## Super Group Cosets of the Centers Subgroup

Submitted by B MacKenzie on Mon, 06/20/2016 - 06:52.Continuing my work with the 3x3x3 super group, I have written a coset solver for cosets of the pure center cubie subgroup. This subgroup is made up of the 2048 even parity center cubie configurations composed with the identity edge and corner configurations. The super group may be partitioned into cosets of the pure centers subgroup, g * [CTR] , where g is an element of the super group and [CTR] is the centers subgroup. The centers subgroup is a normal subgroup of the super group, g * [CTR] = [CTR] * g, and the standard cube group is the quotient group of the super group and the centers subgroup.

## Finally hitting depth 13 consistently with my 5x5x5 solver

Submitted by NoLongerUnsolve... on Fri, 06/17/2016 - 18:52.It finally occurred to me why my hash table was sometimes not finding the shortest solutions 100% of the time. When I upgraded my computer to one with 128 GB of RAM, I had enough to load more positions into RAM. The number of hash table entries exceeded 4.2 billion, which is more than 32 bits. I never adjusted all of my access code to use 64-bit indices which were now necessary. All I had to do was change the data type, and, lo and behold, it found a 13-move solution to this arrangement, which previously it was reporting required 14 moves!

A very happy day for me.

## Super Cube States at Depth

Submitted by B MacKenzie on Wed, 06/15/2016 - 10:36.Super Cube States at Depth

I've been working with the super cube group (the 3x3x3 cube with center cubie orientation). There are two earlier threads here dealing with this group, Lower bounds for the 3x3x3 Super Group and Supergroup knowledge. Neither of these contain any states at depth information. To test my model I have performed a breadth first states at depth enumeration of the group out to depth 10 in the qtm. Can anybody confirm these numbers for me?

## Revisiting Korf's 3x3x3 Counts And Identifying Duplicate Positions

Submitted by NoLongerUnsolve... on Fri, 06/10/2016 - 21:18.Recently I decided to implement the Korf move generator. In my mind, it is really more akin to a 2x2x2 move generator, since every move sequence that is generated can also be played out on a 2x2x2 cube. (Contrast that to some 5x5x5 moves which clearly have no counterpart of smaller cubes.) My move generator started with the solved cube, counted nodes as a function of depth, and placed each unique cube in a hash table, flagging all of the duplicate positions that came next.

## 5x5x5 Solving Programs: The list is growing

Submitted by NoLongerUnsolve... on Sun, 06/05/2016 - 22:06.## New 3x3x3 Corners Data Including Centers

Submitted by NoLongerUnsolve... on Sat, 06/04/2016 - 08:55.I decided to create a database that:

A. requires no single point of reference

B. contains all 24 possible rotated states of the cube's corner arrangements

C. measures the distance the corners are from the solved state with respect to the fixed centers

## IP address for forum changed to 69.165.220.244

Submitted by cubex on Fri, 05/06/2016 - 23:57.My upstream has changed my ip address so now the numeric ip is 69.165.220.244. All the links should work as before once all the dns changes have propagated.

## Cube archives changed

Submitted by cubex on Sat, 03/19/2016 - 07:40.## Confirmation of Results for Edges Only Cube, Face Turn Metric, Reduced by Symmetry

Submitted by Jerry Bryan on Tue, 02/23/2016 - 12:42.Even though it's only confirmation of some rather ancient results, I would like to report that I have succeeded in replicating Tom Rokicki's 2004 results concerning the edges only cube in the face turn metric reduced by symmetry. My goal was not really to solve that particular problem. Rather, it was to use that problem as a way to prototype some ideas I have had for improving my previous programs that enumerate cube space.

The base speed of my program is that I am now able to enumerate about 1,000,000 positions per second per processor core when not reducing the problem by symmetry, and I am now able to enumerate about 150,000 symmetry representatives per second per processor core when reducing the problem by symmetry.

## Refining Bruce Norskog's 4x4x4 five stage analysis

Submitted by Clement Gallet on Fri, 12/11/2015 - 08:03.Except for the last stage where centers need to be solved, every other stage needs to place 8 centers from a single axis (RL, FB or UD centers) in their correct faces (RL, FB or UD faces), in one of the 12 positions that can then be solved using only half turns. However, as pointed by Shuang Chen in his analysis, we don't need to store the exact colours of centers but only if two centers have the same or different colours. This reduces the number of center coordinates by 2.

Also, as opposed to the 3x3x3, 4x4x4 does not have fixed centers, which allows us to do more symmetry reduction. I'm representing a sym-coordinate as:

s1 * g * < H > * s1' * s2'

where g * < H > is a coset, s1 and s2 are symmetries from subgroups S1 and S2 of the group M of symmetries of the cube. s1 is the usual conjugated symmetry, and s2 correspond to a rotation of the cube, which is possible on the 4x4x4. Using carefully chosen subgroups S1 and S2 for each stage, more symmetry reduction is achieved. I will be using the Schoenflies symbols for subgroups of M in the following.