## Presentation for the Mathieu Group M24 from dedge superflip

Submitted by Paul Timmons on Fri, 04/20/2018 - 20:58.*< a,b,c | a ^{2} = b^{2} = c^{2} = 1,
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* (ab) ^{6} = [(bc)^{6}] = [(ca)^{6}] = 1,
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* bacabacacabacababacabac = 1,
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* (ababacbc) ^{3} = 1
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* bababcbcbcbabab = cacabacacabacac** >
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*
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:
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
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Consider a 5x5 sliding puzzle with the solved state
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.
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?
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Back in Jan 1995 Mike Reid found this process for the 12 flip:

R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3 L1 D2 F3 R1 B3 D1 F3 U3 B3 U1 D3 24q

This process has 24 q turns, so I'm wondering could there be a 24 q turn process that evenly distributes the turns so that each side turns 4 q? The idea just seemed elegant to me, 6 faces each turning 4 q turns.

Mark
## Gear cube extreme can be solved in 25 moves

Submitted by Ben Whitmore on Thu, 02/15/2018 - 23:07.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.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.

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

Submitted by Ben Whitmore on Fri, 01/26/2018 - 17:46.1 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.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.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?

## More details about my new program

Submitted by Jerry Bryan on Thu, 06/08/2017 - 14:55.
**Introduction**

On 02/23/2016, I posted a message about a new program I had developed that had succeeded in enumerating the complete search space for the edges only group. It was not a new result because Tom Rokicki had solved the same problem back in 2004, but it was important to me because the problem served as a testbed for some new ideas I was developing to attack the problem of the full cube. I am now in the process of adapting the new program to include both edges and corners. In this message, I will include some additional detail about my new program that was not included in the first message.

## Pattern databases for the 5x5 sliding puzzle

Submitted by stannic on Sun, 04/02/2017 - 11:37.In 2002, Korf and Felner [1] used pattern databases to solve optimally 50 random instances of the 5x5 sliding puzzle. They used a static

## A cubic graph with cubic diameter

Submitted by stannic on Mon, 03/06/2017 - 03:53.The Fifteen puzzle is sometimes generalized to a sliding puzzle on an arbitrary simple connected graph *G* with *n* vertices in the following way. *n* − 1*n* − 1)*G*. At most one piece is placed on each vertex. One vertex of *G* is left unoccupied. A move consists in choosing a vertex *v* adjacent to the currently unoccupied vertex *v*_{0} and 'sliding' the piece at *v* along the edge (*v*; *v*_{0}). The aim is to restore the order so that piece numbered *i* occupies vertex numbered *i*, for *i* = 1 .. *n* − 1*G* is the

## Is there a way to evenly distribute face turns for 12 flip?

Submitted by cubex on Mon, 11/14/2016 - 22:10.R3 U2 B1 L3 F1 U3 B1 D1 F1 U1 D3 L1 D2 F3 R1 B3 D1 F3 U3 B3 U1 D3 24q

This process has 24 q turns, so I'm wondering could there be a 24 q turn process that evenly distributes the turns so that each side turns 4 q? The idea just seemed elegant to me, 6 faces each turning 4 q turns.

Mark