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Rubik's Cube Proof Cut To 25 Moves

Posted by samzenpus on from the too-much-time-on-your-hands dept.

KentuckyFC writes "A scrambled Rubik's cube can be solved in just 25 moves, regardless of the starting configuration. Tomas Rokicki, a Stanford-trained mathematician, has proven the new limit (down from 26 which was proved last year) using a neat piece of computer science. Rather than study individual moves, he's used the symmetry of the cube to study its transformations in sets. This allows him to separate the 'cube space' into 2 billion sets each containing 20 billion elements. He then shows that a large number of these sets are essentially equivalent to other sets and so can be ignored. Even then, to crunch through the remaining sets, he needed a workstation with 8GB of memory and around 1500 hours of time on a Q6600 CPU running at 1.6GHz. Next up, 24 moves."

11 of 386 comments (clear)

  1. Re:1.6ghz? by RabidJackal · · Score: 3, Interesting

    Speaking from personal experience, a Q6600 at 100% load on all four cores gets incredibly hot, even with a good cooler. Perhaps he did not wish to risk overheating at the expense of his experiment?

    Then again, it could just be an error in the article.

  2. Computational proofs by timeOday · · Score: 3, Interesting

    Is this becoming common for proofs to be done by searching through billions of combinations (albeit, reduced to that number only through some clever observations about symmetry) and simply showing that each one is possible because your computer found a solution? Sometimes we talk about the number of steps in a proof, this proof must have trillions of steps. Not complaining, it just seems like a uniquely computer-age technique. I know of no reason to assume that every true thing that can be proven has a concise, elegant proof - in fact I'm sure that cannot be true because there are only so many small numbers to go around!

    1. Re:Computational proofs by Sage+Gaspar · · Score: 4, Interesting

      As far as I know the first "big" computational proof (which another poster alluded to) is the Four Color Theorem. It was initially met with some distrust but it's pretty widely accepted now, and there are people that worked after the original proof to cut down the amount of computer verification needed from a couple thousand to a couple hundred I think.

      I would guess that it is more common in fields like graph theory and other discrete math just because obviously the discrete lends itself well to computers, and many times it's not hard to whittle it down to a finite number of cases to check. The objects of study also tend to admit matrix representations and other things computers are good at working with. Even before computers you'd cut things into lots of cases that you needed to verify but now it's easier to handle proofs that need larger number of cases.

      I've actually seen some really interesting proofs using computers to check things over continuous domains. The basic idea lots of times is if you can check things over a fine enough "net" of cases in some space and you can prove that the variance between each of these points is small enough, then you can cover your entire space by just checking a finite number of cases.

      Given all this people still have a healthy amount of skepticism for computer aided proofs and would rather not if possible in most cases, especially when you're talking about billions of cases. Then again what is the potential for errors in a computer checking billions of cases based on a relatively small amount of code versus some of these enormous human-created decades-long behemoth proofs?

  3. Distributed computing by Anonymous Coward · · Score: 3, Interesting

    Since it took just a few months for someone to do this on one computer, this seems like an ideal candidate for a small-scale distributed computing project. TFA says his program's working on 24, so he already has an algorithm. Assuming it's massively parallelizable, which from the description of the method seems very likely, it shouldn't take too many computers to get to 24 in a matter of days. Anyone care to implement the algorithm in one of the open-source distributed computing frameworks out there?

  4. Re:1.6ghz? by Anonymous Coward · · Score: 5, Interesting

    Practically speaking, this is more a memory intensive than a CPU intensive problem. Given that the Q6600 supports an FSB speed of only 1066 MHz, if the computations generally require a fetch from RAM (i.e. the on-die cache is insufficient to the task, as in most memory bound tasks) then you can't operate at the full speed of the chip since it is constantly waiting on the memory controller.

    In benchmarks, AMD CPUs tend to beat Intel CPUs on memory bound tasks, even though Intel CPUs win at CPU intensive tasks because the AMD CPUs integrate a faster memory controller on-die instead of relying on a slower FSB. Intel's weakness is less noticeable when the CPU is running at a clock speed closer to the FSB, and given that increases in CPU clock speed increase the power and heat usage geometrically, it wouldn't make sense to run the CPU at full clock for a task of this nature.

  5. "God's Algorithm" by wickerprints · · Score: 5, Interesting

    Take every possible unique configuration of the cube (those that are obtainable by legal moves--no rearranging stickers or disassembling allowed). Represent each configuration by a vertex. Now join two vertices by an edge if and only if the configurations represented by those two vertices differ by a single move (we will elaborate on what constitutes a "single move" later). The result is a mathematical object called a graph. A horrendously giant graph.

    One, and only one vertex in this graph corresponds to the solved configuration of the cube.

    Note that this graph represents all possible moves and positions--any scrambled cube is a vertex somewhere in the graph, and solving that cube is equivalent to traversing a path in this graph to the "solved" vertex. In general, many paths to the solution exist, some of which will be shorter than others.

    The question of interest is this: Which vertex/vertices of this graph is/are farthest away (i.e., requiring the most edge traversals) from the solved vertex, and how far is it? As of this latest discovery, this maximum distance is 25. It means that every possible scrambled configuration of the cube can be solved in 25 moves or less.

    Wikipedia notes that we know that at least 20 moves are required to solve the cube for every configuration--that is to say, we know that this maximum distance is at least 20 (there exists some vertex that is at least 20 steps away from the solved vertex). It is believed that the true "least upper bound" is closer to 20 than it is to 25.

    Finally, we should clarify that a "single move" can either mean a rotation of a face by either a quarter- or half-turn, or it could mean a quarter-turn only. These different metrics of what constitutes a "move" leads to different answers.

    1. Re:"God's Algorithm" by insecuritiez · · Score: 4, Interesting

      Most every cuber believes the limit _is_ 20. There is only one known permutation that requires 20 moves and it is called the "super flip". In it, every edge and corner piece are in their correct positions but all the faces have the opposite orientation. It makes for a nice checkered pattern. It is the symmetry of the scramble and the lack of known permutations "harder" than the super flip that lend a strong argument to 20 being the max.

  6. Re:You only need one by tlhIngan · · Score: 5, Interesting

    It's much easier to pull the stickers off. Though less fun I suppose.


    Fun trick: Take a solved cube, and on one of the inner edge pieces (the ones with two stickers), and swap the colors. Mix it up, and give it to someone to solve. Or take a corner piece and rotate it.

    Hint: It's unsolvable. The Rubik's Cube, if taken apart and put back together randomly, will more often than not end up being unsolvable.

    A great way to frustrate that showoff cuber at the office. Especially if they appreciate it when someone scrambles the cube and they'll have it solved in front of everyone. Just go and put it back together randomly, or do one of those devious swaps, and you'll have fun watching him try to solve it.
  7. Solve This! by kramulous · · Score: 3, Interesting

    In light of a certain parallel programming news item a few days ago, I'd like to see him use the same code, same CPU on this one: http://www.youtube.com/watch?v=UrjmeYdVTlc. Hold your breath for that solution.

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  8. Re:Which 25 moves? by Anonymous Coward · · Score: 3, Interesting

    In fact, it is pretty likely every cube can be solved in as few as 21 moves (or less),
    as someone (Dik Winter) has already written a program that does just that.
    Source: http://mathworld.wolfram.com/RubiksCube.html

  9. Re:Wow, it really works by The+Clockwork+Troll · · Score: 5, Interesting

    Unfortunately the "Interesting" moderation is not finest grain.

    There are at least two subtypes of interesting:

    - Interesting to someone with joint degrees in math and computer science
    - Interesting to someone who has smoked two joints

    Any thread involving Rubik's cube is going to pull both, sorry.

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    There are no karma whores, only moderation johns