# Message #581

From: David Smith <djs314djs314@yahoo.com>

Subject: Permutation formula updates

Date: Mon, 22 Sep 2008 02:20:58 -0700

Hi everyone,

First of all, nice to meet you, Mark and Remi, and congratulations to both of you

on your checkerboard solutions! I thought I would update the group with my progress,

since it now seems that many of you are also interested in the mathematics of the

cube.

I’ve updated my formula for the upper bound for the number of reachable positions of an

n^4 Rubik’s cube (it contained some errors), and also finished a similar formula for the

supercube. Until now, I’ve only been using combinatorial arguments and concepts

of higher dimensions in my work. However, I’m currently learning group theory, so I

can hopefully come up with general formulae for the n^d cube, supercube, and

super-supercube. Also, although I mentioned before that I am not particularly

interested in proving these upper bounds to be exact, I believe that with some effort,

I can do so using mathematical arguments without actually specifying any

particular solution algorithm (which would be necessary for the general n^d cases).

I am reading an excellent paper on the 3^n cube ("An n-dimensional Rubik Cube",

by Joe Buhler, Brad Jackson, and Dave Sibley), which without I would be completely

lost.

Once again, nice to meet you Mark and Remi! I hope that my work is of some

interest to the group, and that it was not inappropriate to bring it up here. It is

nice to see how mathematics can be applied to solving the cube and related

problems (such as the optimal checkerboard solutions). I’ll have to try it

someday!

All the Best,

David