reducing the number of moves by 10 only increases the number of algs by 1, then that is a good (unbalanced) trade off). And so, we explore these trade offs until we are able to "juice" a method or it's substeps, until we achieve the most efficient method when it comes to ratios of algs vs moves, efficiency vs human brain power. It's the problem we are still looking to solve.

What do you guys think of the ways we go about this? What are your preferences and priorities in regards to the balances? How does this influence your cubing style and efficiency? Any other comments/thoughts?

 

[tseitsei]

Well Heise is most intuitive because you need 0 algs...

Also just solving only with commutators (like in BLD) is entirely intuitive and doesn't need any algs (except for parity, but that also can be solved intuitively. Just do U-move(or any quarter turn) and fix with commutators)

 

[Tempus]

For example, at the expense of alg count, what kind of methods can be found that solve the cube logically with as few steps as possible? (the most extreme being 43 quintillion algorithms for 1-Look solved from all orientations.) OR at the sacrifice of steps, using as few algorithms as possible, how intuitive can a method be made? What would be the most intuitive?

The opposite extreme of your 1-Look solution system would be a solution system based on the Bogosort sorting algorithm. It would have the longest runtime and the fewest algorithms, which is to say zero. Here's how it would work:

• Step 1: You close your eyes and start scrambling the cube until you feel that it's good and mixed.
• Step 2: You then open your eyes and check to see if it's fully solved. It if is, you're done. If it isn't, you start back at step 1.

I believe that this method is the most intuitive, so much so that people have probably used it in the past without ever knowing that it was a solution system.

I hereby name this system Bogosolve.

 

[pewpewrawr]

There already is a method with perfect balance, that's why I chose it.

 

[Tao Yu]

I was thinking recently that CFOP substeps with high alg counts (ZBLL, OLLCP, RV) could be made a lot more useful if we could somehow highlight all of the nice cases. Like, a wiki page with only easy-to-recognise ZBLL cases would be really really useful. So instead, of learning all of the algs, we just learn the nice ones. So you can save moves without learning all of the algorithms.

Btw, here's a nice WV case that everyone should know: link

someone make this pls

 

[goodatthis]

I think that this has been explored several times already, and I personally think that the perfect method is entirely dependent on the person using it. You could do FMC for your solves, do 25 moves and turn at 2.5 TPS, and average 10 seconds, and likewise you could use the sexy method and turn at 10 TPS for a 100 move solve, and average 10 seconds. But for comparisons like Roux vs CFOP vs ZZ vs Petrus, CFOP and Roux work best for most people because the so called "balances" work the best for the people who use them. The reason why ZZ works best for Phil Yu and why Petrus works best for Erik Johnson is the same. You will never have an ideal balance, because some people will always be wired differently than others. Plus there are lots of other factors.

 

[stoic]

I was thinking recently that CFOP substeps with high alg counts (ZBLL, OLLCP, RV) could be made a lot more useful if we could somehow highlight all of the nice cases. Like, a wiki page with only easy-to-recognise ZBLL cases would be really really useful. So instead, of learning all of the algs, we just learn the nice ones. So you can save moves without learning all of the algorithms.

This is a VERY good idea.
I remember asking a while back on somebody's video how they managed to 1-look the LL and was told "it's an easy case, you should know it". Which left me thinking: fair point, but how?

 

[Hypocrism]

I was thinking recently that CFOP substeps with high alg counts (ZBLL, OLLCP, RV) could be made a lot more useful if we could somehow highlight all of the nice cases. Like, a wiki page with only easy-to-recognise ZBLL cases would be really really useful. So instead, of learning all of the algs, we just learn the nice ones. So you can save moves without learning all of the algorithms.

Btw, here's a nice WV case that everyone should know: link

someone make this pls

I don't think so. If you only learn a subset of an algorithm-based step, you need to first check if you know the 1LLL and then recall and execute, but if you don't know it you need to check, then switch and recognise the different algorithm. That's why you can't, for example, really just learn say half of the cases for each COLL set, because it just adds an extra recognition step.

 

[Tao Yu]

I don't think so. If you only learn a subset of an algorithm-based step, you need to first check if you know the 1LLL and then recall and execute, but if you don't know it you need to check, then switch and recognise the different algorithm. That's why you can't, for example, really just learn say half of the cases for each COLL set, because it just adds an extra recognition step.

It's not that hard. People people like Feliks make use of easy ZBLL cases all the time. I just taking about an easier way to find easy cases.

 

[Dane man]

I think that this has been explored several times already, and I personally think that the perfect method is entirely dependent on the person using it. You could do FMC for your solves, do 25 moves and turn at 2.5 TPS, and average 10 seconds, and likewise you could use the sexy method and turn at 10 TPS for a 100 move solve, and average 10 seconds. But for comparisons like Roux vs CFOP vs ZZ vs Petrus, CFOP and Roux work best for most people because the so called "balances" work the best for the people who use them. The reason why ZZ works best for Phil Yu and why Petrus works best for Erik Johnson is the same. You will never have an ideal balance, because some people will always be wired differently than others. Plus there are lots of other factors.

This is true, which is why I asked about preferences and priorities. For example, someone who has a low TPS but good memory might want a method with more algs and less moves, while someone with bad memory but quick reflexes might want a method with more moves and less algs. The methods that can be useful because of this have more variety, and I'm willing to bet that there are few if any cubers who use exactly the same method and exactly the same algorithms these days. And when it comes to fingertricks, I think the variety in style is even greater.

I was thinking recently that CFOP substeps with high alg counts (ZBLL, OLLCP, RV) could be made a lot more useful if we could somehow highlight all of the nice cases. Like, a wiki page with only easy-to-recognise ZBLL cases would be really really useful. So instead, of learning all of the algs, we just learn the nice ones. So you can save moves without learning all of the algorithms.

Btw, here's a nice WV case that everyone should know: link

someone make this pls

This is a VERY good idea.

This makes a lot of sense. I've thought of it long long ago, but then forgot. I really want to do it.

I don't think so. If you only learn a subset of an algorithm-based step, you need to first check if you know the 1LLL and then recall and execute, but if you don't know it you need to check, then switch and recognise the different algorithm. That's why you can't, for example, really just learn say half of the cases for each COLL set, because it just adds an extra recognition step.

Actually there are ways to mentally organize it efficiently. Where recognition of the 1LLL will be a separate case from the other cases. For example, 1LLL will look one way, and a normal case will appear another, and so you separate them in your mind so that they can't be considered as both at the same time, thus making it just as efficient as any other set of memorized algs.

 

[pewpewrawr]

For example, someone who has a low TPS but good memory might want a method with more algs and less moves, while someone with bad memory but quick reflexes might want a method with more moves and less algs..

What about roux? It's low on algs and moves.

edit: or any highly intuitive method for that matter.

 

[Dane man]

What about roux? It's low on algs and moves.

edit: or any highly intuitive method for that matter.

Roux is slower for those who can't do slice move fingertricks as well as they can do face move fingertricks. And because of it's great intuitiveness, the recog for the construction of blocks and the orientation/permutation of edges only comes through a decent amount of time practicing, and a memory good enough to sustain the habit. Some only have the memory capabilities (or simply the determination) enough to do LBL, on piece at a time with only 4 simple algorithms for the last layer. The ability to intuit and build the habit of intuition isn't a super common gift (hence the reason for so few speedcubers in the world).

The trade off is low algs, low moves (which I personally love), but at the sacrifice of the slice moves and higher intuition required.

 

[Hypocrism]

Actually there are ways to mentally organize it efficiently. Where recognition of the 1LLL will be a separate case from the other cases. For example, 1LLL will look one way, and a normal case will appear another, and so you separate them in your mind so that they can't be considered as both at the same time, thus making it just as efficient as any other set of memorized algs.

It still sounds dodgy to me, at least, that kind of strategy hasn't worked for me in the past. It might work for others, but I am skeptical as to whether it is worthwhile. Say you know COLL and you're learning ZB: you know some ZB cases for one of the COLLs but not others. If a case of this COLL comes up that you don't know the ZBLL for, you need to check to see if it's one you know, and then revert to your fastest alg for that COLL case. That's just what I found when I started to learn COLL and didn't keep going with it: until I finished the entire set of COLLs for one OCLL case, or dropped COLL for that OCLL case, I would falter on that case. In the same way you might get a fast LL when you know the ZBLL that comes up, but on average, your time for edges-oriented LL will drop due to that miniscule faltering. I value the consistency more than the occasional faster solve.

 

[Dane man]

I value the consistency more than the occasional faster solve.

And that's where personal preference comes in. One can do as you do and prefer to be consistent. Or there might be someone who like to be and has the capability to be dynamic in their methodology. As an example, say there are all the COLL cases and your memorized ZB cases. The overlapping cases you can look at and say (alright, now these are ZB cases, and not COLL cases). If you practice looking at the cases as a whole instead of as individual steps in a progression, then you can recognize the ZB cases as if they were separate from and independent of the COLL cases, coexistent as individual cases and algorithms for the same moment in the solve. It's kind of tricky to train the mind that way, but if one knows how to make themselves look at something differently (looking at the last layer as a whole instead of in parts like corners, edges, orientation, permutation, etc), then one can have great success in using this method. But do as you feel best. Each of us works and thinks differently.

 

[pewpewrawr]

Roux is slower for those who can't do slice move fingertricks as well as they can do face move fingertricks. And because of it's great intuitiveness, the recog for the construction of blocks and the orientation/permutation of edges only comes through a decent amount of time practicing, and a memory good enough to sustain the habit. Some only have the memory capabilities (or simply the determination) enough to do LBL, on piece at a time with only 4 simple algorithms for the last layer. The ability to intuit and build the habit of intuition isn't a super common gift (hence the reason for so few speedcubers in the world).

The trade off is low algs, low moves (which I personally) love, but at the sacrifice of slice moves and higher intuition required.

Fair enough, I was just trying to make the point that alg count and move count aren't always relative. I didn't really take determination/practice-time into consideration as a factor, but I suppose for some people that could be an issue. I guess you may be right in saying that there is always a trade off of some sort, but as far as I'm concerned roux is without a doubt the most balanced method for speed.

 

[Dane man]

Fair enough, I was just trying to make the point that alg count and move count aren't always relative. I didn't really take determination/practice-time into consideration as a factor, but I suppose for some people that could be an issue. I guess you may be right in saying that there is always a trade off of some sort, but as far as I'm concerned roux is without a doubt the most balanced method for speed.

I agree. That's also what I was saying when I said the trade off isn't always equal. If reducing the number of moves by 10 only increases the number of algs by 1, then that is a good (unbalanced) trade off. And I love to do roux solves because of their incredible intuitiveness, but I don't use them in speedsolving because I don't have the slice move speed that some do. I could probably try to practice it and get it pretty fast, but it's still akward for my fingers given the way I hold the cube. There are a lot of factors that go into choosing a method.

 

[GuRoux]

Roux is slower for those who can't do slice move fingertricks as well as they can do face move fingertricks. And because of it's great intuitiveness, the recog for the construction of blocks and the orientation/permutation of edges only comes through a decent amount of time practicing, and a memory good enough to sustain the habit. Some only have the memory capabilities (or simply the determination) enough to do LBL, on piece at a time with only 4 simple algorithms for the last layer. The ability to intuit and build the habit of intuition isn't a super common gift (hence the reason for so few speedcubers in the world).

The trade off is low algs, low moves (which I personally love), but at the sacrifice of the slice moves and higher intuition required.

sometimes the slice move is not a sacrifice, and the intuition is not much more than cfop in my opinion.

 

[Renslay]

sometimes the slice move is not a sacrifice, and the intuition is not much more than cfop in my opinion.

I agree.

 

[Christopher Mowla]

I couldn't help but interpret the first post as calculus.

because

It should be apparent without calculus, but, in other words, I don't think there ever will be an agreement between everyone of an optimal balance because people's opinion on what matters to them differs. As shown in the calculus example, we can evaluate one sided opinion limits, but we cannot evaluate a limit which takes opinions from all sides (maybe I should have used a multivariate limit instead to describe more than two options) because not all opinions are the same.

For example, at the expense of alg count, what kind of methods can be found that solve the cube logically with as few steps as possible? (the most extreme being 43 quintillion algorithms for 1-Look solved from all orientations.) OR at the sacrifice of steps, using as few algorithms as possible, how intuitive can a method be made? What would be the most intuitive?

I'm curious as to what you and others consider to be "1-look". Is it solving a cube in such a way that no portion of the cube is permanently solved (so that we do not mess up what we solved previously and then solve it back by solving more of the puzzle) until the last moves?

Even if we do in the future find such a method (if it doesn't exist already and it is just not publicly known yet), I don't think everyone will agree that it is what they consider to be 1-look (I believe I have found such a method which is not the "Bogosolve" method, but I actually consider it to be 2-look or 3-look, depending on if some slices have an odd permutation) because the nature of our problem-solving minds is to break big problems into several smaller ones (for example, the 3x3x3 is solved optimally in 2 phases, which to me sounds like 2-look), and therefore even if it is theoretically possible to solve the cube in 1-look from some people's perspective (and it is not the "Bogosolve" method), some people will come along and claim that this "1-look" can be broken into several substeps.

For example, back to calculus, we can technically find the area under a curve in one step (evaluate a definite integral from a to b), but the definition of the integral says that we find the area under a curve by taking the limit of the sum of an infinite number of sub-intervals.

As another example, if you see the link to the number of OLLs in my signature, I created a formula from cuBerBruce's vector function, which iteratively calculates the number of OLLs, whereas my formula, which is nothing other than an interpretation of cuBerBruce's vector function, technically calculates the number of OLLs in a single step. How can we say (if you see my derivation of my formula) that I am not summing numbers in several steps when using my formula?

Also, we have to agree on what is considered to be an "algorithm". Obviously some people believe that commutators are not algorithms because they are "intuitive", but clearly they are referring to certain 3-cycle commutators.

If we can't agree on the above two concepts, then how can we say that it's possible to solve the cube in 1-look or n-look or determine how many algorithms a method requires?...which brings me back to the first calculus example.

 

[GuRoux]

I couldn't help but interpret the first post as calculus.

because

It should be apparent without calculus, but, in other words, I don't think there ever will be an agreement between everyone of an optimal balance because people's opinion on what matters to them differs. As shown in the calculus example, we can evaluate one sided opinion limits, but we cannot evaluate a limit which takes opinions from all sides (maybe I should have used a multivariate limit instead to describe more than two options) because not all opinions are the same.


I'm curious as to what you and others consider to be "1-look". Is it solving a cube in such a way that no portion of the cube is permanently solved (so that we do not mess up what we solved previously and then solve it back by solving more of the puzzle) until the last moves?

Even if we do in the future find such a method (if it doesn't exist already and it is just not publicly known yet), I don't think everyone will agree that it is what they consider to be 1-look (I believe I have found such a method which is not the "Bogosolve" method, but I actually consider it to be 2-look or 3-look, depending on if some slices have an odd permutation) because the nature of our problem-solving minds is to break big problems into several smaller ones (for example, the 3x3x3 is solved optimally in 2 phases, which to me sounds like 2-look), and therefore even if it is theoretically possible to solve the cube in 1-look from some people's perspective (and it is not the "Bogosolve" method), some people will come along and claim that this "1-look" can be broken into several substeps.

For example, back to calculus, we can technically find the area under a curve in one step (evaluate a definite integral from a to b), but the definition of the integral says that we find the area under a curve by taking the limit of the sum of an infinite number of sub-intervals.

As another example, if you see the link to the number of OLLs in my signature, I created a formula from cuBerBruce's vector function, which iteratively calculates the number of OLLs, whereas my formula, which is nothing other than an interpretation of cuBerBruce's vector function, technically calculates the number of OLLs in a single step. How can we say (if you see my derivation of my formula) that I am not summing numbers in several steps when using my formula?

Also, we have to agree on what is considered to be an "algorithm". Obviously some people believe that commutators are not algorithms because they are "intuitive", but clearly they are referring to certain 3-cycle commutators.

If we can't agree on the above two concepts, then how can we say that it's possible to solve the cube in 1-look or n-look or determine how many algorithms a method requires?...which brings me back to the first calculus example.

i always considered solving blindfold as one looking since you only look once. if you predict your pll before you do your oll, then that's one look. one looking doesn't matter how many algs, moves, or commutators, just that you can solve the whole thing without looking again. and i would say the limit does exist because truth is not bent by what people's opinions are. i would like to think we would get close to the "limit" sooner or later. and most of the time, in any sport or game, there will develop a strong general consensus on what is "best," not always the "limit," despite there being people preaching "no best ... or depends on the person's ..."