## Rubik's Cube Proof Cut To 25 Moves 386 386

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."*
## Re:1.6ghz? (Score:3, Interesting)

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

## I still beat him! (Score:2, Interesting)

## Computational proofs (Score:3, Interesting)

## Distributed computing (Score:3, Interesting)

## Re:Which 25 moves? (Score:3, Interesting)

insane. And no need to hash; that would take extra time and use a non-efficient amount of key space. Just search by the configuration of the cube.## Re:1.6ghz? (Score:5, Interesting)

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:Computational proofs (Score:5, Interesting)

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?

## Re:Wow, it really works (Score:1, Interesting)

essentially similarones?## Re:Theory versus practice (Score:2, Interesting)

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...Proving that

ALLcubes can be solved in 25 moves is not the same as solvingAcube in 25 moves. I'd imagine if he can crunch the entire problem space of 400 billion configurations in 1500 hours, he can solve a single cube pretty damn quick.## "God's Algorithm" (Score:5, Interesting)

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:You only need one (Score:5, Interesting)

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.

## So the comic was right!! (Score:3, Interesting)

## Solve This! (Score:3, Interesting)

## Re:"God's Algorithm" (Score:5, Interesting)

## Re:Which 25 moves? (Score:3, Interesting)

as someone (Dik Winter) has already written a program that does just that.

Source: http://mathworld.wolfram.com/RubiksCube.html [wolfram.com]

## "may be non-intuitive" (Score:3, Interesting)

I actually solved the cube all by myself back in the 80's. I'm amazed that so much effort is still being put into it and some of the methods used by the record breakers need an awful lot of dedication to learn.

## Re:"God's Algorithm" (Score:1, Interesting)

That's not correct, there are a lot of such positions found. Superflip is perhaps the most well known, but certainly not the only one.

## Re:Wow, it really works (Score:5, Interesting)

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.

## Re:Wow, it really works (Score:3, Interesting)

## Re:Wow, it really works (Score:3, Interesting)

Taking into account mirrors and rotations, another program got it down to under 500, iirc. (it may have been slightly over - can't recall for sure).

It was really neat - a tic-tac-toe-playing set of web pages with no javascript, no programming on the server or the client - just 2,000 static pages.