Discussions on the mathematics of the cube

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

I outline an approach which may be able to determine the maximin depth of the Rubik cube R - may be able to prove the answer is 20 face turns - with a feasible amount of computer time.

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 = subgroup (which is known to be possible in at most 12f because of an exhaustive search of R/H) and then solving (which is known to be possible in at most 18f because of an exhaustive search of H) thus proving an upper bound of 30f.

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

In January of this year I set out to find a 21f* position---or, at the very least, extend the set of known 20f* positions. At that time I knew of only three 20f* positions, despite having exhaustively solved several collections of pretty patterns and performed months of optimal cube solutions.

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

I have put Thistlethwaite's 52-move algorithm on my site. This might be of historic interest to those of you who haven't seen a complete copy of it before.

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's Puzzle Page

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

God's algorithm for 4x4x4 Squares Set

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

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.

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

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.

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

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

Rubik can be solved in 35q

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         28144
The group N contains no elements of odd length and the maximum length is 24.

New optimal solutions for an important group

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

Analysis of another two symmetry subgroups of order 4

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: