## Lower bound for the diameter of the 3-gen subgroup <U,F,R>

Submitted by Ben Whitmore on Sun, 09/02/2018 - 22:00.What are the known bounds for the diameter of this subgroup? This thread gives a lower bound of 26 QTM but doesn't discuss HTM. I seem to recall a lower bound of 20 HTM from somewhere, but no upper bound. However, I recently found a position that requires 21 HTM:

U R F2 R U2 F2 U' F' R U F U R' F R2 U F R U2 R F'Is this the first one known? Out of all 186624 positions with all pieces correctly permuted, this position and the inverse are the only ones requiring 21. Here's the full distribution:

Depth Positions

» 13 comments | read more

## Mad Octahedron

Submitted by B MacKenzie on Thu, 06/21/2018 - 09:47.I have written a computer simulation of the octahedral twisty puzzle. It is available as freeware on the Apple App Store:

Mad Octahedron## Future URL recommendation for the forum

Submitted by cubex on Tue, 05/08/2018 - 14:21.Hello everybody :)

In the future I think it would be a good idea if we all used URLs of the form:

http://forum.cubeman.org/?q=node/view/563#comment

rather than

http://cubezzz.dyndns.org/?q=node/view/563#comment

dyndns.org has raised their prices every year and I'm considering

dropping their service. Unfortunately if we do that a lot of old URLs will stop

working, so I'm open to any clever ideas on what is the best way to deal with

this problem. I'm committed to keeping the maxhost.org and

cubeman.org domain names working for the long term, but

I'm very unhappy with dyndns.

In the future I think it would be a good idea if we all used URLs of the form:

http://forum.cubeman.org/?q=node/view/563#comment

rather than

http://cubezzz.dyndns.org/?q=node/view/563#comment

dyndns.org has raised their prices every year and I'm considering

dropping their service. Unfortunately if we do that a lot of old URLs will stop

working, so I'm open to any clever ideas on what is the best way to deal with

this problem. I'm committed to keeping the maxhost.org and

cubeman.org domain names working for the long term, but

I'm very unhappy with dyndns.

## Three Million Positions in Four Metrics

Submitted by rokicki on Mon, 04/23/2018 - 10:52.I optimally solved three million positions in four distinct metrics.

These positions are distinct from the three million positions I ran

some years back. Random numbers were generated with the Mersenne

Twister algorithm. The four metrics I ran were quarter-turn metric,

half-turn metric, slice-turn metric, and axial-turn metric (equivalent

to the robot-turn or simultaneous-turn metric on the 3x3 cube).

The generators for each metric are strict super- or sub-sets of the

generators for the other metrics. The quarter-turn metric has 12

generators, the half-turn metric has 18 generators, the slice-turn

These positions are distinct from the three million positions I ran

some years back. Random numbers were generated with the Mersenne

Twister algorithm. The four metrics I ran were quarter-turn metric,

half-turn metric, slice-turn metric, and axial-turn metric (equivalent

to the robot-turn or simultaneous-turn metric on the 3x3 cube).

The generators for each metric are strict super- or sub-sets of the

generators for the other metrics. The quarter-turn metric has 12

generators, the half-turn metric has 18 generators, the slice-turn

» 7 comments | read more

## Presentation for the Mathieu Group M24 from dedge superflip

Submitted by Paul Timmons on Fri, 04/20/2018 - 20:58.Following on from a highly symmetric presentation I supplied for the miniature Rubiks cube group
Presentation for the 2x2x2 Rubiks cube group
I investigated whether the Mathieu Group M24 could be similarily presented taking full advantage of the
plentiful symmetry inherent in the Rubiks Revenge cube (4x4x4). The answer was indeed yes - here is the presentation I found - again on three involutions:

**< a,b,c | a ^{2} = b^{2} = c^{2} = 1,
**

** (ab) ^{6} = [(bc)^{6}] = [(ca)^{6}] = 1,
**

** bacabacacabacababacabac = 1,
**

** (ababacbc) ^{3} = 1,
**

** bababcbcbcbabab = cacabacacabacac** >

## Gear cube extreme can be solved in 25 moves

Submitted by Ben Whitmore on Thu, 02/15/2018 - 23:07.Write the puzzle as the group <R,F,U,D> where R and F are 180 degree moves. We use a two-phase algorithm to first reduce the state of the puzzle to the subgroup <R3,F3,U,D>, and then finish the solve in the second phase. The subgroup <R3,F3,U,D> is the group of all positions where all of the gears are oriented, because R3 is the same as R' except the gear orientation remains unchanged.

The first phase is easy to compute. There are only 3^8 = 6561 positions because each gear has only 3 different orientations, despite having 6 teeth.

Phase 1 distribution:

The first phase is easy to compute. There are only 3^8 = 6561 positions because each gear has only 3 different orientations, despite having 6 teeth.

Phase 1 distribution:

Depth New Total 0 1 1 1 4 5 2 8 13 3 78 91 4 102 193 5 1064 1257 6 920 2177 7 3576 5753 8 592 6345 9 216 6561 10 0 6561The second phase is harder. The number of positions is 24*8!^2/2 = 19,508,428,800, since it turns out that the permutation of the 3 unfixed edges on the E slice is completely determined by the permutation of the centres. This phase was solved with a BFS and took around 7 and a half hours to complete.

## Kilominx can be solved in 34 moves

Submitted by Ben Whitmore on Sun, 02/11/2018 - 12:58.Last night, I found this thread on the speedsolving forums which proves an upper bound of 46 moves. First, the puzzle is separated into two halves, which takes at most 6 moves. Each half is then solved in at most 20 moves (= 7 moves for orientation + 13 moves for permutation, after orientation is solved), for a total of 6+2*(7+13) = 46. xyzzy writes

The ⟨U,R,F⟩ subgroup, while much smaller than G_0, is still pretty large, having 36 billion states. It's small enough that a full breadth-first search can be done if symmetry+antisymmetry reduction is used, but I will leave this for another time.

» 3 comments | read more

## 5x5 sliding puzzle can be solved in 205 moves

Submitted by Ben Whitmore on Fri, 01/26/2018 - 17:46.Consider a 5x5 sliding puzzle with the solved state

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24We can solve the puzzle in three steps. First solve 1,2,3,4,5,6,7, then solve 8,9,10,11,12,16,17,21,22, and finally solve the 8 puzzle in the bottom right corner. Step 1 requires 91 moves:

depth new total 0 18 18 1 6 24 2 13 37 3 27 64 4 54 118 5 117 235 6 231 466 7 443 909

## God's algorithm for the <2R, U> subset of the 4x4 cube

Submitted by Ben Whitmore on Wed, 01/24/2018 - 22:00.Here I'm using sign notation, so 2R is the inner slice only. There are 10 edges, 10 centres in sets of 2, 2, 2 and 4, and 4 permutations of the corner pieces for a total of 4*10!*10!/(2!2!2!4!) = 274,337,280,000 positions. From July 4th 2017 to July 6th 2017, I ran a complete breadth first search of this puzzle in around 60 hours. God's number is 28.

Depth New Total 0 1 1 1 6 7 2 18 25 3 54 79 4 162 241 5 486 727 6 1457 2184 7 4360 6544

## Do we have a 3x3x3 optimal solver for stm metric?

Submitted by cubex on Thu, 08/10/2017 - 06:46.I thought it might be interesting to run an optimal solver using the slice turn metric (including face turns) on some pretty patterns. I don't remember anyone releasing an optimal solver that uses stm but maybe there is one by now?

Also is it true we don't know if using slice turns plus face turns could reduce God's Number from 20 to less than 20?

Also is it true we don't know if using slice turns plus face turns could reduce God's Number from 20 to less than 20?