The thread on this forum entitled "Non Trivial Identities" has had the most responses of any thread to date (39 responses so far).  The thread contains a great deal of useful information, including a detailed analysis of non-trivial identities of length 12q.  The thread may be found at http://cubezzz.homelinux.org/drupal/?q=node/view/114 (you have to login to see all the responses).

I tend to view the problem of analyzing identities more in terms of duplicate positions than in terms of identities, but you can of course easily transition from one view of the problem to the other.  In any case, the problem of finding a formula for the number of moves at each distance from Start is closely tied to the problem of finding duplicate positions and/or non-trivial identities.

Inspired by Jerry Bryan's recent Cubelovers article, I decided to run a few large randomly-selected cosets such that optimal solutions were found for *all* positions (rather than just proving all positions were within 20, as I usually do). From this, I hoped to gain some insight into how long it would take to enumerate the entire cube space and find counts for every distance. I also hoped to get an estimate for the total count of 20s in the entire cube space.

I first wrote a program to select a random coset, and then set those cosets to running, specifying a maximum search ply of 18. This way

Some observations on the Rubik's Cube Group, its sub-groups and solution algorithms.

When approaching the problem of auto-solving Rubik's cube in my Virtual Rubik program I chose to proceed in a three step process by first solving a 2 x 2 x 2 block using turns of all six faces. Then using turns of the three faces not intersecting the solved block, the block is extended to a 3 x 2 x 2 block, leaving two faces unsolved. And finally the last two faces are solved using just turns of those two faces.

I've been playing around with what it would take to enumerate the entire cube space for the standard 3x3x3 cube, using a large number of computers working together.  I've come up with what I think is a reasonable approach, a bit past the bare edge of what might be possible with today's technology, but perhaps not too far past.  The assumption will be that we want to solve the problem in one year, with all the machines involved in the project fully dedicated to the problem for the whole year.

The following is not a compete description of my proposed approach, but rather is an analysis of the performance characteristics that would be required.  There is not much advantage in coming up with an approach that would require billions of years to compute.

Hello.

A quick introduction: I'm Robert Smith, I'm 18, and I live in the US. I used to be quite into speedcubing, starting around 2003, but I have since moved into the more "theoretical" aspects of the cube. I also like mathematics, and, fortunately, programming. Anyway, that's that.

I have been writing a "general puzzle solver," whose goal is to be able to solve any (permutation) puzzle with (almost) any method. Quite quickly, we see there are definitely limits to this (e.g., we couldn't solve any scrambled 10x10x10 cube in a single phase). But, if some certain requirements are met, solving any puzzle should be an ability with this program.

I've read that the proof of the superflip requiring a minimal solution of 20 moves relied heavily on the symmetry of the position and is purported to be quite clever. I just found it in the cube lovers archive, but haven't had a chance to read it in depth, yet.

Has anyone posited other positions that could be proved to require 20 moves similarly? Has it be postulated that the superflip could be the only position that requires 20 moves?

Thanks,
e.

All elements of the Rubik's cube group may be generated using turns of only five of the six faces, say all but the Up face. This suggests a puzzle variant: a cube with the Up center cubie glued fast. Has any work been done on such a puzzle? I played with this a bit today. Here's an U turn macro using no U turns:

B D' F L' F' B D F' R B' L F' D F B' L' F

Did another long overdue upgrade with apache and php. There's tons of new exploits so it has to be done. We were down for about an hour. Looks like everything still works :)

Mark

With a total of 1.28 million cosets solved, we have shown that every position of Rubik's cube can be solved in 22 or fewer face turns.

This required approximately 50 core-years of CPU time contributed by John Welborn and Sony Pictures Imageworks.

No distance 21 positions were found in this search, despite solving a total of more than twenty-five million billion cube positions.

There is a short article in New Scientist (August 9th edition) on this problem and this result.

The same techniques for the proof of twenty-five moves were used, just on many more computers.

I have found 310 cosets with an upper bound of 18, and about 82,000 with an upper bound of 19 (or less); all the rest have an upper bound of 20 or less.

What is known about the supergroup? (That is, the normal cube but with oriented
center facelets.) Have any computer explorations been performed? Any "hard"
positions known? Any coset explorations?

I know Jaap has a supergroup solver embedded in a Java applet. Does anyone else have
any programs?

A start might be an optimal solution length distribution that take a solved cube to a
solved cube, but just change the center facelet twists. But maybe an optimal solver
for the supergroup would just be too slow.