Domain of the Cube Forum - Cube lovers returns

Square subgroup in QTM

Thu, 28 Mar 2013 20:44:46 -0400

The square group analysis in HTM is as follow :

        Analysis of the 3x3x3 squares group
        -----------------------------------

                                          branching
Moves Deep       arrangements (h only)     factor      loc max (h only)

  0                    1                      --             0
  1                    6                      6              0
  2                   27                      4.5            0
  3                  120                      4.444          0
  4                  519                      4.325          0
  5                1,932                      3.722          0

Subgroups using basic moves

Wed, 20 Mar 2013 21:20:23 -0400

In QTM, the whole cube is generated using U,D,R,L,F,B moves.
If we drop some moves we end up with some subgroups. The subgroups are:

1) I [the identity]
2) U  
3) U,D
4) U,F
5) U,D,F
6) U,F,R
7) U,D,F,B
8) U,D,F,R
9) U,D,F,B,R

I know the depth table for subgroups 1) 2) 3) and 4):
The subgroup 1) generated by "no move", has the following obvious table:

Moves Deep       arrangements (q only)     

  0                    1                   
                   ------
                       1

The subgroup 2) generated only by the move U, has the following table:

Symmetry Reduction of Coset Spaces

Fri, 22 Feb 2013 22:44:55 -0500

Having repeatedly shot myself in the foot by mishandling the symmetry reduction of coset spaces, I finally sat down, laid out the math and put together a set of notes on the matter. These notes follow.

Coset Spaces

Solving Rubik's cube either manually or by computer usually involves dealing with coset spaces. A group may be partitioned into cosets of a subgroup of the group:

     g * SUB          where g is an element of the parent group and 
	              SUB is a subgroup of the parent group

RUF Group Enumeration

Fri, 22 Feb 2013 22:39:46 -0500

I recently bought a new computer and wanted to put it through its paces. I dusted off my RUF three face coset solver and spruced it up a bit. Since I now have three iMacs in my household connected on an airport network, I rewrote the program using a server–client model. With this I can have all three computers working on a problem in parallel with as many as 14 cores. With these tools I have extended the enumeration of the three face group out to twenty q–turns:

Three Face Enumerator Client

Fixed cubies in subgroup: UF, UR, UB, UL, DF, DR, FR, FL, BR.
92,897,280 cosets of size 1,837,080

Server Status:
Three Face Group Enumerator
Sequential coset iteration
Enumeration to depth: 20

Snapshot: Friday, February 22, 2013 9:28:02 PM Central Standard Time

 Depth             Reduced             Elements
   0                     1                    1 
   1                     1                    6 
   2                     4                   27 
   3                    12                  120 
   4                    51                  534 
   5                   213                2,376 
   6                   914               10,560 
   7                 4,038               46,920 
   8                17,639              208,296 
   9                78,234              923,586 
  10               344,175            4,091,739 
  11             1,524,115           18,115,506 
  12             6,722,358           80,156,049 
  13            29,739,437          354,422,371 
  14           131,158,304        1,565,753,405 
  15           578,971,538        6,908,670,589 
  16         2,546,820,524       30,422,422,304 
  17        11,174,670,698      133,437,351,006 
  18        48,528,827,222      579,929,251,620 
  19       205,901,170,504    2,459,821,160,421 
  20       814,027,054,726    9,731,195,124,049 

 Sum     1,082,927,104,708   12,943,737,711,485 

92,897,280 of 92,897,280 cosets solved

Back from the Brink

Wed, 06 Feb 2013 08:40:24 -0500

Well, the server had a hard drive failure and I decided that it was time for an operating system upgrade. Unfortunately in the last 9 years everything had changed, e.g. the new versions of php and drupal and mysql were all incompatible with the old versions, and in various ways.

You can imagine my horror when I realized just how much work would be involved in salvaging the forum and make it usable again. I thought all I could do is make the drupal mysql file available to the web and figure out a way of upgrading later.

Finally as a last ditch effort I remembered the Ultimate Boot CD which has a hard drive cloning program and it was able to copy all the sectors still readable to another hard drive. The fact that the critical files were readable and there were multiple kernels bootable on the old failing hard drive was enough to get the server to at least boot, and I was able to restore the last missing files from another backup.

2x2x2 Cube Antipodes

Wed, 12 Dec 2012 13:34:50 -0500

I have written a GUI NxNxN cube program to which I just added a 2x2x2 cube auto solve function. To test the performance of the solution algorithm I wanted try it on the 14 q-turn antipodes. So I did the depth-wise expansion of the group, found the 276 antipodes and reduced them with M^† symmetry. In the context of the fixed DBL cubie 2x2x2 model, that is the <R U F> group model, M^† symmetry classes are formed by ( c * m' * q * m ) where q is a group element, m ranges over the cubic symmetry group and c is the whole cube rotation needed to place the conjugate back in the group.

How many 26q* maneuvers are there?

Sat, 20 Oct 2012 23:17:11 -0400

How many 26q* maneuvers are there?

Well, obviously we can't say for sure, as it hasn't yet been proved that the 3 known 26q* positions (which are symmetrically equivalent to each other) are the only 26q* positions. In another thread, Herbert Kociemba mentioned that there are "many" such maneuvers, but he did not attempt to generate them all (for the known 26q* positions).

I note that 26q* refers to a maneuver that is 26 quarter turns long and that is known to be optimal in the quarter turn metric. It may also refer to a position that requires a minimum of 26 quarter turns to solve. 26q (without the asterisk) refers to any maneuver 26 quarter turns long, but isn't necessarily optimal for the position it solves.

5x5 puzzle: Comparison between reduction chains (STM, 10000 instances)

Tue, 09 Oct 2012 05:48:02 -0400

The multi-chained approach used in kumi na tano allows to use multiple search chains at the same time. The main advantage is that the best chain can be choosen depending on the instance to be solved, rather than hard-coded into the search algorithm.

For example, the first of the following two 5x5 instances has its leftmost column solved, while second has solved four tiles in top-right corner.

[1]
  1 17  9 10 18    18  3 16  4  5
  6  0  2  3  8    11  7 17  9 10
 11  5 22  7  4    19  2 23  0 21
 16 15 20 23 13     6 20 14 12  1
 21 19 12 14 24    13  8 15 22 24

We cah use multi-chained approach here. The following two partitioning schemes:

One Million Random Twenty-Four Puzzle Instances in the STM metric

Sun, 07 Oct 2012 08:56:55 -0400

I have solved sub-optimally 1,000,000 random instances of 5x5 sliding tile puzzle in STM metric (single-tile moves). The actual running time was about 18,5 hours. The minimum, maximum and average solution length were 73, 171 and 124.48 moves respectively. About 52% of 1,000,000 solutions were in range [118; 132]. There were only 32 instances with (suboptimal) solution length less than 81 (range [73; 80]). Only one instance was solved in 171 moves.

Sliding tile puzzle suboptimal solver

Mon, 24 Sep 2012 09:07:09 -0400

Hello all.
I wrote a program capable to solve (MxN-1) sliding tile puzzles, such as the Fifteen puzzle. The program can solve puzzles from 2x2 to 11x11.
The main thread is on Speedsolving.com:
http://www.speedsolving.com/forum/showthread.php?38689-kumi-na-tano-3-00-sliding-tile-puzzle-suboptimal-solver
- Bulat

Policy Change for New Accounts

Fri, 29 Jun 2012 04:55:22 -0400

Due to the constant spamming I have changed the access rules for new accounts. From now on new users must email cubexyz at gmail dot com and explain why they want an account here. A short note on your specific interests on Rubik's Cube and math should be sufficient.

Also the ban on gmail has been lifted. Sorry for the trouble, but deleting spam entries got tiresome.

Mark

Megaminx needs at least 45 moves

Tue, 28 Feb 2012 17:56:26 -0500

Surprisingly, nobody seems to have done anything else as a rough analysis of the number of moves to solve the Megaminx puzzle, especially no analysis which includes the commutativity of some moves.

A Hamiltonian circuit for Rubik's Cube!

Mon, 20 Feb 2012 21:30:04 -0500

I have found a Hamiltonian circuit for the quarter-turn metric Cayley graph of Rubik's Cube! In fact, it only uses turns of five of the six outer layers of the cube.

In more basic terms, this is a sequence of quarter moves that would (in theory) put a Rubik's cube through all of its 43,252,003,274,489,856,000 positions without repeating any of them, and then one more move restores the cube to the starting position. Note that if we have any legally scrambled Rubik's Cube position as the starting point, then applying the sequence would result in the cube being solved at some point within the sequence.

Regularities in maximum WD values

Sat, 14 Jan 2012 15:26:12 -0500

Regularities in maximum WD values

This post is about any mathematical laws inside the Walking Distance heuristic. It seems like WD is not just puzzle to be computed. Maybe the whole WD heuristic is some math structure.

A Hamiltonian Circuit for the 2x2x2

Mon, 26 Dec 2011 13:33:59 -0500

I have found a Hamiltonian circuit for the 2x2x2 cube group (3674160 elements). I have posted the solution on the speedsolving.com forum. Link: http://www.speedsolving.com/forum/showthread.php?34318