#### Herbert Kociemba

##### Member

- Joined
- Nov 29, 2008

- Messages
- 189

I am currently working on a general NxNxN cube solver for large N. For large N the main work is to fix the 6*(N-2)^2 centerpieces so I will concentrate on fixing the centers. I also ignore even N in the moment and hence I assume that N is odd for the time being.

A corner piece has three visible facelets, an edge piece has two visible facelets and a centerpiece has only one visible facelet. When talking about the centers I will use the terms "facelet" and "centerpiece" interchangeable.

Using an appropriate coordinate system for each of the 6 faces with coordinates ranging from (0,0) to (N-1,N-1) a facelet at position (x,y) will go to a position (y,N-1-x), (N-1-x,N-1-y) or (N-1-y,x) on the same face when applying a face move and on a different face when applying a slice move. This will lead to 24 different positions of a facelet, except in the case x=y=(N-1)/2 where we only have 6 different positions.

For center facelets the 24 different positions correspond to 24 different centerpieces which constitute the center cluster C(x,y). If we assume without loss of generality 1<=x,y<=(N-1)/2 each center cluster is uniquely determined by x and y.

In the first step we fix all center clusters C(x,y) such that the U and D centers are in the U and D face.

First we solve the "+cross" clusters C(x,(N-1)/2), 1<=x< (N-1)/2. Since there are only Binomial[24,8] different positions, this is done in almost no time.

For x<>y we solve C(x,y) and C(y,x) simultaneously. This is the hardest part because there are Binomial(24,8)^2 different states and there is the additional restriction that the other clusters may not be affected. Usually it takes something from a few seconds to several minutes to solve a single pair C(x,y) and C(y,x).

At the end we solve the "x-cross" clusters C(x,x), 1<=x< (N-1)/2. Again there are only Binomial[24,8] states and the solution can be computed almost instantaneously,

The two pictures show an example for N=51. It took 4211 single face or slice moves within about 4 hours to fix the 4800 U and D facelets in 600 clusters.

In the second step we put the R and L centers into the R and L faces using only 180 degree turns for the R, L, F and B face and slice moves so that the U and D faces will not be destroyed again. More to come...

A corner piece has three visible facelets, an edge piece has two visible facelets and a centerpiece has only one visible facelet. When talking about the centers I will use the terms "facelet" and "centerpiece" interchangeable.

Using an appropriate coordinate system for each of the 6 faces with coordinates ranging from (0,0) to (N-1,N-1) a facelet at position (x,y) will go to a position (y,N-1-x), (N-1-x,N-1-y) or (N-1-y,x) on the same face when applying a face move and on a different face when applying a slice move. This will lead to 24 different positions of a facelet, except in the case x=y=(N-1)/2 where we only have 6 different positions.

For center facelets the 24 different positions correspond to 24 different centerpieces which constitute the center cluster C(x,y). If we assume without loss of generality 1<=x,y<=(N-1)/2 each center cluster is uniquely determined by x and y.

In the first step we fix all center clusters C(x,y) such that the U and D centers are in the U and D face.

First we solve the "+cross" clusters C(x,(N-1)/2), 1<=x< (N-1)/2. Since there are only Binomial[24,8] different positions, this is done in almost no time.

For x<>y we solve C(x,y) and C(y,x) simultaneously. This is the hardest part because there are Binomial(24,8)^2 different states and there is the additional restriction that the other clusters may not be affected. Usually it takes something from a few seconds to several minutes to solve a single pair C(x,y) and C(y,x).

At the end we solve the "x-cross" clusters C(x,x), 1<=x< (N-1)/2. Again there are only Binomial[24,8] states and the solution can be computed almost instantaneously,

The two pictures show an example for N=51. It took 4211 single face or slice moves within about 4 hours to fix the 4800 U and D facelets in 600 clusters.

In the second step we put the R and L centers into the R and L faces using only 180 degree turns for the R, L, F and B face and slice moves so that the U and D faces will not be destroyed again. More to come...

Last edited: Sep 19, 2016