Discussions on the mathematics of the cube

Some Thoughts on Representing the Cube

I wanted to post a number of miscellaneous items about representing the cube, and I will also include a few other related items.

I'll start with the group S3 as an example.  I will treat the group S3 as acting on the set {0, 1, 2}.  As I have been doing recently, I'll use the notation (a b c) to represent the permutation 0→a, 1→b, 2→c. 

In this notation, the entirety of S3 can be listed as follows:

(0 1 2)
(0 2 1)
(1 0 2)
(1 2 0)
(2 0 1)
(2 1 0)

This basic idea, or something very similar to it, is probably the way most people represent the cube in a computer program.  Variations on the theme could include an S54 model, an S48 model, an S24 × S24 model, and some sort of wreath product model.  The most common wreath product model would probably be something like (S8 wr C3) × (S12 wr C2).  In the wreath product model, S8 and S12 represent the corner cubies and the edge cubies, respectively, and C3 and C2 represent the twists of the corner cubies and the flips of the edge cubies, respectively.  Of course, none of these various models are isomorphic to the cube group.  Rather, the cube group is a subgroup of whatever group is chosen as the computer model.

Solving the Rubik's Cube in Sub 13 Algs, BLD!

Ok, this method is called Simul Block, or Shotgun. It's a very powerful bld method, using 8 new algs, and putting some old algs to new uses, or using uncommon variations of a common algorithm.

This is an advanced version of Pochmann, the method has three key steps.

Solve the F/B face + 1 S Edge (the UL as Default)
Roux Cycle the last three edges
Parity Fix.

The parity is something common in bld, so most of you will laugh at this. My method never encounters the 2 Corner 2 Edge swap parity, because that's what my system is based on.

It's something that 4 Step Solvers encounter, and they deal w/ it first.

How to Compute Optimal Solutions for All 164,604,041,664 Symmetric Positions of Rubik's Cube

Using some new ideas, techniques, and computer programs, we
have successfully found optimal solutions to all symmetric positions
of Rubik's cube in the face turn metric (FTM). Furthermore we have
maneuvers for 1,091,994 20f* (positions whose optimal solutions
have 20 face turns) cubes and proven that there are no symmetric
21f* cubes. So if there are any cubes at depth 21 then these must
be unsymmetrical. To the best of our knowledge, at the start of this
investigation in January, only a few such positions were known (less
than a dozen). Expressions for all these cubes can be found on
Rokicki's home page http://tomas.rokicki.com/all20.txt.gz.

The 4x4x4 can be solved in 79 moves (STM)

I have done a five-stage analysis of the 4x4x4 cube. My analysis considers the four centers for each face to be indistinguishable. It also assumes that there is no inner 2x2x2 cube in the middle of the cube.

Like Morwen Thistlethwaite's well-known four-stage 3x3x3 analysis, my five-stage procedure consists of multiple stages where each successive stage only allows use of a subset of the moves allowed in the previous stage, with the final stage only allowing half turns. So far, I have completed analyses of the five stages using the slice turn metric (STM). Use of other metrics is possible. (In fact I have done some other metrics for some of the stages.) My analyses for each individual stage are optimal with respect to the specified move restrictions for each stage. The results indicate that the 4x4x4 can be solved using a maximum of 79 slice turns.

Suboptimal solvers for the 4x4 and 5x5 cubes?

I am looking for a suboptimal solution algorithm for the 4x4 and 5x5 cubes.

I am pondering about prepending a third phase to Kociemba's two-phase algorithm for the 3x3 cube. The initial phase performs two-layer twists on a 4x4 cube or a 5x5 cube until the stickers on the edge parts and on the side parts line up to form a 3x3 cube. Then Kociemba's two phase algorithms takes over and solves the 3x3 cube.

Does anyone have experience with such an algorithm? I currently don't know how to create the pruning tables for the initial phase. Also I am not sure, if my approach will work at all.

Using latex2html utility for posting

Silviu Radu has suggested using the latex2html utility to generate html for use in messages posted to the forum. Let's try a test:


The files would have to be hosted somewhere but this would provide a way of handling all those mathematical symbols although no doubt some minor hand-turning would be necessary.


the search for a 21f, an idea for some candidates

There is a unique Rubik cube position maximally far from Start *PROVIDED*
you only look at edges and ignore corners - it was found by J.Bryan and
is all edges flipped in place, composed with a mirror reflection of the whole

That suggests, taking this one edge position and exploring all possible configurations
(there are about 3 million) of the 8 corners to get 3 million cube positions.
If you are seeking a cube configuration with 21f or more distance to start,
these 3 million candidates seem tolerably likely to include a winner.

Because 3 million is a lot of searching, you might try a cheaper approach like just

A plan to settle the maximin distance problem so we can all go home

I outline an approach which may be able to determine the maximin depth of the Rubik cube R - may be able to prove the answer is 20 face turns - with a feasible amount of computer time.

Because R has 4.3*10^19 configurations, exhaustive search is not feasible.

At present at least 10000 configurations are known (including the "superflip") that require 20 face-turns (20f) to solve.

Silviu Radu has a proof at at most 27f are necessary.

So the answer is somewhere in [20, 27]. What is it?

H.Kociemba's "two phase solver" works by first getting into the H = subgroup (which is known to be possible in at most 12f because of an exhaustive search of R/H) and then solving (which is known to be possible in at most 18f because of an exhaustive search of H) thus proving an upper bound of 30f.

In search of: 21f*s and 20f*s; a four month odyssey.

In January of this year I set out to find a 21f* position---or, at the very least, extend the set of known 20f* positions. At that time I knew of only three 20f* positions, despite having exhaustively solved several collections of pretty patterns and performed months of optimal cube solutions.

At this point, I have found no 21f* positions, but with Herbert Kociemba and Silviu Radu, have found 11,313 (mod M+inv) 20f* positions. This set represents 16,510 mod M positions, and 428,982 overall cube positions. The majority of the positions were found by Silviu using a spectacular coset solver that he will write about soon.

Thistlethwaite's 52-move algorithm

I have put Thistlethwaite's 52-move algorithm on my site. This might be of historic interest to those of you who haven't seen a complete copy of it before.

David Singmaster had a copy that he scanned in, and put on his Singmaster CD6. That is a cd with all his notes and research on all kinds of recreational mathematics, which he makes available to anyone who is interested. I have converted those scans to text and put it all on my site.

Jaap's Puzzle Page