# Message #644

From: David Vanderschel <DvdS@Austin.RR.com>

Subject: Re: [MC4D] a short diversion into sticker and cubie counts

Date: Wed, 04 Feb 2009 18:45:18 -0600

On Tuesday, February 03, "David Vanderschel" <DvdS@Austin.RR.com> wrote:

>On Monday, February 02, "Roice Nelson" <roice3@gmail.com> wrote:

>>I found n = f neat because a priori, why should the

>>number per side have any relationship whatsoever to

>>the the number of faces? (maybe this surprise is just

>>the fact that the number of faces = 2d in disguise.)

>Roice, I understand your surprise. …

>I suspect there may be deeper insights which are

>eluding us and which would make these relationships

>appear more plausible.

I take it back. This one is easy to see. I think I

can explain it in a way that is more obvious than what

David Smith wrote:

Given an order-m n-puzzle, we note that there is a 1-1

correspondence between the set of m^(n-1) stickers

which lie in a face and the m^(n-1) n-cubies which

make up the corresponding external slice. Consider

cloning that set of cubies and set them aside. Do

this for each of the 2n faces and pile up the cloned

slices. The height of the cloned-cubie pile is 2n and

the number of cubies in it is the same as the number

of stickers on the puzzle. The slices of the puzzle

are the same size ( m^(n-1) n-cubies ) as those in the

clone pile and its height is m cubies. So now

comparing the number of stickers to the number of

cubies is just a matter of comparing the heights (2n

and m) of the two piles.

It really does come down to the fact that the number

of cloned slices is the number of faces.

Regards,

David V.