## 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

## 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.