## 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. QuantumG says:

My cat’s breath smells like cat food.

2. Raging_Azn says:

This is exactly how I would solve a Rubik’s cube if I didnt’ want to solve it in a reasonable amount of time at all. Even children blessed with an additional chromosome may be able to solver it faster.

3. foo says:

These berries taste like burning…

4. Nathan Myers says:

A “Q6600 CPU” appears to mean a quad-core Intel P4, and 1500 hours means two months. It seems strange (i.e. silly) to say that “there are no configurations that can be solved in 21 moves.” I wonder what the correct statement would be.

5. James Method 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.

6. Erik says:

So how DO you solve it in 25 moves? I want to know so I can try it.

7. Some Guy says:

Take a solved Cube and make 21 random moves. In your hand you then hold a configuration that can be solved in 21 moves.

8. Simon says:

With a sledgehammer

9. name says:

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

you have to remember that mathematicians and computer scientists are fond of actually speaking precisely, and not of perverting language to make it sound more exciting (like an english professor or journalist would).

so take what was said in the above summary at face value.

10. Dan Tastic says:

Chances are, that cube you put 21 turns on can be solved in less moves.

11. anjan bacchu says:

hi all,

if the stanford scientist want, he could rent CPU hours from Amazon’s EC2 large instance feature(http://www.amazon.com/b/ref=sc_fe_c_0_201590011_2?ie=UTF8&node=370375011&no=201590011&me=A36L942TSJ2AJA), he could use 10 computers for 1 month to get the solution.

The problem : it will set him back by 3 grand. However, if there are enough people interested in getting it solved sooner, then we/they could pool in and get it done earlier! 🙂

BR,
~A

12. mb says:

the statement “it’s also known that there are no configurations that can be solved in 21 moves” is incorrect. if you look at the paper, you’ll find the actual statement is “there are no known configurations that require 21 moves” i.e more than 20.

13. […] Rubiks cube can be solved in 25 moves March 26th, 2008 If you think I’m nerdy, this guy spent his time proving that rubiks cube can be solved in 25 moves or less regardless of the configuration. http://arxivblog.com/?p=332 […]

14. There are a few important errors in this article.

1. “He’s shown that there are no configurations that can be solved in 26 moves, thereby lowering the limit to 25.” Wrong. He shows that there are no configurations whose solution *requires* 26 moves. In fact, his results imply that every configuration can be solved in 26 (and, indeed, 25) moves.

2. “…it’s also known that there are no configurations that can be solved in 21 moves.” Wrong. There are plenty of configurations that can be solved in 21, or even fewer moves. (Take, for example, the trivial configuration, where the cube is already solved.) What you probably mean here is that there are no configurations which are known to require 21 moves.

3. “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.” Set theorists would probably have little to say on the subject. This is closest to the domain of combinatorics, with group theory coming in a close second. Unfortunately, no one is likely to come up with better bounds by human argumentation alone, since proofs (particularly ones involving high-dimensional objects) often unavoidably involve huge numbers of cases, something only a computer has the patience or speed to deal with. Indeed, an old metamathematical theorem states that the number of lines in the shortest proof of almost every true statement (i.e., theorem) is more than the number of protons in the universe, and so is inevitably beyond the purview of the pen.

15. a says:

Why would this be strange? If it is solvable in 20 moves, then ANY 21st move would un-solve it.

16. Noah says:

Figures. Typical math geek. Give us the theorem and proof with no practical example. Show me a video of a Rubik’s cube with you solving it in 25 moves and I’ll believe it. Otherwise I call “bullshit”.

17. Ibrahim says:

Interesting, and you’ve been slashdotted btw. Also, why is that Q6600 at 1.6Ghz? The stock clock on that is 2.4Ghz, and they overclock like mad I’ve heard.

18. […] Rubik’s Cube Solved In 25 Moves Mar27 27 March 2008, WebMaster @ 6:19 am 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. […]

19. Char-Lez says:

He did show you, in fact, that *any* can be done in 25 or fewer moves. So go pick your own example.

What you probably mean is “give me the solution to specific configuration X”, which is a different problem, and you can do that yourself by taking a cube and putting 25 twists on it.

Knowing the lower bound to a problem is useful because it helps you evaluate various algorithms for finding solutions to specific problems. Considering that there are 43,252,003,274,489,856,000 possible starting positions (Per wikipedia) knowing which algorithm performs better could save lots of time/money/effort by not picking less efficient ones.

20. Dave says:

Speaking as an english professor who likes precision, name is a strictly speaking, a maloderous douchebag.

21. LetnI says:

Following the though pattern of the comment section…

When the article mentions the Intel Quad Core Q6600 is only running at 1.6Ghz, this actually means that the CPU was running a minimum of 1600 Mhz. This does not mean that the CPU did not indeed actually run the rated speed of 2.4Ghz at any time during the test. In fact the literal fact that the CPU is running 2.4Ghz, but really has 4 cores really means that it is technically running 9.6Ghz… but due to the Rubix Cube possibility of there not being any solutions at 21 moves forces the CPU to underclock itself to 400mhz, effectively rendering the CPU to only quantativley run at 1.6Ghz. Sheez.. isn’t it obvious???

LetnI

22. John says:

A++ Seller. Item exactly as described. Would buy again.

23. Verb Ate Him says:

His actual abstract was very helpful. It doesn’t have the ambiguities that the article had…

“How many moves does it take to solve Rubik’s Cube? Positions are known that require 20 moves, and it has already been shown that there are no positions that require 27 or more moves; this is a surprisingly large gap. This paper describes a program that is able to find solutions of length 20 or less at a rate of more than 16 million positions a second. We use this program, along with some new ideas and incremental improvements in other techniques, to show that there is no position that requires 26 moves.”

24. Ralph says:

hi supernintendo chalmers!

25. […] обнаружили, что кубик

26. i love bacon says:

You only need from 0 to 26 moves to solve any configuration. Take a configuration that requires 26 steps, those 26 steps provide the minimum steps needed to arrive at the answer. That’s what is ‘proved’ until someone finds the optimal maximum to be lower than 26. If you start with another configuration, it could only need 4 moves…

Now, your using 359 moves just means you suck. How about you just give up and buy a Jeep Rubicon. Those things solve themselves. I mean, sell themselves.

LetnI, saying that 4 cores running at 2.4GHz is equivalent to one core running at 9.6GHz is an oversimplification, at best. As the saying goes; “Nine women can’t make a child in one month”.
Depending on the parallellism of the running code, the equivalent speed can be anything from 9.6GHz to 2.4GHz. In this case, I’d say it’s safe to assume that it’s somewhere closer to the upper limit, but I doubt it is precisely 9.6GHz.

28. Martijn says:

@Dave the ‘english professor’:

I like how you misspelled malodorous. And made two more errors in your post. Teaching English isn’t what it used to be, hmm?

29. Fred X. Quimby says:

Move 1: Twist top left 1/4 turn.
Move 2: ???
Move 3: Profit!!!

In Soviet Russia, Rubic’s Cube solves YOU!

30. panefsky says:

I am happy to volunteer some of my cycles. Please use quicksort.. 🙂

31. James Beam says:

How about proving that Rubik’s Cube theorems can be solved in less than 21 _WEEKS_ ?

32. […] If you’re a Rubik’s Cube enthusiast interested in decreasing your solve times, go ahead and read the full article. […]

33. […] proven last year by a group at Northeastern University. Tomas Rokicki, a mathematician by school, uses a rather nifty piece of computer science. Rokicki’s proof is a neat piece of computer science. He’s used the symmetry of the […]

34. […] 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. (link) […]

35. Rubik’s Cube I tried to solve it once and it took me 25 days. How you do it in 25 moves it sounds like a good way to win some money in a bet.

36. Jason says:

If you have a solved cube… that IS a configuration so that would be ZERO moves.

Also starting from ANY configuration and using 25 moves… after the first move the cube is still in ANY configuration and now you can start over and solve it in 24 moves…. repeat and now it is solvable in 23 moves….etc…

So this proves that the cube is solvable in ANY number of moves equal to or greater than zero

I don’t get the point of the article.

37. cap2501 says:

It is also important to note that the 1500 hours of time it took to solve is most likely measured in cpu time (as is the standard measure in this industry) so on a quadcore Q6600 it would be 375 real hours.

38. cap2501 says:

Also, to clear up confusion what he has developed is not an algorithm that will solve one configuration in 25 moves but an algorithm that is proven to solve any conceivable configuration of a rubics cube in 25 moves or less.

39. Jason says:

And what i said still stands… if a cube can be solved in 25 moves from ANY conceivable configuration… then after the first move from any conceivable configuration it only takes 24 moves… etc… so my point still stands…

The Rubik’s cube can be solved from ANY conceivable configuration in ANY number of moves equal to or greater than zero

40. Jason says:

But there is a problem with all of this…

Because of the The dichotomy paradox, you cannot even start to solve the cube

“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”

41. […] to have proven that a scrambled Rubik’s Cube (the standard 3×3×3 version, mind you) can be solved in no more than 25 moves, regardless of configuration. This trumps a study done last year suggesting a 26-move solution from […]

42. Joe says:

Correct. I believe the implication is that there is no known configuration which _requires_ 21 moves to solve. Therefore, any move (including attempts to scramble the cube further) from any configuration known to require 20 moves is a step _towards_ the solution.

43. VLandi says:

No “name”, James Method and Some Guy are correct. Of course you have configurations that can be solved in 21 moves, as Some Guy proves, that does not imply that 21 is the minimum number of moves to solve such configurations. Since the article is about finding the minimum number of moves, the slight “abuse” of language, where you see “can/cannot be solved in X moves”, is acceptable, in my opinion. It wouldn’t be acceptable in a scientific publication though.

44. nick says:

Jason said, “if a cube can be solved in 25 moves from ANY conceivable configuration… then after the first move from any conceivable configuration it only takes 24 moves… etc… so my point still stands…

The Rubik’s cube can be solved from ANY conceivable configuration in ANY number of moves equal to or greater than zero”

but your assuming that your first move always makes a unique configuration… it doesnt

i.e. several of the different starting positions will have the same configuration after the first move

45. Pao e Vinho says:

“The stock clock on that is 2.4Ghz, and they overclock like mad I’ve heard.”
Cause if it’ll take 1500h (2 months) of interruptible processing time. I guess you don’t wanna take *any* chance of burning the CPU or even crash. That would mean weeks of delay in the project.

46. hatesmorons says:

It’s funny reading all the opinions from people that don’t understand any math greater than algebra.

Damned fools.

47. Greg says:

I hope Noah is kidding. If not, he’s an idiot. How is a proof, for all configurations, less meaningful than a solution for a single configuration?

48. Move 26: throwing the rubix’s cube into the staduim of adoring fans!

49. Tom says:

Good to know our brilliant scientists are putting so much effort into solving the Rubik’s f-ing Cube rather than any of our real problems.

The generalist approach would be to ask:

What is the maximum number of required moves to bring an NxNxN cube from any pattern to any other pattern if N is the edgewise number of divisions, and there are six colors of NxN squares each, where when N is odd, the center squares of each face will be of the six unique colors, and that there is for any NxNxN one pattern that puts only one color per cube face? One might also generalize this for all dimensions. For example, in a hyperspace of 4 linear dimensions. We disregard any non-required moves, as otherwise, the maximum number is without bound. A required move is defined as any move that allows the operator to achieve the second pattern in the fewest number of moves. (The semantics of minimum/maximum can get confusing here. We can also talk about: are there any two patterns with a minimum number of required moves greater than X, where X is some number, in this case 25.)

For a 1x1x1 cube there is only 1 pattern and therefore the maximum number of required moves is 0.

For a 2x2x2 cube there are going to be many more patterns, but probably not a huge number. I suspect without doing any effort that the required maximum number of moves is either 2 or 3.

The solvers here are working on a 3x3x3 cube.

To argue that because if you make a move the required number of step decreases by one and therefore it can be deduced that any solution requires zero moves is wrong, because we are finding the maximum number of required moves. Any pattern may or may not have the maximum number of required moves, all the way down to zero moves. Therefore, with each move the goal is to reduce the required number of moves by 1. You cannot do any better, but you can do worse, either by moving to a pattern requiring the same number of moves or a pattern that requires one more move up to the maximum. On a 3x3x3 cube there are 3x3x3=27 “slices”, each with 3 moves or 81 potential moves per move. It may be that only one of those 81 will reduce the requirement, but it is guaranteed that there will be at least one.

On a 2x2x2 cube there are actually only 1x1x1 = 1 relative slices, each with 3 moves, so only 3 potential moves per move. This is because the left and right slice can only move relative to each other, making a left or right move equivalent. Each face can have any combination of 8 cubes, each with 4 potential orientations, held four at a time. This is, I believe, 8x7x6x5x4/(2x3x4) combinations on one side and the other side can only have 4 rotations of the remaining four cubes, giving 8x7x6x5x4x4/(2x3x4) or 1120 total configurations. 3^7 is 2187, therefore all patterns can be achieved by making 7 distinct moves. This does not give you the maximum number of required moves, however, which will be less.

Note that not all patterns that can be “painted” on a cube are possible through “moves”, because of the limitations of the mechanics of the cubes. A digitized version of the cube would not have that limitation, though the combinatorics of play are based precisely on the limitations of the mechanics of the cube. Therefore, any set theory proof would also be limited by the inference of the mechanics. One may therefore also investigate the bounds of mechanical limitations on play and move limits. It is probably so that the more flexible the mechanics of the cube, the fewer number of required moves despite a huge increase in the number of possible patterns.