Discussions on the mathematics of the cube

Lower bounds for the 3x3x3 Super Group

For quite a while I was looking for an optimal solution for the 'Pure Superflip' (a Superflip pattern were all centers remain untouched). Such an algorithm would also allow to define the lower bound for the 3x3x3 Super Group. I don't know if there already exist lower and upper bounds for this cube group.

Years ago I computed all optimal solutions of the 'Superflip' pattern in ftm (face turn metric) to see if any of the 4416 algorithms may leave the centers unchanged. Unfortunately all these algorithms twist either 4 or 5 of the centers.

I figured out that such an algorithm must be within the range of 23 and 24 moves, but I never was able to prove it. It took just too long for solvers these days to compute a solution.

God's Algorithm out to 18q*: 368,071,526,203,620,348

Almost exactly four years after 17q* was announced by Thomas
Schuenemann, we have calculated the number of positions at a distance
of exactly 18 in the quarter-turn metric. This is more than one in
twenty positions.

This number matches (mod 48) the count of distance-18 symmetric
positions; this provides a bit of confirmation that it is correct
(or rather, about 5.6 bits of confirmation).

The approach we used does not permit us to calculate the number of
positions mod M or mod M+inv without significantly increasing the
amount of CPU required; these computations will have to wait for

Fifteen Puzzle MTM

I wrote a fifteen puzzle simulation back in 2010 which I recently went back to and updated before submitting it as freeware to the Apple App Store.

Playing around, I then plugged my model into my coset solver framework and performed a states at depth enumeration in the multi-tile metric out to depth 23:

XV Puzzle Enumerator Client(bdm.local)

XV Coset Solver
	Fixed tokens in subgroup: 0, 1, 2, 3, 4, 8, 12, 15.
	518,918,400 cosets of size 20,160

Cosets solved since launch: 165,364,141
Average time per coset: 0:00:00.001

Server Status:
XV Puzzle Enumerator Server
Enumeration to depth: 23

Snapshot: Thursday, June 12, 2014 at 11:11:29 AM Central Daylight Time

 Depth             Reduced             Elements
   0                     1                    1 
   1                     3                    6 
   2                    11                   18 
   3                    29                   54 
   4                    87                  162 
   5                   253                  486 
   6                   752                1,457 
   7                 2,213                4,334 
   8                 6,379               12,568 
   9                18,205               36,046 
  10                51,785              102,801 
  11               145,489              289,534 
  12               405,728              808,623 
  13             1,118,586            2,231,878 
  14             3,043,537            6,076,994 
  15             8,153,139           16,288,752 
  16            21,464,200           42,897,301 
  17            55,475,870          110,898,278 
  18           140,272,410          280,452,246 
  19           346,202,190          692,243,746 
  20           831,610,844        1,662,949,961 
  21         1,938,788,875        3,877,105,392 
  22         4,370,165,315        8,739,560,829 
  23         9,490,811,983       18,980,345,944 

 Sum        17,207,737,884       34,412,307,411 

518,918,400 of 518,918,400 cosets solved

27 QTM Moves Suffice

Every position of the Rubik's Cube can be solved in at most
27 quarter turns.

This work was supported in part by an allocation of computing time
from the Ohio Supercomputer Center. It was also supported by
computer time from Kent State University's College of Arts and
Sciences. In order to obtain this new result, 25,000 cosets of
the subgroup U,F2,R2,D,B2,L2 were solved to completion, and
34,000,000 cosets were solved to show a bound of 26. No new
positions at a distance of 26 or 25 were found in the solution
of all of these cosets.

Twenty-Eight QTM Moves Suffice

Every position of the Rubik's Cube can be solved in at most
28 quarter turns. The hardest position known in the quarter-turn
metric requires only 26 moves, so this upper bound is probably
not tight.

This new upper bound was found with the generous donation of
computer time from Kent State University's College of Arts and
Sciences. In order to obtain this new result, 7,000 cosets of
the subgroup U,F2,R2,D,B2,L2 were solved to completion. Each
coset took approximately an hour on a 6-core Intel CPU. No new
positions at a distance of 26 or 25 were found in the solution
of all of these cosets.

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.