## Rubik can be solved in 27f

Submitted by silviu on Sat, 04/01/2006 - 16:39.In this paper we give a proof that Rubiks cube can be solved in 27f.

The idea is to eliminate the 476 cosets at distance 12 in the group H=< U,D,L2,F2,R2,B2 >.

In this way we never have to consider in the 2 phase algorithm that a coset is at distance 12.

So we only solve cosets at distance 11. Together with my earlier result of 28 this gives a proof of 27.

The same idea was used by Bruce Norskog in his 38q proof.

However we do not really need to compute all 476 cosets. In fact we only need to compute 7 cosets of the group

T = Intersection ( < U,D,L2,F2,R2,B2 > , < F,B,L2,U2,R2,D2 > , < L,R,F2,B2,U2,D2 > )

The group H is not invariant under all symmetries. But the group T is invariant under all 48.

The idea is to eliminate the 476 cosets at distance 12 in the group H=< U,D,L2,F2,R2,B2 >.

In this way we never have to consider in the 2 phase algorithm that a coset is at distance 12.

So we only solve cosets at distance 11. Together with my earlier result of 28 this gives a proof of 27.

The same idea was used by Bruce Norskog in his 38q proof.

However we do not really need to compute all 476 cosets. In fact we only need to compute 7 cosets of the group

T = Intersection ( < U,D,L2,F2,R2,B2 > , < F,B,L2,U2,R2,D2 > , < L,R,F2,B2,U2,D2 > )

The group H is not invariant under all symmetries. But the group T is invariant under all 48.

» 10 comments | read more

## Two more classes with exacly 4 symmetries done - most 20f* are antisymmetric

Submitted by Herbert Kociemba on Thu, 03/30/2006 - 12:09.I finished the analysis of two more classes with 4 symmetries now. The computation took more than two weeks. All can be solved within 20 moves.

The definition of the classes D2(edge) and C2v(b) are explained on this page. Here you also can get some more information about these and other classes.

What is interesting, that from the 12 20f*-cubes of the class D2 (edge), 10 also have antisymmetry and from the 94 20f*-cubes of the class C2v(b) 92 also are antisymmetric.

The definition of the classes D2(edge) and C2v(b) are explained on this page. Here you also can get some more information about these and other classes.

What is interesting, that from the 12 20f*-cubes of the class D2 (edge), 10 also have antisymmetry and from the 94 20f*-cubes of the class C2v(b) 92 also are antisymmetric.

» 8 comments | read more

## Results of two more cosets of the H group, this time face turn metric.

Submitted by rokicki on Fri, 03/24/2006 - 23:35.After seeing Silviu have such success with H group (U, D, F2, B2, R2, L2)

cosets, I decided to give it a shot in the face turn metric. So far

I've completed the identity coset and the flip8 (upper and lower edges)

cosets; the superflip coset and flip4 (middle edges) are still running.

The identity, flip4, flip8, and superflip are the four centers of the

H group. I've also shown that of the approximately 234,101,145,600

positions represented by these four cosets, none have a depth greater

than 21. This exploration covers more than 5/1,000,000,000 of the total

cube space.

The identity coset has the following depths. For comparison on the right

cosets, I decided to give it a shot in the face turn metric. So far

I've completed the identity coset and the flip8 (upper and lower edges)

cosets; the superflip coset and flip4 (middle edges) are still running.

The identity, flip4, flip8, and superflip are the four centers of the

H group. I've also shown that of the approximately 234,101,145,600

positions represented by these four cosets, none have a depth greater

than 21. This exploration covers more than 5/1,000,000,000 of the total

cube space.

The identity coset has the following depths. For comparison on the right

» 20 comments | read more

## Rubik can be solved in 35q

Submitted by silviu on Wed, 03/22/2006 - 10:19.Let H be the group < U,D,L2,F2,B2,R2 > and let N be the subgroup of H that contains
all even elements in H.
I have run an exhaustive search on the coset space G/N and got the following table:

0q 1 1q 9 2q 68 3q 624 4q 5544 5q 49992 6q 451898 7q 4034156 8q 35109780 9q 278265460 10q 1516294722 11q 2364757036 12q 235188806 13q 28144The group N contains no elements of odd length and the maximum length is 24.

» 3 comments | read more

## New optimal solutions for an important group

Submitted by silviu on Wed, 03/15/2006 - 03:32.I have computed optimal solutions for every element in the group <U,D,L^2,R^2,F^2,B^2>.
For this task I made some minor modifications to Reid's solver. I would like to thank him for sharing it.

0q 1 1q 4 2q 10 3q 36 4q 123 5q 368 6q 1,320 7q 4,800 8q 15,495 9q 54,016 10q 194,334 11q 656,752 12q 2,222,295 13q 7,814,000 14q 26,402,962 15q 89,183,776

» 13 comments | read more

## Analysis of another two symmetry subgroups of order 4

Submitted by Herbert Kociemba on Wed, 03/08/2006 - 12:46.The symmetry class C4 defines a 1/4-rotational symmetry around a face (I chose
the UD-axis). It took about 8 days to show that all 36160 cubes, which exactly
have this symmetry (M-reduced) are solvable in at most 20 moves. There are 39
20f*-cubes. 35 of them also have antisymmetry, 4 only have symmetry, so reduced
wrt M+inv there are 37 cubes.

The class D2 (face) consists of all cubes which have a 1/2-rotational symmetry around all faces. Up to M-symmetry there are 23356 cubes, which exactly have this symmetry. It took about 4 days to show, that all cubes of this symmetry class can be solved in 20 moves. There are only 4 cubes which are 20f*, all of them also are antisymmetric. Here are the results:

The class D2 (face) consists of all cubes which have a 1/2-rotational symmetry around all faces. Up to M-symmetry there are 23356 cubes, which exactly have this symmetry. It took about 4 days to show, that all cubes of this symmetry class can be solved in 20 moves. There are only 4 cubes which are 20f*, all of them also are antisymmetric. Here are the results:

» 5 comments | read more

## Regarding God's Algorithm Calculations and lower bounds on face metric

Submitted by xintax on Mon, 03/06/2006 - 10:35.Hi...

Im kinda new here. Just a few intros: Im sheelah from the philippines and im doing an overnight research on the length of God's Algorithm... due tomorrow.

I was kinda ready but then I encountered this page that says that the real size of the cube universe is not 43 quintillion but rather 901 quadrillion.

For one, the repercussions of this on my project are:

1.) Theoretically less runtime and

2.) A different criteria for 'similarity'.

I was trying to read the calculations but I got lost between the numbers. Can someone please clarify this for me?

Secondly, I want to ask about the current lower-bound of the length of God's Algorithm in face turn metric. I keep seeing 20f* but I was wondering if there is a higher lowerbound...

Im kinda new here. Just a few intros: Im sheelah from the philippines and im doing an overnight research on the length of God's Algorithm... due tomorrow.

I was kinda ready but then I encountered this page that says that the real size of the cube universe is not 43 quintillion but rather 901 quadrillion.

For one, the repercussions of this on my project are:

1.) Theoretically less runtime and

2.) A different criteria for 'similarity'.

I was trying to read the calculations but I got lost between the numbers. Can someone please clarify this for me?

Secondly, I want to ask about the current lower-bound of the length of God's Algorithm in face turn metric. I keep seeing 20f* but I was wondering if there is a higher lowerbound...

» 9 comments | read more

## Permutations of the corners and edges: FTM

Submitted by Bruce Norskog on Tue, 02/21/2006 - 17:00.I have finally completed the face-turn metric version of the analysis of the permutations of the corners and edges of the 3x3x3 Rubik's Cube. That is, I have generated a table of the Cayley graph distances for the positions where the permutation of the corner cubies and the permutation of the edge cubies are considered, but not the orientation of either the edge or corner cubies. This is a set of 8!*12!/2 or 9,656,672,256,000 positions. As with the quarter-turn metric analysis, I used symmetry in the corner permutations to reduce the number of stored distances to 984*12!/2 or 235,668,787,200.

» 12 comments | read more

## 39 cubes with 20f moves in a class with 4 symmetries

Submitted by Herbert Kociemba on Tue, 02/21/2006 - 14:58.After the analysis of cubes with more than 4 symmetries I now try to analyze cubes with 4 symetries. The smallest class has 15552 cubes wrt to M-symmetry. All cubes of this class can be solved in 20f moves. All positions which could not be reduced with the two phase solver to less than 20f moves within a day were solved with my optimal solver within about 4 days. I did not check if the positions are local maxima, so they still are candidates for a 21f maneuver when appending a move.

D L2 U' B2 L2 B2 D B2 U L2 U2 R2 B F L' D U F2 L' R' (20f*) //C2v (a1)

D L2 U' B2 L2 B2 D B2 U L2 U2 R2 B F L' D U F2 L' R' (20f*) //C2v (a1)

» 14 comments | read more

## Another four interesting cosets

Submitted by rokicki on Fri, 02/03/2006 - 22:38.These are the distributions of the optimal solution lengths for the cosets where the edges start at the given M-symmetric position, using the quarter turn metric. The first run took longer than the other three combined; not sure why. Only a handful of positions at length 24; none at 26 or higher. I'm currently running the coset where the edges begin in the known length-26 position. It's interesting to note that with the edges superflipped and reflected across the center, all solutions took either 18, 20, or 22 moves.

» 6 comments | read more