Discussions on the mathematics of the cube

## Classification of the symmetries and antisymmetries of Rubik's cube

In 2005, Mike Godfrey and me computed the number of of essentially different cubes regarding the 48 symmetries of the cube (group M) and the inversion, see here for details.
We used the Lemma of Burnside to find this number. Since then I wondered if it would be possible to confirm this number by explicitly analyzing all possible symmetries/antisymmetries of the cube.

## Solving the 4x4x4 in 57 moves(OBTM)

According to my computation, the 4x4x4 cube can be solved no more than 57 moves.
The solving algorithm is based on tsai's 8-step method which can be found here: link

The only modification is that I merged step 3 & step 4 to one step.

For some reasons, the algorithm cannot be defined to the conversions between subsets, but can be defined to the conversions between sets:
```S0: <U R F D L B Uw Rw Fw Dw Lw Bw>
step 1 =>
S1: <U R F D L B> * <U R F D L B Uw2 Rw Fw2 Dw2 Lw Bw2>
step 2 =>
S2: <U R F D L B> * <U R2 F D L2 B Uw2 Rw2 Fw2 Dw2 Lw2 Bw2>
step 3 & step4 =>
S3: <U R F D L B>
3x3x3 solver =>
S4: Solved State
```

## Five generator group of the 3x3x3 cube

As is well known we can dispense with one of the Singmaster generators to still realise the whole of the 3x3x3 cube e.g. using the generating set . Apologies if this has come up before - I was wondering if there has been any analysis on the likely diameter with these five generators including inverses in the QTM? I am guessing that it will be in excess of 26.

## About 490 million positions are at distance 20

Even though we now know the diameter of Rubik's Cube group in the half-turn
metric, there is still much yet to be discovered. The diameter in the QTM
and the STM are unproved (although they are almost certainly 26 and 18,
respectively). The exact count of positions at distance 16, 17, 18, 19,
and 20 in the half-turn metric is unknown. This note reports some progress
on an estimate for the count of 20's in the half-turn metric.

It is fascinating to me how problems of distinctly different difficulty
exist around the 3x3x3 cube in the half-turn metric. Initially, back in
the early days, we could solve individual positions non-optimally.

## Symmetrical Twist Assignment, a chimera

A corner cubie may be moved to any of the eight corner cubicles in three different ways; untwisted, with clockwise twist or with counterclockwise twist. The standard convention is to assign the twist with reference the orientation of the cubie's U/D facelet vis-a-vis the cubicle's U/D face. If the cubie's U/D facelet is on the cubicle's U/D face the cubie is untwisted. If the cubie's U/D facelet is rotated 120° clockwise from the cubicle's U/D face the cubie has clockwise twist and vice versa. The disturbing thing about this convention is that it is unsymmetrical. Under this definition the 12 q-turns have different effects on the twist of the cubies turned. Turns of the U and D faces have no effect on the twist of the cubies while a q-turn of any of the other four faces twist two cubies clockwise and two cubies counterclockwise. Since all the faces are symmetry equivalent it has always seemed to me that there ought to be a way of defining corner cubie orientation which preserves this equivalence.

## 3x3x3 edges only calculations restored

I restored Tom's 3x3x3 edge only calculations from 2004 back to the God's Algorithm Calculations file. It was the waybackmachine to the rescue this time. Somehow the newer version of the file was overwritten. Hopefully it will be updated again soon.

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

I hacked up a quick simulator for the 3x3x1 Rubik's Square today and I noticed that the number of positions at each level was the same for the ( U2, D2, F2, B2 ) subgroup of the normal 3x3x3 cube.
```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
---
192
```
Everyone agree? I'm not sure if this has been pointed out before.

## Square subgroup in QTM

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

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```