# Message #641

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

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

Date: Tue, 03 Feb 2009 14:18:31 -0600

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

>Why do the number of 0-colored pieces grow so much

>faster than the others, even taken together?

>Consider a puzzle with very large n. In the limit,

>0-coloreds are the only piece type that are filling

>up the full dimension of the cube. 1-coloreds fill up

>the faces of dimension d-1, 2-coloreds fill up the

>d-2 spaces, etc. And higher dimensional spaces are

>more voluminous, so it makes sense 0C will win out in

>the end.

As I see it, the phenomenon here is just a quantized

scale effect. The cubies with any stickers at all

(without regard to how many stickers) are all on the

‘surface’. The size of that surface (in cubie count)

is proportional (asymptotically) to n^(d-1).

Meanwhile, the others fill a volume the size of which

(again in cubie count) is proportional

(asymptotically) to n^d.

>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. It is similar to

my surprise about reorientations of a 3-cube: Every

one of the 23 possible reorientations is achievable by

rotation about one of the three types of rotation axis

(through the origin and through a corner, the middle

of an edge, or the middle of a face) and that axis is

uniquely determined for any reorienting

transformation.

I suspect there may be deeper insights which are

eluding us and which would make these relationships

appear more plausible.

>I also wonder why the existence of puzzles where the

>number of stickers and cubies coincide should even be

>guaranteed, another fact not a priori obvious to me.

Indeed.

>P.S. I want to defer to the group on the use of m^n

>verses n^d. In this email, I wanted to say

>"n-dimensional" at one point, but that would have

>conflicted with my usual labels. It got me

>second-guessing myself. Any opinions? Maybe we

>should make a poll?

I prefer m^n. The "nth dimension" is too well

ingrained in me. I, too, often wish to say

"n-dimensional". I also would tend to say something

like, "an order-5 4-puzzle", to imply 5^4.

Regards,

David V.