Discussions on the mathematics of the cube

2x2x2 Cube

I recently added the 2x2x2 cube to my Virtual Rubik app. Playing around with the code I threw together a breadth first god's algorithm calculation using anti-symmetry reduction. This is old stuff but I thought I would post the results just the same.

2x2x2 States At Depth
Depth   Reduced(Oh+)       States
 0             1             1
 1             1             6
 2             3            27
 3             4           120
 4            13           534
 5            35         2,256
 6           126         8,969
 7           398        33,058
 8         1,301       114,149
 9         3,952       360,508
10        10,086       930,588
11        14,658     1,350,852
12         8,619       782,536
13         1,091        90,280
14             8           276
15             0             0

Group Order: 3,674,160

  1 R R U R F R' U R R U' F U' F' U'

  2 R R U R R F' U R F' R F' U R' F'

  3 R R U R' U F U' R F R' U' R U R

  4 R R U F F R' U' R F' R F' U U F

  5 R R U R' F R R U' R F R' U R R

  6 R R U R R U F U' R F' U R U' R

  7 R R U R R U' R F' U R' U R U U

  8 R U R R F' R F R U' F' U R U' F

Old Domain Names now restored

Hi Everybody,

I've re-activated the old domain names cubezzz.dyndns.org and cubezzz.homelinux.org so all the old links to the Domain of the Cube forum should be working now.

Now that I've thought about it more it actually feels good to get the original URL working again.

You can all thank Tom for coaxing me into it. I still wish dyndns.org could have helped us more, but I guess you get what you pay for.


Domain name changed (again)

Hi folks,

I'm reverting back to http://cubezzz.dyndns.org/drupal

The other URLs should also work but this one is the canonical URL for the cube forum.

Note that it should also be possible to access the forum directly via


All 164,604,041,664 Symmetric Positions Solved, QTM

Perhaps the most amazing feat of computer cubing was Silviu Radu and Herbert Kociemba's optimally solving all 164,604,041,664 positions in the half-turn metric back in 2006. Computers were much slower and had much less memory back then, and handling so many different subgroups can be tricky. Radu used GAP to help with the complexity of the group theory, and Michael Reid's optimal solver to provide the fundamental solving algorithms, and Kociemba used his Cube Explorer optimal solver to handle both the smaller subgroups and the positions left over after Radu's subgroup solver ran.

Symmetries and coset actions (Nintendo Ten Billion Barrel tumbler puzzle)

Jaap Scherphuis suggested that I post this result here, and it seems very relevant to the discussions taking place about symmetries in Cube solutions.

I recently calculated a solution for the Nintendo Ten Billion barrel puzzle that solves any position within 38 moves. Forgive the chatty presentation there - though there are GAP files linked, there is very little actual mathematics included in that blog post. This would be a more appropriate place to discuss the details. As far as I know, this is the first result of its kind for this puzzle, but I'm very convinced I missed a far better result.

Classification of the symmetries and antisymmetries of Rubik's cube

In 2005, Mike Godfrey and me computed the number of of essentially different cubes regarding the 48 symmetries of the cube (group M) and the inversion, see here for details.
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)

According to my computation, the 4x4x4 cube can be solved no more than 57 moves.
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

As is well known we can dispense with one of the Singmaster generators to still realise the whole of the 3x3x3 cube e.g. using the generating set . 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

Even though we now know the diameter of Rubik's Cube group in the half-turn
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

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.