## A plan to settle the maximin distance problem so we can all go home

Submitted by WarrenSmith on Fri, 05/19/2006 - 20:50.Because R has 4.3*10^19 configurations, exhaustive search is not feasible.

At present at least 10000 configurations are known (including the "superflip") that require 20 face-turns (20f) to solve.

Silviu Radu has a proof at at most 27f are necessary.

So the answer is somewhere in [20, 27]. What is it?

H.Kociemba's "two phase solver" works by first getting into the H =

## In search of: 21f*s and 20f*s; a four month odyssey.

Submitted by rokicki on Sun, 05/07/2006 - 12:21.At this point, I have found no 21f* positions, but with Herbert Kociemba and Silviu Radu, have found 11,313 (mod M+inv) 20f* positions. This set represents 16,510 mod M positions, and 428,982 overall cube positions. The majority of the positions were found by Silviu using a spectacular coset solver that he will write about soon.

## Thistlethwaite's 52-move algorithm

Submitted by jaap on Wed, 04/19/2006 - 01:31.David Singmaster had a copy that he scanned in, and put on his Singmaster CD6. That is a cd with all his notes and research on all kinds of recreational mathematics, which he makes available to anyone who is interested. I have converted those scans to text and put it all on my site.

Jaap

Jaap's Puzzle Page

## God's algorithm calculations for the 4x4x4 "squares set"

Submitted by Bruce Norskog on Mon, 04/03/2006 - 21:46.I have computed God's algorithm for the set of Rubik's Revenge (4x4x4 cube) positions that are reachable by the following set of moves: { U^2, u^2, d^2, D^2, L^2, l^2, r^2, R^2, F^2, f^2, b^2, B^2 }. Actually, not all of the above moves need to be included in order to generate the whole set. Since these moves are expressed as squares of other basic moves in the group theory notation, the set of positions reachable by these moves is referred to as the Squares Group. In my analysis thus far, I have considered the four centers of a given face to be indistinguishable. That is, I am considering only the "plain" 4x4x4, not the 4x4x4 supercube (where all centers are taken to be distinguishable from the others). With this simplification, this set of positions does not actually form a mathematical group, so I will refer to it as the Squares Set here. 19 slice turns was found to be sufficient to solve any of the positions in this set.

## Rubik can be solved in 27f

Submitted by silviu on Sat, 04/01/2006 - 16:39.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.

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

Submitted by Herbert Kociemba on Thu, 03/30/2006 - 12:09.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.

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

Submitted by rokicki on Fri, 03/24/2006 - 23:35.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

## Rubik can be solved in 35q

Submitted by silviu on Wed, 03/22/2006 - 10:19.0q 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.

## New optimal solutions for an important group

Submitted by silviu on Wed, 03/15/2006 - 03:32.0q 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

## Analysis of another two symmetry subgroups of order 4

Submitted by Herbert Kociemba on Wed, 03/08/2006 - 12:46.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: