## Rubik’s cube proof cut to 25 moves

Last year, a couple of fellas at Northeastern University with a bit of spare time on their hands proved that any configuration of a Rubik’s cube could be solved in a maximum of 26 moves.

Now Tomas Rokicki, a Stanford-trained mathematician, has gone one better. He’s shown that there are no configurations that can be solved in 26 moves, thereby lowering the limit to 25.

Rokicki’s proof is a neat piece of computer science. He’s used the symmetry of the cube to study transformations of the cube in sets, rather than as individual moves. 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.

But Rokicki isn’t finished there. He is already number-crunching his way to a new bound of 24 moves, a task he thinks will take several CPU months. And presumably after that, 23 beckons.

Where is this likely to finish? A number of configurations are known that can be solved in 20 moves but it’s also known that there are no configurations that can be solved in 21 moves.

So 20 looks like a good number to aim at although that will still be an upper limit. No news yet on whether 20 might also be the lower limit, which would give the answer a satisfying symmetry.

What this problem is crying out for is a kindly set theorist who can prove exactly what the upper and lower limits should be without recourse to a few years of CPU time (although it may take a few years of brain time). Any takers?

Ref: arxiv.org/abs/0803.3435: Twenty-Five Moves Suffice for Rubik’s Cube

### 99 Responses to “Rubik’s cube proof cut to 25 moves”

1. […] Rubik’s Cube article here. […]

2. […] Cube solved in under 45,032 moves…about 45,007 […]

3. […] However lately, I saw this Slashdot article about a Stanford-trained mathematician named Tomas Rokickitrimmed, who trimmed down the 41 moves to just 25 moves using a

4. […] 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 […]

5. HellCat says:

What was proved is that any configuration can be
solved in 25 moves or less. He proved this by making
a computer go through a set of “typical” configurations
and showing that there is no solution that requires
26 moves. The prover did not give
a solution because each configuration requires a
different set of moves.

Let us say you have a solved cube in your hand. If
you make 1 move, then you have in your hand a cube that
requires at least 1 move to be solved. If you take
a solved cube and make 10 moves, then you have
a cube that will take ten moves to solve. However,
it is possible that, because of the way you made your
moves, that the solution will take less than ten moves.
There is also a solution that will take more than
ten moves and that has been proven.

However, let’s say that you take that solved cube
and made 30 moves. You now have a cube that will
be solvable in 30 moves. What has been proven above
is that there is a set of moves that will also solve
that configuration in 25 moves or less. You don’t
need to reverse the 30 moves you made to get the
solved configuration, there is solution (which
you need to find) where you only need to make 25
moves at most.

6. […] הפתרון לקובייה ההו

7. […] colleague told me about a news post I should read, detailing that a new upper boundary for solving the Rubik’s Cube has been found – now 25 […]

8. JimFive says:

A fairly good explanation but I notice an error
A 3x3x3 cube has 9 slices(top, zcenter, bottom, front, xcenter, back, right, ycenter left) with 3 moves each or a maximum 27 possible moves.

The 2x2x2 cube has 6 slices using the same counting method (left, right, top, bottom, front, back), for a maximum of 18 possible moves. Symmetry of course cuts down the possible move count in the 2x2x2 cube as any move of one slice has an equivalent move of its partner slice leaving us 9 possible moves.

JimFive

9. Cody Hayes says:

Learning to solve these really isn’t hard, most Rubik’s Cubes actually have a booklet that teaches you a very basic solution. I got hooked about a year ago and coming close to averaging under twenty seconds. Surprisingly that’s very slow to a lot of cubers.

10. Cody Hayes says:

Actually, Helpful, the 2x2x2 cube has over three million possible configurations, and I believe the maximum amount of moves for an [optimal] solution was around 14 moves. Can it be solved in more than 14 moves? Yes, but it wouldn’t be optimal. Some configurations also have lower optimal solutions, it could be only five moves away. And as something some of you may not know, the turning of any face twice (180 degrees) without turning another face before it has been turned, cubists count that as one move. That would be notated as U2 or R2 (dependent on which face was turned.)

Now my reply to Jason who appears to be saying that “The Rubik’s cube can be solved from ANY conceivable configuration in ANY number of moves equal to or greater than zero”.

That is true yes, but not what these people are searching for. If you were to turn the rubik’s cube fifty times, fifty turns would not be the optimal solution. What they are looking for is the maximum amount of moves it takes to solve the rubik’s cube in an optimal solution. What they have just finished proving is that the most moves there can be in an optimal solution is 25, and now they are working to see if they can lower that to 24. So what they’ve proven as of now, is that every single configuration out of the 43 quintillion possible configurations, an optimal solution will take no more than 25 moves.

11. PRCROW says:

So when you take the stickers off… does that count as a move?? because then i would suggest the optimal solution is 2 moves…

1) stickers off from wrong places
2) stickers on in right places

ashes bloody ashes

12. […] the physics arXiv blog » Blog Archive » Rubik’s cube proof cut to 25 moves […]

13. […] Adobe’s new Photoshop Express offering * Some laptops from the future * Rubik’s Cube and other 1980s icons that’ve stood the test of time * New Ubuntu, new Vista SP1 * A […]

14. […] Last week a mathematician called Tomas Rokicki proved the new upper-limit for solving any scrambled cube, in 25 just moves. It’s a theoretical limit, the how-to is not given. Guess I’ve still got some training to do to actually solve my cube in 25 moves, let alone solving it in the recent speed record of 9.18s…;) […]

15. gary shandling says:

me fail english? that’s unpossible!

16. Luke says:

the point of this was to establish that a cube no matter how scrambled it is, can be solved with 25 moves. this doesnt necessarily mean that it can be solved with less, it’s just stating that there is a 25 move way to solve every cube.

even if you only moved one side.

17. I misread “theorist” as “terrorist” in the article.

18. […] Dünyada bu bulmacayı en hızlı çözme yarışmaları bile yapılıyor. Rekor ise 3×3×3’lük küp için şu an 9.18 saniye. 2007 yılında ABD’deki Northeastern Üniversitesi’nde bir grup araştırmacı, Rubik Küpü’nün çözümünün en fazla 26 hamlede yapılabileceğini ispatlamışlardı. Matematikçi Tomas Rokiçki ise bunu bir adım daha ileri götürerek çözümün en fazla 25 hamlede yapılabileceğini ispatladı. BU hesaplama için Rokiçki bir bilgisayardan yararlanmış. Bulmaca ne kadar basit görünse de çözüm için milyonlarca olasılık var. Bu yüzden Rokiçki, 16 GHz hızında, 8 GB hafızası olan bir bilgisayarla bu hesaplamayı 1500 saatte yapmış. Rokiçki’nin bundan sonraki amacı ise 24 ve 23 hamleyi denemek. Derleyen: Sinan Erdem Haber tarihi: 24 Mart 2008 Kaynak: http://arxiv.org/abs/0803.3435 http://arxivblog.com/?p=332 […]

19. […] 此他将还原最少步骤降至25步(论文预印本)。Rokicki的证明完全依

20. Frank says:

Wow. Just wow.

Reading these comments has proved to me that just reading arXiv (and/or the arXiv blog) does not exempt you from stupidity.

21. Marie says:

@JimFive
To make the move count for the 3x3x3 cube even smaller, I don’t think the center slices are generally considered “movable” in these tricks. I am pretty sure a move consists of rotating one of the six surface slices by 90, 180, or 270 degrees. You can achieve the same effect as moving the center slice by rotating the two parallel surface slices in the other direction. This is sort of intuitive if you are holding a cube because it physically trickier to move the center slice and you effectively end up making two moves instead anyway. So the number of moves from any given state is 6*3, or 18.

To try to clarify the point of all of this:
If I am holding a scrambled Rubik’s cube and happen to really know what I am doing (say I am God or an as-yet non-existent computer program), I only need ‘n’ moves or fewer to solve it. Previous proofs have said that ‘n’ is definitely at least 20 because there are actual scramblings that truly need that many moves to solve them. People have tried to find simpler solutions and the mathematicians say that no, you need 20 moves to solve these particular cubes. Other previous proofs have said that ‘n’ is definitely no more than 27, and then no more than 26, and then this one says no more than 25. The algorithm being used to do these proofs is NOT solving all of the Rubik’s cubes, because that would take much too long. So don’t bother asking the Mr. Rokicki for the perfect 25 move solution for any scrambling, he doesn’t know it and can’t find it for you. If he could, he would be able to tell you for sure that there is some cube that needs 25 moves, and the game of how-low-can-you-go would be done. So now he will go after the next proof, the 24.

He might be wanting to do all of the computing on his own machine for a few reasons. For one, he might want to be tracking huge quantities of statistical data, which nobody would enjoy piping to him over the internet, and he probably wouldn’t enjoy receiving. Also, the processor-time-for-\$\$ schemes involve breaking down your project into manageable pieces that can all be run in parallel. If your program doesn’t parallelize, then they have to just run it consecutively on one computer anyway, and you might as well do it at home. It is tricky to build programs to take advantage of massively parallel processing, and even if he did do this to some extent to work with his own four processors, that doesn’t mean that it will scale up to x processors (and since they probably won’t share other resources like memory, it gets even trickier). So it is probably more work than it is worth to him to make it runnable on a system where they crunch it out quickly. On the other hand, the problem probably goes up at least geometrically for each number, so he may reconsider if this next round comes through so that 23 doesn’t take a decade or a century.

I speculate that once we figure out that magic number of moves, it will be easier to create a program to find optimal solutions. But I am just guessing there. Sorry about the crazy length on this post.

22. Bill says:

Marie gives a good analysis, but I would differ in saying that problems like this are usually more easily paralleizable than most. If one thinks of the prover as a tree search, then different branches of the tree at the topmost level can be handed off to different processors/computers without much difficulty. They then do the same thing, down to a level such as, say, 18 moves, at which point we are talking about millions of processors working in parallel on their own 18 moves.

Further, for any given configuration, given that the search space is at most 25 deep, it shouldn’t take too long to find an optimal solution to that one configuration. Probably no more than hours or days, given some smart branch-and-bound conditions. However, since there are billions of such configurations, that’s what takes so long to prove that there are no configurations which require 26 moves.

23. […] – Rubik’s cube proof cut to 25 moves […]

24. Clemente says:

Seven moves, every time:

1. Scrub thoroughly with wire brush and Goo-Gone ™;

2 – 7: Paint each face a different color.

25. benaldo says:

Q6600 cpu is an intel core 2 quad ,
not a “P4” and i havent seen sunlight in 3 days

26. […] and links to paper. Via The Physics arXiv via […]

27. Matt J. says:

I don’t understand why the article asks for a _set_ theorist when obviously it is a _group_ theorist who is needed.

28. Tony says:

Let me take a face value the statement that there are no configurations that require 21 moves to solve. If this were true in the simplest way then the upper bound on solutions must be twenty. Here’s the proof.

Suppose that there is a configuration which requires more than 21 moves to solve. If we follow the optimal solution for that cube, we at some point will have 21 moves left. Because we are following an optimal solution, the cube is in a configuration which requires 21 moves to solve. Since there is no configuration requiring 21 moves we have reached a contradiction. Therefore the maximum number of moves must be 20 or the cube isn’t soluble. the latter has been ruled out.

Can someone who understands the “no cube in 21 moves” statement fully please post an explanation? This is such a trivial proof that it is clear that I have misunderstood the statement.

29. […] Rokicki, um matemático da Universidade de Stanford, já mostra que é possível resolver qualquer cubo mágico em no máximo 25 passos. Agora ele surpreende demonstrando que é totalmente possível

30. Tor says:

Another way to put it – because of the proved impossibility of exactly 21 moves.

Suppose a configuration takes 20 moves to solve. Since no configuration takes exactly 21 moves to solve, any one move will make the new configuration soluble in 20 or less moves. Do another move. The new configuration still requires at most 20 moves to solve.

If you continue doing this, you will see that no matter how many moves you do, the new configuration must have a solution requiring 20 or less moves.

So, provided that the solution does exist within the range of reachable configurations, any configuration that can be reached by any number of moves must have a solution that requires at most 20 moves

Also, any configuration can be reached from any other configuration in at most 20 moves.

..this is just another way to put Tony’s post, but I thought this way was easier 🙂

31. […] Rokicki a demonstrat că un cub Rubik poate fi rezolvat în doar 25 de mişcări, indiferent de configuraţia iniţială. Fostul “record” era de 26 de mişscări, […]

32. Yahoo says:

me too.

33. qinwamascot says:

The correct statement should be “there are no known cube positions for which 21 or more moves are needed.” Every cube that has been solved optimally has been 20 or fewer moves, but there could potentially be positions that take 21 or more moves. The upper bound has recently been reduced to 23, but if an individual cube position was found requiring either 23 or 22 moves to solve, by pigeon-holing principle, there would also need to be a number of positions requiring 21 moves exactly as an optimal solution.

The way it is phrased is entirely wrong. The only thing we know about 21, 22, and 23-move configurations is that none have been found, but they could exist mathematically. If they do exist, they would have to be extremely rare though (precisely because none have been found after many millions of tests).

34. qinwamascot says:

just noting that the statement above, that ‘any configuration can be reached from any other configuration in a maximum of 20 moves’ is not mathematically proven yet. All known tests agree with this and most mathematicians would say it is likely, but the truth of this statement is unknown.

35. Solving a Rubik's Cube says:

How I solved that thing…

36. […] already know that there is no position that takes exactly 21 moves to solve. (My first reaction to this was, “What do you mean? Take a finished Rubik’s Cube and […]

37. supercube says:

this is old news. the most moves it takes to solve any cube is 22. sony pictures donated the super computer time to a college student. google it. I cant believe people are argueing hypothetical stuff that they know nothing about. real mathematicians and rubik’s fans arent reading this site cause they dont have to. it was news when it happened a few months ago.

38. Michael A says:

Surely there must exist combinations that can only be solved in minimum 21 moves. You simply take a combination that can only be solved in minimum 22 moves, make one correct move, then give it to someone else and tell him to solve it. He will then solve it in 21 moves. =p

39. Tom says:

Some Guy seems to be thinking quite rationally… Good job!

40. Mr. Fozzie Bear says:

this is of course assuming you have hands.

41. Sean OBrien says:

Many years ago I read that someone had proven that any 2 configurations of the cube could be mapped onto each other with only 17 moves (or it might have been 19).

So if any 2 configurations can be transformed in 17 moves then what if one of those 2 is a solved cube? This says that a solved cube maps onto any configuration with only 17 moves.

Perhaps that old proof was shown to be invalid.

42. Christoph says:

“I’m guessing “there are no configurations that can be solved in 21 moves” means There are no configurations that are solved in a minimum of 21 moves.”

You could be one move away from solving it. Just turn the cube to match the last rows.

43. Christoph says:

“no, it means, literally, that from any starting position, you can not solve the cube using exactly 21 moves.”

That doesn’t make any sense.

If you had a position that could be solved in 22 moves, you would start with making a move. This new position call it 21 would be different than position 22. Then you would make 21 more moves to solve the cube.

So, yes, it is possible.

I think what they mean is there are some positions that can’t be solved in 21 moves, for sure.

Just like there are some positions that can’t be solved in 26 moves, for sure.

Whereas the above number crunching proves ALL positions can be solved in 25 moves or less.

44. Schmoo says:

I’m obviously missing something – if there’s no configuration that requires more than 20 moves, why’s he wasting time number-crunching 25, 24, etc.?

45. Borja says:

Mmm interesting, but I think there is a better way to spend these minds and resources, don’t you think?

46. Matt says:

yes that configuration can be solved in 21 moves, but 21 moves isnt the smallest number it can be soved in ie. every configuration can be solved in an infinite number of moves – moving one side back and forth

47. Ben says:

Can’t believe nobody has mentioned the Rubot. Check it out on YouTube.

48. […] Rokicki, um matemático da Universidade de Stanford, já mostra que é possível resolver qualquer cubo mágico em no máximo 25 passos. Agora ele surpreende demonstrando que é totalmente possível