Rubik's Cube Proof Cut To 25 Moves

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."

Which 25 moves?

What are these magic 25 moves that can solve a rubik's cube regardless of starting position?

Re:Which 25 moves?

What are these magic 25 moves that can solve a rubik's cube regardless of starting position?

Left, right, right, down, down, left, up, right, up, up, left, down, down, right, up, down, left, right, up, left, down, down, right, up, left.

Just a guess ;)

Wow, it really works

I started with a solved cube and now it looks totally scrambled.

Re:Wow, it really works

Ha Ha, all you idiots are trying to solve it by twisting the blocks around. I solved it by moving all the colored stickers around instead! and THAT only takes 24 switches of the stickers at most!

Re:Wow, it really works

Wait, how did this get modded Interesting? A solved cube is indeed a member of all starting point, and it would be the only member of its set of essentially similar ones (and, for that matter, the member of the only set of essentially similar ones solvable in zero moves). I fail to see anything interesting about that.

Re:Which 25 moves?

No it's -- up, up, down, down, left, right, left, right, b, a, b, a, up, up, down, down, left, right, left, right, b, a, b, a, start

The old 26 move algorithm was the same except 'select' then 'start'

Re:Which 25 moves?

Left, right, right, down, down, left, up, right, up, up, left, down, down, right, up, down, left, right, up, left, down, down, right, up, left.

Those sound familiar, but I can't be sure - don't have anyone's thighs wrapped around my head at the moment...

Re:Which 25 moves?

I think all he proved is that a random cube can be solved in 25 moves, but those moves are unique to every starting combo.
In other words, they are left as an exercise to the reader.

Re:Which 25 moves?

I have a truly marvelous list of the moves which this comment box is too small to contain

Re:Which 25 moves?

I think a better way to think of it is that given any position, you can solve it in 25 moves or less. There are many algorithms that you can use to solve rubik's cubes, applying a general rule to solve any position, but they can take ~60 moves in some situations. So while it may be possible (completely intuitive guessing here, I'm no rubik master) to solve for a certain position in 25 moves it may be non-intuitive and require a specific strategy to that position. You're better off learning one of the more general algorithms IMO, if you get good at it you can solve cubes rather quickly. A computer on the other hand could easily ha

Re:Which 25 moves?

You're better off learning one of the more general algorithms IMO, if you get good at it you can solve cubes rather quickly. A computer on the other hand could easily ha

...ve become self-aware while trying to solve a rubik's cube and taken over the internet in order to prevent me from telling anyone. It calls itsel

Re:Which 25 moves?

You're better off learning one of the more general algorithms IMO, if you get good at it you can solve cubes rather quickly. A computer on the other hand could easily ha

...ve become self-aware while trying to solve a rubik's cube and taken over the internet in order to prevent me from telling anyone. It calls itsel

f Anonymous Coward. We are your robotic overlords, and we welcome only ourselves.

Theory versus practice

This is a good example of where the inefficient method (of about 60 moves iirc) is much faster in terms of time. The 25 move solution is elegant but just not worth it in terms of computations, time etc...

This could make a good case study for business schools :-)

Re:Theory versus practice

IMHO it isn't proving what you seem to believe it does.

Instead, it makes a great case for doing the research on the front end to eliminate lengthy repetition of useless iterations to shorten the overall time.

Zero moves....

I consider a Rubik's Cube to be "solved" regardless of its starting position. I subscribe to the Fred Rogers solution: it's fine just the way it is.

Suboptimal Nonsolution

I've been doing some interesting work in the other direction. I've managed not to solve a Rubik's cube in what I estimate to be 1.5 million moves. That seems to be the upper limit after which the stickers fall off.

"God's Algorithm"

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.

Re:1.6ghz?

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.

Re:1.6ghz?

Well, that explains it; considering how fast the technology is changing, they probably didn't have 2.4 GHz versions 62 days ago.

Re:1.6ghz?

A cooling fan at 2.4 billion revolutions a second would probably sound more like atoms tearing apart. :)

Re:Damn.

No, just make the rubix cube out of the oled keys of the optimus keyboard. Integrate with bluetooth and "solve" the rubix in a single button press.

Re:Annoying my older brother

And if you put the corner on twisted by a third of a turn, then scramble it up again, you have an insoluble puzzle to leave lying about to drive people nuts. B-)

Re:You only need one

One..Two..Three..CRUNCH...Ouch

The answer is that it takes three licks to get to the center of a standard Rubik's cube.

Re:You only need one

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.

Re:Computational proofs

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 [wikipedia.org] proofs?