Message #584

From: Remigiusz Durka <>
Subject: Re: [MC4D] Re: Permutation formula updates
Date: Tue, 23 Sep 2008 10:49:33 +0200

Can I ask about formula as the input for program Mathematica (or Maxima, etc)… Or just see actual numbers for any hypercube we have?

all teh best,

—– Original Message —–
From: David Smith
Sent: Tuesday, September 23, 2008 2:50 AM
Subject: Re: [MC4D] Re: Permutation formula updates

Hi Thibaut,

    Thanks for your comments and suggestions!  I put a lot of thought into what<br>
    you recommended to me about the formulas, but I have decided to keep them the<br>
    same.  While I value your opinion, and almost did decide to modify the formulas,<br>
    I think that it is more concise and elegant to represent the answers with only one<br>
    formula.  While simplicity can be elegant, to me the formula is already so<br>
    (although this is of course my biased opinion as its discoverer).  I actually<br>
    never seperated the formulas into even/odd cases, and while many may not,<br>
    I like the use of the &quot;n mod 2&quot; terms and how I applied them.  If anyone on the<br>
    group is interested in a basic explanation as to the derivation of these formulas,<br>
    I would be glad to email them one.  I'm looking forward to using more advanced<br>
    reasoning for proving these formulas exact, and for trying my hand at the n&#94;5<br>
    and n&#94;d cases.  I'll let the group know when I get the super-supercube formula.

    All the Best,<br>

    --- On Mon, 9/22/08, thibaut.kirchner &lt;; wrote&#58;

      From&#58; thibaut.kirchner &lt;;<br>
      Subject&#58; &#91;MC4D&#93; Re&#58; Permutation formula updates<br>
      To&#58; 4D&#95;<br>
      Date&#58; Monday, September 22, 2008, 10&#58;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<br>
      into two formulas, one for the odd-sized hypercubes, and on for the<br>
      even-sized hypercubes. I'm sure the two formulas would be easier to<br>
      read than this one, and I'm not sure it's interesting to group the two<br>
      formulas into a single one.<br>
      Also, if you put the factors associated with a single type of piece by<br>
      row, it could help the reader (and group somewhere the constraints<br>
      which link several type of piece).