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

Submitted by Herbert Kociemba on Sat, 10/05/2013 - 16:38.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)

Submitted by CS on Mon, 09/30/2013 - 13:31.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

Submitted by secondmouse on Tue, 09/24/2013 - 19:53.__. 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

Submitted by rokicki on Sat, 09/14/2013 - 13:43.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

Submitted by B MacKenzie on Sat, 08/10/2013 - 00:22.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

Submitted by cubex on Sat, 07/27/2013 - 13:01.## 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