• Welcome to the Speedsolving.com, home of the web's largest puzzle community!
    You are currently viewing our forum as a guest which gives you limited access to join discussions and access our other features.

    Registration is fast, simple and absolutely free so please, join our community of 40,000+ people from around the world today!

    If you are already a member, simply login to hide this message and begin participating in the community!

[Proposal] Shorter scrambles for big cubes

Roman

Member
Joined
Jan 10, 2013
Messages
686
Location
Dubai, UAE
WCA
2012STRA02
YouTube
Visit Channel
I want to start by saying that the conception of the random state scrambles is really good. For those of you who doesn't know a lot about scramble generating, in short: a 3x3 scramble isn't just a random sequence of moves, it's the inversed solution of the generated random state. That is the reason we can't just start using 10-moves scrambles instead of what we have - not all the random states are achievable with 10 moves.

But in practise (right now I now mean the home practising) we never need a fairly random state. If I want to solve a 3x3, I always start by figuring out an easy cross or an x-cross. When I see a scrambled cube, find a cross solution and start solving it, at this point it doesn't matter if the whole cube was 10 or 20 moves away from the solved state at the beginning.
While scrambling the 3x3 doesn't bother me too much, it gets worse with bigger cubes, because the scramble length is unsatisfying. When I practise 5BLD, the only thing I care about in the cube state are the number of targets, the number of cycle breaks, corner twists etc. - and not how many moves it took to build a cube state with this set of features. If I could use 20 moves scrambles instead of 60 moves, with the number of solved pieces remained average (i.e. approximately the same as if I were using the random state scramble), that would be so great!
In fact, that's exactly what I did. I made a little analysis of the BLD features on a 1000 near-random-state scrambles (300 random moves each). Then I generated 20-move scrambles that for each feature have the value close to the corresponding value on the average scrambles. From now on, I will only use these 20-moves scrambles for my practising.
The only issue with the short scrambles I noticed is: some of them make the cube look somewhat regular. But that turned out to be easily solvable too, more info under the spoiler.

The condition of having little or no solved pieces doesn't guarantee that the cube will look random. Trivial example: (M E M' E') has no pieces being on their original positions. Scrambling the 5x5 with wide-moves only also creates a state that looks like... well, the 5x5 scrambled using the wide-moves only. Short scrambles tends to leave blocks like (corner+wing) paired, or (corner + 2 adjacent wings + x-center between them), or 1x1x3 center blocks. A random-state scramble should make the cube look more random.

As a quick dirty solution I defined the "regularity index" and made the generator discard any scramble leading to the state with too big value. It's simply a weighted sum of things like: the amount of adjacent stickers being the same color, the amount of solved centers, the number of similar wings (e.g. blue-red and red-blue) being around one middle edge and some other.
The result is quite satisfying. Here are examples of short (18 htm) scrambles with low regularity index:
https://alg.cubing.net/?puzzle=5x5x5&alg=D-_Lw_F_U_R-_D_F-_L_Dw_Fw_Bw_Rw2_Uw2_Rw_Lw-_Fw_L_R
https://alg.cubing.net/?puzzle=5x5x5&alg=U_L2_D2_L_R_B_F-_L2_D2_Rw-_F-_Bw-_Dw-_U2_Lw2_Fw_Bw2_Lw-
https://alg.cubing.net/?puzzle=5x5x5&alg=Bw_F2_U2_Bw_Rw_Fw2_Bw2_Rw_B-_Fw-_L_Fw2_Bw_R_Lw-_B_D_U
With more moves (22htm) we can achieve more randomness:
https://alg.cubing.net/?alg=B2_R_Bw...-_Dw_R_Lw-_Uw2_Bw2_D-_Fw-_Dw-_Uw&puzzle=5x5x5
https://alg.cubing.net/?alg=Uw_Lw2_...w2_Dw_Uw_Bw-_Rw_Uw_Dw-_L2_R_F2_U&puzzle=5x5x5
https://alg.cubing.net/?alg=B-_R2_L...w2_Lw-_Bw_F-_Uw-_L_D-_U_R_Lw2_B-&puzzle=5x5x5

I want to leave the 3x3 scrambles discussion for later. Currently I feel that 5x5+ scrambles can be improved.
The way I see it:
1) We collectively define a set of features by which we can estimate the cube state in different ways, e.g. number of solved pieces, size and number of NxM center blocks, the regularity index, the shortest center(3x3 block in the middle of the 5x5 face) solution length etc.
2) For each feature, calculate its distribution using very long random-moves scrambles.
3) For obtaining the short scramble, the program would generate a random-move N-moves scramble, calculate its features, and if at least one of them has the value too far away from its distribution mean, it would discard the scramble and generate the next one.

One objection that springs to mind is: the scramble should be solving-method-independant, and hence no amount of "features" can cover all theoretical solving methods that people can come up with. And although 99.99% of cubers use reduction method for solving the big cubes, I understand it can't be an argument. But just think about it: can the awareness of the fact that the cube is 25 moves away from the solved state really give the competitor any advantage?

In conclusion, if the scrambles will become at least 2 times shorter, imagine how much faster and easier the big cube rounds on the competitions will go. The probability of the mistake made by a scrambler will also be lower. I think it is worth considering.

Any thoughts?
 

Duncan Bannon

Member
Joined
Sep 9, 2017
Messages
1,817
Location
Here
I like that! I don’t do many big cubes. But when I do the scrambles are annoying long. Thanks for all your trainers and your continued work on Cubing!
 

Pyjam

Premium Member
Joined
Oct 8, 2010
Messages
2,142
Location
La Baule, France
A 5x5x5 cube is like a supercube with an inner 3x3x3 cube whose each side are the centers of an outer 3x3x3 cube. Each move used to scramble the inner cube also scrambles the outer one. Morevover, each move used to solve the centers also continues to scramble the outer cube.

So... I'll say 20 moves to scramble the inner cube (wide moves) mixed with 10 additionnal moves for the outer cube should be enough to get something that looks like a legal scramble.
 

xyzzy

Member
Joined
Dec 24, 2015
Messages
2,878
Meanwhile I practise with longer scrambles because 60 moves (hell, even 100 moves) is not enough to destroy all the regularity in the centres…

A 5x5x5 cube is like a supercube with an inner 3x3x3 cube whose each side are the centers of an outer 3x3x3 cube. Each move used to scramble the inner cube also scrambles the outer one. Morevover, each move used to solve the centers also continues to scramble the outer cube.

So... I'll say 20 moves to scramble the inner cube (wide moves) mixed with 10 additionnal moves for the outer cube should be enough to get something that looks like a legal scramble.
See also: https://www.speedsolving.com/forum/...ith-random-move-scrambles.64312/#post-1227796

(but also there's a weird caveat I didn't notice when I first wrote that post; if you try to use shorter scrambles with this "scramble with wide moves of a lower-order puzzle, then do normal random moves" method, there's a tiny increase in the number of colour bars between the centres and edges compared to random moves.)

3) For obtaining the short scramble, the program would generate a random-move N-moves scramble, calculate its features, and if at least one of them has the value too far away from its distribution mean, it would discard the scramble and generate the next one.
I think the issue with this is that outliers can and should happen, and if you're looking at sufficiently many features, the chances of one of them being an outlier (with ideal, random-state scrambles) can be pretty high. Simply filtering scrambles that have features "too far off" from average doesn't make the resulting distribution the correct one.

I guess something that bothers me about this (as well as the original transition from 25 random moves to random state for 3×3×3) is that we don't have a consensus of how close our scrambling method's distribution has to be to random state in order for it to be a good scrambling method. People like to bring up how "with n random moves it's three times more likely to have all edges oriented than all edges flipped", but this looks a lot less extreme if you also consider that the probability of those extremes happening is only 1/1024 to begin with.
 
Last edited:

Roman

Member
Joined
Jan 10, 2013
Messages
686
Location
Dubai, UAE
WCA
2012STRA02
YouTube
Visit Channel
I think the issue with this is that outliers can and should happen, and if you're looking at sufficiently many features, the chances of one of them being an outlier (with ideal, random-state scrambles) can be pretty high. Simply filtering scrambles that have features "too far off" from average doesn't make the resulting distribution the correct one.

That's true. Upon creating my post I was so sleepy I haven't took this into consideration. Okay, I would continue working on my project merely for reducing the scrambling time for the home practicing, and maybe will create a new online tool or something.
I have just read your analysis, great work. I think the WCA should discuss switching to that wide-moves-first scrambling method for 5x5 and bigger cubes.
 

AlphaSheep

Member
Joined
Nov 11, 2014
Messages
1,083
Location
Gauteng, South Africa
WCA
2014GRAY03
I think for home practice for blindfolded, it doesn't matter that the distribution isn't correct. All that matters is that you're getting practice on a typical number of targets and flipped/twisted pieces and getting a reasonable distribution of letters. Even if you only filter easy scrambles, having a bias toward harder scrambles during practice might be a good idea. Perhaps even being able to adjust that bias in an online timer may be a cool feature. Start off on easier scrambles when first learning, and then practice on typical scrambles when you have the idea, then introduce harder scrambles when you're comfortable.
 
Top