Shouldn't 17 Scramble Moves be Enough?

[Lucas Garron]

Maybe the question should be whether it affects it in a meaningful way, and indeed how we define randomness for the purpose of the question.

I don't think it's useful to try to define "meaningful" in order to make yourself feel better about miss-scrambles. If you know you made a mistake while scrambling, you should not treat it as any more random than a hand scramble.

And I insist that you cannot call it "purely academic" until you've (academically?) determined how bad the error is.
20-move random-move scrambles seem fine, but turn out to have a bad edge distribution that definitely affects solves, especially record times. You cannot simply dismiss the effect of scrambling errors out of hand, regardless of how deliberate or accidental they are.

 

[mark49152]

20-move random-move scrambles seem fine, but turn out to have a bad edge distribution that definitely affects solves, especially record times.

A study of how edge distribution is affected by inserting errors into random state scrambles would be interesting to see.

 

[SF]

It turns out that Mr. Kociemba from Cube Explorer actually provides a list of 100,000 optimal solutions to random state cubes on his website (http://kociemba.org/cube.htm). They're in a text file in the "God's Number is 20" section under the first "Optimal Cube Solver" graph. His average number of moves was 17.7. There were no 20-move solutions in the 100,000 solves, which makes sense since the odds of getting a random cube state with a 20 move optimal solution is only about 1 in 88 billion.

Anyway, I'm assuming I can just apply his solution moves to a solved cube and end up with a random cube state each time. I'm going to start with number 1 and just go down the list. In this way, I'll get the shortest possible scrambles which will save time, and because there are less moves, it should reduce the chances of my making errors and invalidating the random cube state.

Very happy about this! Just hope I'm not missing something that would invalidate my assumption that these would be true random state scrambles.

 

[shadowslice e]

They may be fairly random scrambles if you choose them randomly but as you are limiting yourself to 100,000 scrambles (some of which may be pretty easy) they you only have a 1/100,000 chance of getting any scramble as opposed to the 1/43 quadrillion with a truly random state scrambler. In addition, it is entirely possible that all of those scrambles have an element in common hence why Kociemba uploaded those specific one as they may have been the first few scrambles he generated (this is also likely why there are no 20 move solutions cause they didn't get to solving most of those until later because that used exponetially more processing power and time).

 

[AlphaSheep]

As far as I know, they are randomly generated, so they should be fine. Although shorter doesn't necessarily mean faster to scramble. Often a longer scramble that has easier move combinations will be faster than a shorter one with awkward move combinations. I think you're worrying about optimisation beyond what is reasonable. Rather solve faster to save time - it is more directly relevant to your end objective.

 

[SF]

As far as I know, they are randomly generated, so they should be fine. Although shorter doesn't necessarily mean faster to scramble. Often a longer scramble that has easier move combinations will be faster than a shorter one with awkward move combinations. I think you're worrying about optimisation beyond what is reasonable. Rather solve faster to save time - it is more directly relevant to your end objective.

True dat. Especially your last sentence.

Wouldn't it be fun though if some of the really fast speedcubers out there had an informal competition using the Kociemba list? Obviously, each person would be self-policing their times, DNFs, etc., but they could report their stats for the first 100 scrambles, the first 500, or any series in the list of 100,000. You can copy the text from the Kociemba file, paste it into Excel, and number the algs from 1 to 100,000 (I did this last night). Then you can add a column to the right for recording solve times for each row.

Since each person would be solving the identical cube states, the results would be a true apples to apples comparison. The benefit for me and maybe for others who are trying to improve is that we could compare our much slower times on a line-by-line basis to see if there is any correlation between the relative difficulty of each solve for the experts vs. the relative difficulty for us.

I guess we'd have to determine certain things in advance, like color orientation during the scrambles.

 

[Herbert Kociemba]

I just read this discussion about the 100000 scrambles. Yes, these are random positions using the Mersenne-Twister random number generator.

 

[SF]

Thanks Mr. Kociemba. I appreciate your replying and confirming these are a random sample. I've been using the 100M scrambles for practicing solves (just going in order, I'm on #374). I did a little bit of analysis of the scrambles (I should say "solutions"), and the moves are very evenly spread throughout each series. I did notice that there is a slightly heavier weighting on D, L, & B moves as the first move, and a significantly heavier weighting on the U moves as the last step in the longest solutions (19 moves). Of the 3,303 19-move solutions, 2,900 are U', 332 are U2, 52 are U, 15 are R', 3 are R2, and 1 is B'. I find that interesting since I know your program for finding optimal solutions doesn't use the CFOP method that many of us humans use. Yet, using CFOP, I most often end up with a U (AUF) move at the end of solves, especially if I don't do the AUF prior to the PLL (though obviously not as often as your 19 move solutions do).

The other thing that surprised me (and maybe it shouldn't have) was how often two opposing faces are turned consecutively. I just assumed optimal scrambles would have fewer of these since I figured they'd be less efficient in solving the cube. Though now that I think about it there are quite a few OLL & PLL algs that contain either slice moves or consecutive opposing face moves.

 

[Herbert Kociemba]

I did notice that there is a slightly heavier weighting on D, L, & B moves as the first move, and a significantly heavier weighting on the U moves as the last step in the longest solutions (19 moves).

In principle, the optimal solver generates all possible maneuvers for depth 1, 2, 3, 4 ... in the lexicographical order U, R, F, D, L B. Most cubes are solvable within <= 18 moves, but there are so many 19 move maneuvers that for those cubes which need 19 moves there usually are many solutions, surely evenly distributed concerning the first move. But my program just takes the first solution which then of course often happens to be an U-move.

 

[SF]

Thanks for the explanation. Interesting.

 

[efattah]

For what its worth, I found this discussion interesting, and I usually scramble my cubes manually (no PC generated scrambles). So I tried doing an average with cube explorer generated scrambles and guess what: the cubes were easier to solve. One average isn't enough, but this is what I noticed:
1. When I scramble my cubes manually, I use around 40 moves, but I watch the cube and I purposely and deliberately break up any patterns or clusters that I see. This means when I actually start the solve, I rarely can find anything to work with in terms of pre-built sections.
2. When I used the PC generated scrambles, there were VERY OFTEN pieces together, making the start of each solve easier.

This reminds me of when I watched Chris Olsen's video on making efficient 2x2 faces. I think in 11 of 12 example solves (each with PC generated scrambles), the cube started with a random bar of two face pieces or more. I was rolling my eyes in his 'luck' since when I scramble my own cubes I never get those 'lucky' situations because I purposely break up any patterns or blocks.

I think it also goes without saying that one person's 'lucky' scramble is another person's nightmare; depending on which method they use, which color(s) they prefer, and so on. A CFOP solver could be given a cube with all the corners already solved, and they would gain nothing from that, whereas a corners first solver would be a third through his solve. Or vice versa in terms of pre-built crosses.

Eric Fattah
Vancouver, BC

 

[00]

Every single scramble can be done in 20 or fewer moves. The way the WCA (and the general cubing community) agreed that a 20 move scramble was enough is because it was god's number.

no.

For this very reason, scrambles are easier and not the difficulty wished upon by the WCA. If the WCA was okay with excluding the permutations that take 18 or more moves to create, than they would make the scrambles 17 moves. In essence, the WCA is controlling the difficulty by changing the number of moves.

no.

If they made a scramble 17 moves, it would be easier, and not what the WCA wants.

no.

I suggest you look up what random state scrambles are.