# Message #683

From: David Smith <djs314djs314@yahoo.com>

Subject: n^d General Formula Found!

Date: Sun, 19 Jul 2009 18:13:43 -0000

I am happy to announce that after starting this project almost four

years ago (when I first calculated the number of configurations of a

standard 3x3x3 Rubik’s Cube), I have finally discovered a formula for

the mathematically justifiable upper bound of the number of

configurations of an n^d Rubik’s Cube!

I would like to thank everyone in this group for their support and

interest in my work. Before writing the papers, I am going to take a

long break and work on other projects for a while. When I return, I

think I will work on all of the analogous formulas for the super and

super-supercube variants, and after that work on the papers.

For reference, here are all of the higher-dimensional cube formulas I

have found for regular Rubik’s Cubes:

n^4:

http://www.gravitation3d.com/david/n%5E4_Cube.pdf

n^5:

http://www.gravitation3d.com/david/n%5E5_Cube.pdf

n^6:

http://www.gravitation3d.com/david/n%5E6_Cube.pdf

3^d:

http://www.gravitation3d.com/david/3%5Ed_Cube.pdf

(I put this together after finding the general formula, it is

equivalent to the formulas for a 3^d cube in "The Rubik Tesseract"

and "An n-dimensional Rubik Cube".)

and n^d:

http://www.gravitation3d.com/david/n%5Ed_Cube.pdf

I believe that these formulas are final, but there is always the

possibility of errors remaining in them. I have corrected such

errors in the past after finding them, they were almost always simple

oversights that were easily corrected, and probably resulted from

working too fast and not double-checking my results enough. If any

mistakes do remain, they will definitely be found when writing my

papers. However, I do believe that the formulas are correct, as I

have worked through each one term by term, and checked them

multiple times.

A special thanks goes out to Roice as always, for the multiple ways

he supports me and my work. I am also grateful to Melinda, Don, and

Jay Berkenbilt for creating the program and concept which inspired

this group and all of my work.

Thanks everyone, and I’ll let you know when I get back to making any

further progress in the research of higher-dimensional permutation

puzzles.

All the best,

David