# Message #3284

From: joelkarlsson97@gmail.com

Subject: Re: Permutation formula updates

Date: Sun, 17 Jan 2016 06:43:46 -0800

Hi David,

I am writing a paper regarding n^d cubes and have a few questions to you.

You earlier wrote:

"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)."

Have you, or anyone else, succeeded with this? Didn’t Joe Buhler, Brad Jackson and Dave Sibley accomplish something similar in their paper "An n-dimensional Rubik’s Cube"?

"and while many may not, I like the use of the "n mod 2" terms and how I applied them".

I totally agree, that was very clever and I personally think that it’s more elegant with one formula even though two separate might be simpler and easier to understand.

"Roice suggested to me that if I ever get the formulas for n^d cubes, I might want to write a program that takes as input a dimension and side length and displays

the number of permutations. I would definitely do that, probably using C with

an arbitrary-precision library."

That would be truly amazing, especially if you prove them to be exact and not only a upper bound. Have you done this or have anyone else done it by now?

I’m writing a paper regarding n^d cubes and will appreciate all answers that I get.

Best regards,

Joel