# 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