# Theoretical World Record Prediction for Big Cubes

#### obelisk477

##### Member
Got bored and created an excel spreadsheet based on current WR averages for 2x2 - 7x7, and it fits the regression curve nicely (r^2=.998). It predicts probable world record averages for cubes from 8x8 (4:11) to 20x20 (39:19). For those of us playing with big cubes, it can give numbers to shoot for.

2x2- 0:00:02
3x3- 0:00:06
4x4- 0:00:26
5x5- 0:01:00
6x6- 0:01:49
7x7- 0:02:53
8x8- 0:04:11
9x9- 0:05:45
10x10 0:07:34
11x11 0:09:37
12x12 0:11:56
13x13 0:14:29
14x14 0:17:17
15x15 0:20:20
16x16 0:23:38
17x17 0:27:11
18x18 0:30:59
19x19 0:35:01
20x20 0:39:19

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#### brian724080

##### Member
Cool idea, I'm far from those times, so I won't even think about it, but I've never actually thought that you can fit a curve on times like these.

Btw, you registered four years ago and this is your first post?

#### Rocky0701

##### Member
Got bored and created an excel spreadsheet based on current WR averages for 2x2 - 7x7, and it fits the regression curve nicely (r^2=.998). It predicts probable world record averages for cubes from 8x8 (4:11) to 20x20 (39:19). For those of us playing with big cubes, it can give numbers to shoot for.

2x2- 0:00:02
3x3- 0:00:06
4x4- 0:00:26
5x5- 0:01:00
6x6- 0:01:49
7x7- 0:02:53
8x8- 0:04:11
9x9- 0:05:45
10x10 0:07:34
11x11 0:09:37
12x12 0:11:56
13x13 0:14:29
14x14 0:17:17
15x15 0:20:20
16x16 0:23:38
17x17 0:27:11
18x18 0:30:59
19x19 0:35:01
20x20 0:39:19
This is awesome, and could really help people who are getting into big cubes. It won't really help for 11x11 and up, because the technology just isn't there yet.

#### qqwref

##### Member
What kind of regression did you use for this? Depending on the curve it might make a lot of sense or zero sense to extrapolate it this far. I'd also like to point out that for larger cubes it gets very hard to find pieces (with a Reduction method, anyway) but that doesn't kick in until well after the 7x7.

#### sukesh12

##### Member
Nice to see this kind of thing. I'm aiming for the 5x5 world record thing. We could see 39:19 on 20x20 only if we anybody has one. Theoretically, that figure is true though.
Looks like you did a lot of work. Good job.

#### IRNjuggle28

##### Member
The curve is less steep than reality, I think. Even if a 20x20 could be created that functioned well, less than 40 minutes is pushing it. I guess it could make a small difference that your numbers aren't that exact (2x2 average is actually 1.69, less than 2 seconds, while 3x3 is 6.54, greater than 6 seconds. In this case, rounding them steepened the curve slightly) and are in some cases wrong. 4x4 is 28 seconds; 5x5 is 55 seconds; 6x6 is 1:51, and 7x7 is 2:52.

The number of pieces to solve increases pretty fast. I don't have the numbers, but it's not even close to linear and it's more than your numbers would suggest, I think. The difference between the number of pieces in the 19x19 and the 20x20 is 222, if I counted right.

Another reason these numbers would be skewed is that the puzzles get harder to hold at an increasing rate. The difference between turning a 3x3 and a 4x4 is much less than the difference between a 7x7 and an 8x8, in my opinion. 7x7 is the size that is just approaching the capacity of two normal sized human hands. A physical 20x20 would have a ridiculously slow turn speed. This makes me wonder how much more realistic the theoretical average would be on computer cubes instead of physical puzzles. EDIT: I'd be interested to hear qqwref's opinion on that last idea, since he's done a lot of virtual cubes.

So, the number of pieces increases at an increasing rate, and the turn speed decreases at an increasing rate. That accounts for some of the graph's inaccuracy when compared to actual UWRs up to 11x11, but that's exactly why you called it "theoretical." It's not directly comparable.

I'm sure a more complex graph could be made. It could take into account the number of pieces in the puzzle (and also note the solving time of each subcategory, since the solving of centers, edges, and corners is very different) as well as the rate that turn speed decreases.

This post was sloppier than it could've been. This is a thought provoking topic, and I may decide later that I have more to say, but I don't have the energy to articulate these ideas more at the moment. Lol.

Nice first post, and welcome to the forum.

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#### qqwref

##### Member
As far as computer cubes go, the best bigcube method I had uses a pretty much constant time per piece. It helps that you can turn any layer quickly as long as you can get your mouse there. An NxNxN cube has N^3 - (N-2)^3 = 6N^2 - 12N + 8 pieces (subtract 6 on odd cubes if you don't want to count the fixed centers). So the time grows quadratically. Sample PBs are 9:29 for 10x10, 34:13 for 20x20, and 2:38:22 for 40x40 (about 4 times increase for each doubling of cube size).

For keyboard cubes, where you can't quickly turn whatever layer you want, the time grows faster. Sample PBs are 42ish seconds for 5x5, 7:38 for 10x10, and 1:11:48 for 20x20 (about 10 times increase for each doubling of cube size).

I imagine real cubes would be somewhere in between - as long as your hands are big enough to hold the thing (which has been a problem for me ) turning specific layers on bigger cubes does get harder, but probably not as much as it does for keyboard cubes.

Oh, by the way, don't forget that when considering UWRs for real cubes of size 8x8+ they are probably not as high-quality as the UWRs for smaller cubes, due to lack of competition, and not as much motivation to really get the best possible time.

#### ryanj92

##### Member
Thanks for this, you inspired me to try the same thing myself!
I fit both the current single and average world records to a generic power law fit (A*t^B) within MATLAB, and extrapolate the data in a similar way. To be conservative and remain within current mass production cubes, I have only extended the fit to cubes of order 11.

The power law fits both came back with A = ~0.25, B = ~3.3 (both with R-sq ~0.996), indicating an almost cubic (lol) growth in time as cube order increases.
Spoiler has results. Lower and upper bound predictions come from varying B to +- 0.5 of a standard deviation.

Single for cube of order 8 is 4:12.78
Lower bound is 3:25.76
Upper bound is 5:10.38

Average for cube of order 8 is 4:30.01
Lower bound is 3:35.63
Upper bound is 5:38.11

Single for cube of order 9 is 6:15.37
Lower bound is 5:02.00
Upper bound is 7:46.30

Average for cube of order 9 is 6:36.87
Lower bound is 5:12.92
Upper bound is 8:23.34

Single for cube of order 10 is 8:54.65
Lower bound is 7:05.68
Upper bound is 11:11.11

Average for cube of order 10 is 9:20.12
Lower bound is 7:16.63
Upper bound is 11:58.52

Single for cube of order 11 is 12:16.25
Lower bound is 9:40.69
Upper bound is 15:32.90

Average for cube of order 11 is 12:44.95
Lower bound is 9:50.19
Upper bound is 16:31.45

Graph of fit: http://puu.sh/9iI6b/f1a04023f2.png

Hope this is interesting to you

EDIT: for fun, I extended the results to cubes of order 12 to 17

Single for cube of order 12 is 16:26.01
Lower bound is 12:51.01
Upper bound is 21:00.15

Average for cube of order 12 is 16:56.72
Lower bound is 12:57.09
Upper bound is 22:10.23

Single for cube of order 13 is 21:29.96
Lower bound is 16:40.73
Upper bound is 27:41.69

Average for cube of order 13 is 22:00.91
Lower bound is 16:40.89
Upper bound is 29:03.25

Single for cube of order 14 is 27:34.32
Lower bound is 21:14.02
Upper bound is 35:46.70

Average for cube of order 14 is 28:03.13
Lower bound is 21:05.17
Upper bound is 37:19.16

Single for cube of order 15 is 34:45.48
Lower bound is 26:35.13
Upper bound is 45:24.68

Average for cube of order 15 is 35:09.10
Lower bound is 26:13.57
Upper bound is 47:06.87

Single for cube of order 16 is 43:09.99
Lower bound is 32:48.41
Upper bound is 56:45.45

Average for cube of order 16 is 43:24.65
Lower bound is 32:09.78
Upper bound is 58:35.54

Single for cube of order 17 is 52:54.57
Lower bound is 39:58.26
Upper bound is 69:59.14

Average for cube of order 17 is 52:55.75
Lower bound is 38:57.53
Upper bound is 71:54.55

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#### obelisk477

##### Member
It fits a quadratic equation with r^2=.998, which makes sense because as puzzle size increases linearly (2x2 to 3x3) with respect to each side, what you're really saying its that the volume of the cube is increasing (cubically?) per increase in side length. Or x^3 / x (volume / side length) = x^2, which means that the puzzles should get harder with respect to how many more pieces there actually are. Could fit a cubic regression nicely as well, I suppose.

Also, given the r^2 value that is so close to 1, even if the curve isn't exactly the right one, it should do fine for extrapolating no further than the number of cubes posted here. I would think that the time wouldn't deviate by any more than 30 seconds at the upper end given how closely the function matches the current world record for actual cubes.

By the way the function is y = 7.4489x^2 - 32.941x + 38.187 (where x is the length of one side of a cube), and y is the theorectical world record average in seconds.

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#### obelisk477

##### Member
Good thoughts, my only reply is in regards to the rounding that was necessary for some of the smaller cubes. The numbers I got were actually more accurate than represented in the results here for the smaller cubes, but since it only shows whole seconds, the numbers make a little more sense once you get into bigger cubes where milliseconds dont matter so much.

#### 10461394944000

##### Banned
It fits a quadratic equation with r^2=.998, which makes sense because as puzzle size increases linearly (2x2 to 3x3) with respect to each side, what you're really saying its that the volume of the cube is increasing (cubically?) per increase in side length. Or x^3 / x (volume / side length) = x^2, which means that the puzzles should get harder with respect to how many more pieces there actually are. Could fit a cubic regression nicely as well, I suppose.
big cubes are harder to hold and harder to turn as fast, so these estimates are probably under estimates

#### obelisk477

##### Member
big cubes are harder to hold and harder to turn as fast, so these estimates are probably under estimates
definitely, this would just be in an ideal situation where you could turn as easily as your brain could process what you're seeing (which is basically the case for up to 7x7). I remember solving an 11x11 at nationals a few years back, and my hands were killing me by the time I finished. It would be interesting to find a way to fit some kind of mathematical term in for hand fatigue, given more data of course.

#### uberCuber

##### Member
definitely, this would just be in an ideal situation where you could turn as easily as your brain could process what you're seeing (which is basically the case for up to 7x7). I remember solving an 11x11 at nationals a few years back, and my hands were killing me by the time I finished. It would be interesting to find a way to fit some kind of mathematical term in for hand fatigue, given more data of course.
Just solving 6x6 makes my hands tired >_>

#### obelisk477

##### Member
What kind of regression did you use for this? Depending on the curve it might make a lot of sense or zero sense to extrapolate it this far. I'd also like to point out that for larger cubes it gets very hard to find pieces (with a Reduction method, anyway) but that doesn't kick in until well after the 7x7.

#### obelisk477

##### Member
Thought I'd update my first post ever with the latest stats. Here it is:

8x8: 3:00.70
9x9: 4:05.76
10x10: 5:20.862
11x11: 6:46.00
12x12: 8:21.17
13x13: 10:06.39
14x14: 12:01.64
15x15: 14:06.93
16x16: 16:22.26
17x17: 18:47.62

#### Loser

##### Member
Thought I'd update my first post ever with the latest stats. Here it is:

8x8: 3:00.70
9x9: 4:05.76
10x10: 5:20.862
11x11: 6:46.00
12x12: 8:21.17
13x13: 10:06.39
14x14: 12:01.64
15x15: 14:06.93
16x16: 16:22.26
17x17: 18:47.62
What are the lower cubes out of curiosity?

#### obelisk477

##### Member
What are the lower cubes out of curiosity?
The regression is so good that it matches up so much with the current records, so I didn't think it was worth posting.

#### Loser

##### Member
My curiosity is killing me

#### Christopher Mowla

My curiosity is killing me
I guess we will await his response!

Since you are curious about his, I thought I would share my nearly 3-year-old formula (which is really 6 years old). Although before your membership join date on here, I put this formula in this post (a link to this post among links to my other cubing contributions can be found in this post).

Substituting 5.80 (the current official 3x3x3 WR avg time) into the formula (which outputs the solve times in seconds), I get this for the 4x4x4 through 17x17x17.

Of course, to get my formula to get a result close to the current 4x4x4 UWR of 16.13, we should input 5.1155 in place of 5.8. This gives us this set of times for the 4x4x4 through 17x17x17 UWRs. So for example, it's claiming that if the 4x4x4 UWR is 16.13, then the 17x17x17 UWR should be 3206/60 = 53:26:00 (but the current UWR is 56:51.03).

And for everyone who knew me before, after a very long hiatus, I'm back! (I haven't been logged in for about 2 years. I have of course been updating the 4x4x4 parity algorithms speedsolving wiki page recently, but today I decided today is the day I finally return!)

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