## Three Million Positions in Four Metrics

Submitted by rokicki on Mon, 04/23/2018 - 10:52.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

## 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,
**

** (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.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