Can the Rubik's Cube be solved with a specific color pattern?

[collinsmark]

Yes, they scramble the cube from the solved position, however, the thing to remember is the cube is NEVER, EVER, EVER more than 13 steps from being solved. If it takes longer that means you don't have the ideal solution and doesn't necessarily reflect how scrambled the cube is.

If during the official scrambling, only 13 moves are used, then Yes, I agree that the cube is never, initially more than 13 moves from being solved.

What the links are saying however, is if the initial scrambling has more than 20 moves, then there does potentially exist states that take up to 20 (minimum) moves to solve. 20 is the maximum, minimum number of moves that it takes to solve a Rubik's cube.

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On a different note, I find interesting is that the entropy of a Rubik's cube (according to the link) is maximum at 18 ideal moves away from the solution. But there does exist some states that have a lower entropy, yet a larger number (19 or 20) of ideally solvable moves away from the solution (if we are to believe the link). In other words, if the cube starts in one of these states, the solver must first increase the cube's entropy to get it to 18 moves away from the solution before decreasing its entropy.

 

[BruceW]

@wuliheron: well collinsmark's links say otherwise. They say it can take 20 steps to be solved, even if you have the ideal solution.

 

[BruceW]

On a different note, I find interesting is that the entropy of a Rubik's cube (according to the link) is maximum at 18 ideal moves away from the solution. But there does exist some states that have a lower entropy, yet a larger number (19 or 20) of ideally solvable moves away from the solution (if we are to believe the link). In other words, if the cube starts in one of these states, the solver must first increase the cube's entropy to get it to 18 moves away from the solution before decreasing its entropy.

yep, that is interesting. So, I guess a completely 'shuffled' cube will be most likely 18 moves away from the solution? And I suppose entropy doesn't always agree exactly with the idea of how 'not useful' the state of the system is. I think entropy is one of those things where the mathematical definition is good, but the interpretation is not always clear.

 

[collinsmark]

yep, that is interesting. So, I guess a completely 'shuffled' cube will be most likely 18 moves away from the solution?

Yes, if we are to believe the link, it means that if you shuffled a Rubik's cube for a long time -- not just 13 moves, but shuffle it randomly until the cows come home -- it's most likely to fall into a state that takes 18 moves to solve, minimum.

It also means that if you were the shuffler, and you wanted to put the cube in a state that takes 19 or 20 moves to solve, minimum, you would probably have to put some thought into the process, particularly the 20 move configuration. (It's very unlikely that the cube would ever end up in the 20 move minimum configuration by pure random shuffle. Possible yes, but very unlikely.)

Putting it yet another way, it means that you were attempting to solve the cube, and you were able to recognize that it were in one of these configurations (19 or 20 ideal moves away from the solution), you would have a very high probability of getting closer to the solution by first making one or two totally random moves.

And I suppose entropy doesn't always agree exactly with the idea of how 'not useful' the state of the system is. I think entropy is one of those things where the mathematical definition is good, but the interpretation is not always clear.

I can think of a simple analogy to this situation. Suppose you had six coins lined up in a row, all heads. Randomly roll a six sided die and flip the corresponding coin over. Repeat indefinitely. You will find that after several rolls the coins converge to having around 3 heads and 3 tails (these are the states of the highest entropy). Although improbable, it is possible on occasion that all 6 coins will show tails. This is another lowest-entropy state, even though it's the furthest number of moves from the first lowest entropy state with all heads.

 

[BruceW]

I don't really understand the analogy. But I do understand the bit you were saying before then. The 'difficult' configurations (which take 19 or 20 moves to solve) are quite rare compared to the other configurations. And this is what seems weird in my intuition of entropy, because I would normally associate the most likely set of configurations with being the most difficult. For example, an ideal gas, the high entropy set of states are where the molecules take up all the space, flying in all directions, all mixed up. (i.e. difficult). But really, I guess is to be expected that what I think of as 'difficult' does not always match up with high entropy.

 

[wuliheron]

I haven't bothered to read the website, but I'm sure they are talking about THEIR solution and not the ideal solution. Evidently they used a computer to devise an algorithm which provides the best possible solution short of actually calculating how to solve it in 13 moves. However, even with their solution dumb luck still plays a part and it should be possible to solve the thing in 13 moves once in awhile. twice I've just been randomly turning faces on a cube and solved it without trying.

If you are interested the cube was developed to teach group theory permutations, the same mathematics in quantum mechanics. It's a closed system or contextual system or fuzzy logic system or whatever the heck you want to call it and those are the rules that govern its behavior, not some website talking about their particular solution.

 

[BruceW]

no, I think they actually used brute force computation to calculate that the 'most difficult' initial configuration can still be done in a minimum of 20 moves. Which is quite impressive. I guess that the number of different possible configurations of a rubik's cube is not impossibly great. (although it is still going to be a very large number, it will be no-where near as great as the number of configurations of a chess game for example, which cannot be 'completely solved' by brute force with today's computers).

Edit: also, yes it is possible to solve it in 13 moves occasionally. But that depends on the initial configuration. In the most difficult initial configuration, it takes 20 moves to solve. And in the most easy initial configuration, it takes zero moves to solve.

 

[wuliheron]

no, I think they actually used brute force computation to calculate that the 'most difficult' initial configuration can still be done in a minimum of 20 moves. Which is quite impressive. I guess that the number of different possible configurations of a rubik's cube is not impossibly great. (although it is still going to be a very large number, it will be no-where near as great as the number of configurations of a chess game for example, which cannot be 'completely solved' by brute force with today's computers).

Edit: also, yes it is possible to solve it in 13 moves occasionally. But that depends on the initial configuration. In the most difficult initial configuration, it takes 20 moves to solve. And in the most easy initial configuration, it takes zero moves to solve.

It is a closed system! Closed, as in no alternatives whatsoever are possible. If you can't understand what that means then I suggest reading up on group theory permutations because that website is only confusing you in my opinion.

 

[BruceW]

You're right. I don't know what you mean by 'no alternatives whatsoever are possible' please explain. hehe, sorry for being sarcastic, but I would like to know what you mean.

edit: at a guess, I'd say you mean that by making moves, you cannot leave the finite set of possible configurations. But I don't know what that has to do with what I was saying.

 

[Borek]

I'm not sure what is confusing you. Computer programs exist which can solve the cube from the most scrambled state in 13 moves every time.

Yes, they scramble the cube from the solved position, however, the thing to remember is the cube is NEVER, EVER, EVER more than 13 steps from being solved.

Can you give any links/quote any sources for these statements?

 

[wuliheron]

A quick check with Google didn't turn up anything, however, I must point out these website are talking about algorithms which are by definition rules for how to solve something rather than specific solutions and there is nothing that insists an algorithm must provide the most parsimonious solutions.

A merry-go-round is a closed system. Your only option is to go round and round and if you get on it thinking it will go someplace else you are a fool. The cube goes round and round and has a minimum and maximum of number of steps required to solve it at any given time. A quarter turn of one face is the minimum that can be solved, and 13 steps (including 180 degree face turns which count as double steps) is the maximum it can require. Algorithms or rules for trying to solve it will always fall short of providing the ideals solutions in some cases by the very nature of their rules limiting the possibilities.

 

[BruceW]

yeah, they are saying that any algorithm cannot beat a 20-move algorithm, in the worst case scenario. That is any algorithm. So it doesn't matter what rule you want to invent, it is proven that you cannot do quicker than 20 moves, given the most difficult initial configuration. Also, why do you say 13 steps?

 

[wuliheron]

yeah, they are saying that any algorithm cannot beat a 20-move algorithm, in the worst case scenario. That is any algorithm. So it doesn't matter what rule you want to invent, it is proven that you cannot do quicker than 20 moves, given the most difficult initial configuration. Also, why do you say 13 steps?

Algorithms are merely convenient shortcuts you can use that may or may not be shorter than something else depending on the specific circumstances. Some days highway traffic is light, while other days the back roads are faster and exactly how the cube is scrambled decides what is the fastest method for unscrambling it rather than any particular algorithm being better than another for every single instance. For example, if just one face is twisted out of sync your algorithm would still work, but not be the shortest solution.

I wish I could still remember the mathematics, but there are 26 pieces, the six center pieces don't move in any way to affect the pattern on the standard cube and, so there are 8 corner pieces and 12 edges to account for which can each only be moved in units of three at a time with their own counterparts. Corners also have to be twisted to orient them and edge pieces flipped which can be done in two at time or more in units of twos. Anyway, what I can remember is that 13 is the absolute minimum number of moves possible and at no time is the cube more than 13 steps from being solved because it is a closed system and the algorithms merely provide approximations. What these algorithms are doing is providing some alternative moves that combine spinning, flipping, and moving pieces simultaneously in some cases when, in actuality, they can almost always be done simultaneously.

 

[BruceW]

http://en.wikipedia.org/wiki/Rubik's_Cube
the section "optimal solutions". I know wikipedia is not always reliable, but until I see a scientific journal stating 13 is the minimum number of moves for the solution of a Rubik's cube, I'm going to assume it is actually 20. No offense intended.

 

[wuliheron]

http://en.wikipedia.org/wiki/Rubik's_Cube
the section "optimal solutions". I know wikipedia is not always reliable, but until I see a scientific journal stating 13 is the minimum number of moves for the solution of a Rubik's cube, I'm going to assume it is actually 20. No offense intended.

Hey, believe whatever you want. However, "optimal solution" does not necessarily mean the "shortest" solution and I caution against reading too much into such things.