Message #594

From: David Smith <djs314djs314@yahoo.com>
Subject: Super-supercube formula
Date: Sat, 27 Sep 2008 19:39:16 -0700

Hello All,
 
I have just finished obtaining the formula for the upper bound of the
number of permutations of an n^4 super-supercube.  It took longer
than I expected, but still not a very long time.
 
It is very clear to me now that I will not be able to prove equality
for the upper bounds of my formulas.  There are simply far too
many variables to consider, and it is clearly beyond my ability.
The 3^3 non-constructive proof alone is quite complicated,
requiring the use of several results of group theory.  Nevertheless,
I will still try to find the n^d formulas, beginning with the n^5 cases.
First however, I will concentrate on the 4D variants that David V. and
others have brought to my attention.
 
It occurred to me after I wrote my last post that there is a small
addition to be made in what I said to David.
 
I wrote:
 
> If you launch MC2D, it will be immediately clear that
> there is only one move, and that is to swap two
> adjacent corners. Thus, (1,2)(3,4) is actually two
> moves, one reflection of the North face and one of
> the South face. If you were to actually reflect
> the entire puzzle, the centers would also reflect,
> and thus would just be a reorientation of the puzzle,
> and not a move at all.
 
There is another move that can be considered - a slice move
which swaps two opposite centers.  This move is entirely
possible, and is equivalent to swapping the corners on two
opposite sides and reorienting the cube by reflecting it.
Thus, (1,2)(3,4) can be accomplished in one move if
one allows this slice reflection.  If we allow this, all
permutations of MC2D can be produced by either an even
or odd number of moves.  However, (1,3) is still not
possible with a single move.  If it were possible in the
manner David intended, then it would also change the
orientation of corners 2 and 4, which is not possible in
MC2D.
 
David, just as you kindly informed me that you do
not mind being corrected if you have made a mistake,
I would like to make the same declaration here.  If
you or anyone on this forum finds an error that I have
made in my statements, feel free to correct me; I would
appreciate it very much.  As you said, we can learn
much from our mistakes! :)
 
All the Best,
David