I am happy to announce that after starting this project almost four=20
years ago (when I first calculated the number of configurations of a=20
standard 3x3x3 Rubik's Cube), I have finally discovered a formula for=20
the mathematically justifiable upper bound of the number of=20
configurations of an n^d Rubik's Cube!
I would like to thank everyone in this group for their support and=20
interest in my work. Before writing the papers, I am going to take a=20
long break and work on other projects for a while. When I return, I=20
think I will work on all of the analogous formulas for the super and=20
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"=20
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=20
possibility of errors remaining in them. I have corrected such=20
errors in the past after finding them, they were almost always simple=20
oversights that were easily corrected, and probably resulted from=20
working too fast and not double-checking my results enough. If any=20
mistakes do remain, they will definitely be found when writing my=20
papers. However, I do believe that the formulas are correct, as I=20
have worked through each one term by term, and checked them=20
multiple times.
A special thanks goes out to Roice as always, for the multiple ways=20
he supports me and my work. I am also grateful to Melinda, Don, and=20
Jay Berkenbilt for creating the program and concept which inspired=20
this group and all of my work.
Thanks everyone, and I'll let you know when I get back to making any=20
further progress in the research of higher-dimensional permutation=20
puzzles.
All the best,
David