Discussions on the mathematics of the cube

## Subgroups using basic moves

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

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

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

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

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?

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)

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

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

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

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