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