Using Calculus to Determine the Speed Frontier?

[lerenard]

I recently began solving the Square-1. My first ao5 was 6 minutes. Only a few days later I was averaging 1 minute.
Last month on 4x4 I was averaging 1:45 and now I've only shaved 15 seconds off my times.
A non-cuber might think I'm absolutely awful at 4x4, given that I've improved by such a paltry amount compared to Square-1 even with a much longer timeframe.
A cuber sees this is probably due to the fact that I had already been speedsolving the 4x4 for quite some time, and in fact I have. This illustrates that improvement is fast in the beginning, and slower later. While some say the only set limit is sub-zero, I would propose that for any individual person using any set method, assuming they only practice that and don't learn anything new, we could use calculus to determine how fast they could get. This would require them to do an ao100 (at least) once a day for a long time (a year?). We could then plot their progress on a graph with x as the number of days since they began and y as their time in seconds and find the line of best fit (probably a logarithmic equation). Then we find the limit of x as x approaches infinity and voila, we've found their limit. Of course, If they learn something substantially new (A new alg set would be an example, but something like lookahead would not be, since it happens naturally) then they could decrease their times even more.

I would further propose that if we took a sufficiently large sample of all cubers and performed this experiment we could discover the limit of everyone's efforts put together. I'm not 100% sure the math would work and of course there would always be the possibility that the community would discover something new and substantially more effective at improving, but it would be interesting.

Another option would be to draw from only very fast cubers, as drawing from even amateurs and people who don't practice very much could skew the results.

I don't think this would be very practical because people are busy and don't practice very consistently most of the time, but it's still an interesting idea. What do you think? Is there a limit to cubing, or at least a limit for a particular person practicing a particular amount with a particular method? What do you think it is? Jessica Fridrich once said that because God's number is 20 and max tps is around 10, a normal human physically could not average below 2 seconds (even if they could see the entire optimal solution in 15 seconds!) But could anyone ever get that low? I've been plotting my progress, but I don't have a very large sample yet, only about 2 months worth of averages, and practical complications have gotten in the way of my consistency, so I don't know how useful the data are.

 

[mDiPalma]

you could try taking some ancient cuber's WCA results and see their predicted minima.

i'd be interested to see what you find

 

[qqwref]

If you extrapolate in the way you're suggesting, there's no reason to think the prediction will be accurate. It'll be a prediction, sure, but it is based on the person's improvement curve and not on their actual theoretical best time. I'd also point out that at the lowest and highest speeds people don't improve in a nice clean way, and certainly not in a way that is based on the best possible time for that puzzle. At very slow speeds (such as the 6 minute sq1 time) you mainly improve by becoming way more efficient in leaps and bounds (through learning algs) and by getting past the "I have no idea what I'm doing" hump. And at very fast speeds, some people might get constant improvement, but a lot of people improve very slowly and gradually with huge standard deviations due to having good/bad days with their hands or hardware. So, in short, you could get a prediction of what would happen if someone continued to improve as they did, but actual cubers don't improve in the same way over time, because the reason for improving changes. You can improve from 6 minutes to 1 minute in Square-1 by learning algs and getting a basic understanding of the puzzle, but you're never going to replicate that kind of improvement - even going from 60 seconds to 10 is only possible for a few of the most dedicated and physically fast solvers.

For what it's worth, this isn't a calculus question at all, but a statistics one (that's where concepts such as line of best fit, standard deviation, etc. come from, and where they are used).

 

[AlphaSheep]

As qqref says, this is statistics, not calculus.

Data from individual cubers is too erratic to be useful. You'd probably get a better idea by taking the average of something fairly consistent... Maybe take the average time of the top 100 ranked cubers in an event.

Have a look at this: https://www.worldcubeassociation.org/results/misc/evolution/

 

[mDiPalma]

As qqref says, this is statistics, not calculus.

Actually, it's both.

 

[Kit Clement]

Actually, it's both.

Of course, nearly all statistical procedures use calculus or a numerical procedure to emulate it, as many other fields do. But I would never say that I fit a calculus model to my data.

 

[qqwref]

Just because you can use integrals in a field doesn't mean that field is calculus. (To be clearer, continuous distributions in statistics are an application of calculus, not part of calculus.) But talking about best-fit lines implies discrete real-world data, which you should never be using an integral for.

 

[unsolved]

There is one thing I should warn you about predictions though: They involve the future.

 

[lerenard]

You would use statistics to collect the data, but then using the line of best fit from statistics to determine the limit would be calculus, would it not?

Either way, I acknowledged in the OP that it would be difficult to procure reliable data and so I guess my question was more theoretical than practical.

I do like the idea of taking the top100 in an event, as that might emulate an improvement curve.

I've also noticed that std tends to be higher as a percentage of time when the cuber isn't as skilled. Do you think there's anything there that could help us?

 

[qqwref]

You would use statistics to collect the data, but then using the line of best fit from statistics to determine the limit would be calculus, would it not?

It would not. It's just normal extrapolation using an elementary function, and has very little to do with the concept of a limit from analysis - you aren't going to be dealing with anything that isn't continuous over the domain you're interested in.

I've also noticed that std tends to be higher as a percentage of time when the cuber isn't as skilled. Do you think there's anything there that could help us?

Time = pauses + (moves / tps). Pauses decrease (as a percentage of time) with skill, so their effect on standard deviation gets smaller and smaller as skill increases; and if two random variables X and Y are independent, stdev(X+Y) = sqrt(stdev(X)^2 + stdev(Y)^2). The second part will always have some standard deviation due to the method/algs used to solve the cube. It's very hard to extrapolate how good movecount and TPS can get, but it's fair to say that a theoretical best solver would have zero pauses.

 

[lerenard]

It would not. It's just normal extrapolation using an elementary function, and has very little to do with the concept of a limit from analysis - you aren't going to be dealing with anything that isn't continuous over the domain you're interested in.


Time = pauses + (moves / tps). Pauses decrease (as a percentage of time) with skill, so their effect on standard deviation gets smaller and smaller as skill increases; and if two random variables X and Y are independent, stdev(X+Y) = sqrt(stdev(X)^2 + stdev(Y)^2). The second part will always have some standard deviation due to the method/algs used to solve the cube. It's very hard to extrapolate how good movecount and TPS can get, but it's fair to say that a theoretical best solver would have zero pauses.

Here's the best I've gotten so far, but I think this is a dead end. I couldn't figure out how to transform the data into the shape of graph I was expecting with the math skills I have.

So assuming you can't mathematically calculate the limit of 3x3 times from the data we have available, perhaps we could use the formula qqwref gave? Assuming pauses are 0, time = moves/tps.
I don't think people could solve a cube in less than 40 moves with more than 10 tps, so I think we can safely say no one will be able to consistently solve a cube in less than 4 seconds on average given 15 seconds of inspection time.

But are 40 move averages and 10 tps even possible? There's generally a tradeoff between lookahead/fingertricking and efficiency. So, could we set the limit even higher?

 

[Meep]

I work in a brain research lab that does a lot of learning curve-related experiments, and from the regression models I've used, the steady-state value is always really wonky; the lab cares more about the learning/exponential decay rate.

Here's a fit to the 3x3 WR averages over time using one of the common learning curve models:

The theoretical steady state 3x3 average is -3.9 seconds.

 

[lerenard]

I work in a brain research lab that does a lot of learning curve-related experiments, and from the regression models I've used, the steady-state value is always really wonky; the lab cares more about the learning/exponential decay rate.

Here's a fit to the 3x3 WR averages over time using one of the common learning curve models:

https://dl.dropboxusercontent.com/u/10477021/cube/wrfit.png

The theoretical steady state 3x3 average is -3.9 seconds.

So the data isn't very helpful :/ I'm still curious whether we could determine the limit based on the limits of a particular method (movecount, lookahead, etc.)

 

[cmhardw]

It may be a good experiment to have someone do an average of 100 solving Cube Explorer generated optimal Rubik's cube solutions using unlimited inspection time for each "solve".

To be more accurate, have lots of people do this, or lots of people who we would classify as "fast".

 

[Lucas Garron]

It may be a good experiment to have someone do an average of 100 solving Cube Explorer generated optimal Rubik's cube solutions using unlimited inspection time for each "solve".

To be more accurate, have lots of people do this, or lots of people who we would classify as "fast".

The Prepared Speedsolve Challenge has been running for 7.5 years. ;-)

You could also do it "speed BLD" style (indefinite memo, only one attempt), but I think the lower bound from the PSC is stronger.

 

[Stefan]

The Prepared Speedsolve Challenge has been running for 7.5 years. ;-)

Winning!