## 3x3x1 rubik square is isomorphic to ( U2, D2, F2, B2 ) cube subgroup

Submitted by cubex on Sat, 06/29/2013 - 14:02.Analysis of ( U2, D2, F2, B2 ) ------------------------------ Level Number of Positions 0 1 1 4 2 10 3 24 4 53 5 64 6 31 7 4 8 1 --- 192Everyone agree? I'm not sure if this has been pointed out before.

## Square subgroup in QTM

Submitted by mdlazreg on Thu, 03/28/2013 - 19:44.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

Submitted by mdlazreg on Wed, 03/20/2013 - 20:20.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 ------ 1The subgroup 2) generated only by the move U, has the following table:

## Symmetry Reduction of Coset Spaces

Submitted by B MacKenzie on Fri, 02/22/2013 - 22:44.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

Submitted by B MacKenzie on Fri, 02/22/2013 - 22:39.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

Submitted by cubex on Wed, 02/06/2013 - 07:34.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

Submitted by B MacKenzie on Wed, 12/12/2012 - 13:34.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

## How many 26q* maneuvers are there?

Submitted by Bruce Norskog on Sat, 10/20/2012 - 22:17.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)

Submitted by stannic on Tue, 10/09/2012 - 04:48.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]117 9 10 18 18 3 164560 2 3 8 11 7 17910115 22 7 4 19 2 23 0 211615 20 23 13 6 20 14 12 12119 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

Submitted by stannic on Sun, 10/07/2012 - 07:56.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.