## Lower bounds for the 3x3x3 Super Group

Submitted by Walter Randelshofer on Tue, 07/22/2014 - 13:45.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.

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.

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

Submitted by rokicki on Sat, 07/19/2014 - 14:32.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

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

Submitted by B MacKenzie on Sat, 07/05/2014 - 07:38.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

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## 27 QTM Moves Suffice

Submitted by rokicki on Mon, 06/23/2014 - 21:13.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.

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

Submitted by rokicki on Fri, 06/06/2014 - 09:48.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.

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

Submitted by B MacKenzie on Sat, 05/17/2014 - 15:19.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 Antipodes: 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

Submitted by cubex on Mon, 05/12/2014 - 06:16.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.

Mark

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.

Mark

## Domain name changed (again)

Submitted by cubex on Wed, 05/07/2014 - 12:20.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 http://204.225.123.154

Mark

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 http://204.225.123.154

Mark

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

Submitted by rokicki on Sat, 04/12/2014 - 19:54.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.

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

Submitted by loseyourmarblesblog on Thu, 10/17/2013 - 20:03.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.

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.

» 53 comments | read more