# Modeling subsets of the cube that involve ignoring certain cubies

Submitted by Jerry Bryan on Sun, 04/24/2005 - 21:41.

I'm in the process of developing a C++ class library for modeling various Rubik's cube problems, including some old problems that have already been run on a computer and some new problems that haven't been run before. One of the capabilities I want to include in my class library is the ability to ignore certain cubies. In a certain sense, we already do so when we solve "corners only" or "edges only" problems or some such. But I want a more general facility where the cubies to be ignored could be some of the corner cubies, some of the edge cubies, or both.

I'm having a little difficulty with some of the group theory underpinnings. For example, consider the corners only group and suppose I want to consider the positions of only six of the eight corner cubies. I would like for what I'm modeling to be a group because if it is, a lot of useful group theory concepts come into play such as conjugates, symmetry, Sims tables, and the like. Essentially, the way to model six of the eight cubies is to consider two positions equivalent if they are the same except for the possible transposition and/or rotation of the two particular corner cubies to be ignored. The set of transpositions and rotations of two particular corner cubies can be thought of as a subgroup the corners group, call it H. I'm thinking that what I need to consider is the factor group G/H, where G is the corners group. Trouble is, G/H is only a group if H is normal in G. And I'm not convinced that all possible subgroups H derived in the manner described (ignoring one or more corner cubies) are normal in G.

I may not be looking at the problem quite right. Any comments or suggestions gratefully accepted.

I'm having a little difficulty with some of the group theory underpinnings. For example, consider the corners only group and suppose I want to consider the positions of only six of the eight corner cubies. I would like for what I'm modeling to be a group because if it is, a lot of useful group theory concepts come into play such as conjugates, symmetry, Sims tables, and the like. Essentially, the way to model six of the eight cubies is to consider two positions equivalent if they are the same except for the possible transposition and/or rotation of the two particular corner cubies to be ignored. The set of transpositions and rotations of two particular corner cubies can be thought of as a subgroup the corners group, call it H. I'm thinking that what I need to consider is the factor group G/H, where G is the corners group. Trouble is, G/H is only a group if H is normal in G. And I'm not convinced that all possible subgroups H derived in the manner described (ignoring one or more corner cubies) are normal in G.

I may not be looking at the problem quite right. Any comments or suggestions gratefully accepted.