Message #583

From: David Smith <>
Subject: Re: [MC4D] Re: Permutation formula updates
Date: Mon, 22 Sep 2008 17:50:40 -0700

Hi Thibaut,
Thanks for your comments and suggestions!  I put a lot of thought into what
you recommended to me about the formulas, but I have decided to keep them the
same.  While I value your opinion, and almost did decide to modify the formulas,
I think that it is more concise and elegant to represent the answers with only one
formula.  While simplicity can be elegant, to me the formula is already so
(although this is of course my biased opinion as its discoverer).  I actually
never seperated the formulas into even/odd cases, and while many may not,
I like the use of the "n mod 2" terms and how I applied them.  If anyone on the
group is interested in a basic explanation as to the derivation of these formulas,
I would be glad to email them one.  I’m looking forward to using more advanced
reasoning for proving these formulas exact, and for trying my hand at the n^5
and n^d cases.  I’ll let the group know when I get the super-supercube formula.
All the Best,

— On Mon, 9/22/08, thibaut.kirchner <> wrote:

From: thibaut.kirchner <>
Subject: [MC4D] Re: Permutation formula updates
Date: Monday, September 22, 2008, 10:53 AM

— In 4D_Cubing@yahoogrou, David Smith <djs314djs314@ …> wrote:
> 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.

I’m amazed by the complexity of the formula. I suggest you to split it
into two formulas, one for the odd-sized hypercubes, and on for the
even-sized hypercubes. I’m sure the two formulas would be easier to
read than this one, and I’m not sure it’s interesting to group the two
formulas into a single one.
Also, if you put the factors associated with a single type of piece by
row, it could help the reader (and group somewhere the constraints
which link several type of piece).