## The 4x4x4 can be solved in 77 single-slice turns

Submitted by Bruce Norskog on Thu, 07/26/2007 - 00:17.Previously I announced that the 4x4x4 cube could be solved in 79 single-slice turns
by solving it in five stages,
in a manner similar to the Thistlethwaite 4-stage solution for the 3x3x3.
(See
The 4x4x4 can be solved in 79 moves (STM).)
However, I have now realized my solution to the 2nd stage could have allowed
the use of more basic turns than I used.
I have realized that:

< U,u,D,d,L^{2},l^{2},R^{2},r^{2},F^{2},f,B^{2},b >
= < U,u,D,d,L^{2},l,R^{2},r,F^{2},f,B^{2},b >

So with l and r replacing generators l^{2} and r^{2},
you still can not reach any additional positions.
As a result, I should have included the moves { l, l', r, r' }
along with the other 24 allowed slice turns for that stage.

## Rubik's cube can be solved in 34 quarter turns

Submitted by silviu on Mon, 07/02/2007 - 10:24.http://www.risc.uni-linz.ac.at/publications/download/risc_3122/uppernew3.ps

## UFR / UF Coset Space

Submitted by B MacKenzie on Sat, 06/30/2007 - 16:52.Following up on an earlier thread I have explored the UFR/UF coset space. The number of UF cosets in the UFR group is given by:

^{6}= 2,322,432

The first three factors are the number of ways the corner cubie position and the two edge cubie positions which are not on the UF faces may be configured. The factor of 6 is for the corner position permutation. Of the 720 corner position permutations on the UF faces achievable using the UFR face turns only 120 are achievable using the UF face turns. Thus the corner permutation may be one of six cosets. The flip of the seven UF edge cubies is constrained under the UF face turns, one cannot perform a double edge flip in this group. This gives rise to a factor of two to the sixth for the edge flip (flip parity determines the flip of the seventh edge cubie). As Bruce Norskrog pointed out in the earlier thread, the above number is the same as the order of the UFR group divided by the order of the UF group, so things check out.

## Antisymmetry, Corners of the 3x3x3 Cube, quarter turn metric

Submitted by Jerry Bryan on Thu, 06/21/2007 - 12:18.Distance Positions Positions Positions from reduced reduced Start by by Symmetry Symmetry and Anti-Symmetry 0 1 1 1 1 12 1 1 2 114 5 5 3 924 24 17 4 6539 149 96 5 39528 850 469 6 199926 4257 2289 7 806136 16937 8768 8 2761740 57848 29603 9 8656152 180787 91688 10 22334112 466220 235710 11 32420448 676786 342593 12 18780864 392342 199610 13 2166720 45600 23818 14 6624 163 110 Total 88179840 1841970 934778

As I have written before, my programs have seldom worked with positions. They have nearly always worked with representative elements of M-conjugate classes. In the table above, the summary of representative elements is labeled "Positions Reduced by Symmetry". The goal of this approach is to obtain a 48 times speedup in processing time, and also to obtain a 48 times reduction in storage requirements.

## Representation of edge permutations and move table

Submitted by Herbert Kociemba on Mon, 06/18/2007 - 15:02.This MoveTable would have 12!*18 4 Byte entries when we take the coordinate from 0..12!-1 and of course is far too big. Of course we could reduce this by 48 symmetries, but then we still would have a very large table.

## Two Face Group

Submitted by B MacKenzie on Wed, 06/06/2007 - 09:47.Has the Rubik cube subgroup generated by the turns of two orthogonal faces been exhaustively expanded? My computer runs out of physical memory and bogs down after 18 q turns:

Shell Classes Elements 0 1 1 1 1 4 2 3 10 3 6 24 4 15 58 5 35 140 6 85 338 7 204 816 8 493 1970 9 1189 4756 10 2863 11448 11 6862 27448 12 16324 65260 13 38550 154192 14 90192 360692 15 206898 827540 16 462893 1851345 17 992268 3968840 18 1973209 7891990 Totals 3792091 15166872

## 26f now claimed proven sufficient

Submitted by Bruce Norskog on Thu, 05/31/2007 - 20:52.http://www.physorg.com/news99843195.html

## More on Branching Factors

Submitted by Jerry Bryan on Mon, 05/21/2007 - 23:32.I was pretty sure that I posted an article to this site about Starts-With and Ends-With, but if so I can't find it. In any case, for a position x we define Starts-With(x) to be the set of moves with which a minimal process for x can start, and Ends-With(x) to be the set of moves with which a minimal process for x can end. If Ends-With(x)=Q (the set of quarter turns), then x is a local maximum. A similar formulation of the same idea is that if |Ends-With(x)|=12, then x is a local maximum.

## Number of maneuvers for Rubik's Cube

Submitted by Herbert Kociemba on Sat, 05/19/2007 - 10:35.Let r = Sqrt(6) and k the maneuver length, then we have

N(k) = [(3+r)(6+3r)^n + (3-r)(6-3r)^n]/4

which gives 1, 18, 243, 3240, 43254, ...

Round[(3+r)(6+3r)^n] is a good approximation even for small n. and we see that 6+3r = 13.348... is the asymtotic branching factor.

## Disjoint Cycles and Twist/Flip Parity Rules

Submitted by B MacKenzie on Wed, 05/16/2007 - 00:13.(-1,-1,-1) being the coordinate for the (left,down,back) cublet through to (1,1,1) being the coordinate for the (right,up,front) cublet. The transformed position and orientation of each cublet is then specified as an element of the O symmetry point group, there being a one to one correspondence between the 24 elements of the O point group and the 24 states a cublet may assume via Rubik cube face turns: 12 edge positions with two flip states each or 8 corner positions with 3 twist states each.