# Message #645

From: rev_16_4 <rev_16_4@yahoo.com>

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

Date: Thu, 05 Feb 2009 04:09:51 -0000

— In 4D_Cubing@yahoogroups.com, David Vanderschel <DvdS@…> wrote:

>

> On Tuesday, February 03, "rev_16_4" <rev_16_4@…> wrote:

> >Order-m n-puzzle, I’ve been trying to think of that

> >terminology for a while. Thank you! That’ll simplify

> >most of my posts in the future! I think I prefer

> >order-m n-cube, but n-puzzle is a little more

> >descriptive. However, I think with the existence of

> >the magic 120-cell, which is a 4-puzzle, we shouldn’t

> >assume the puzzle is an n-cube.

>

> I can debate that. I don’t think that there is all

> that much terminology from higher dimensional Rubik

> analogues that applies directly to other twisty

> puzzles. One can push generalization so far that you

> wind up having to qualify the special cases too much.

>

> I do not like to call an instance of a puzzle a "cube"

> because it really isn’t _one_. It is a pile of little

> cubes with some constraints on what arrangements are

> possible. Admittedly, the shape of the pile is

> cubical; but it is its transformability that is of

> real interest.

>

> Regards,

> David V.

>

David V.-

I would actually make the opposite argument. Attempting to make a

general formula for all possible permutations of every type puzzle in

every dimension would be nearly impossible (however David S. has made

great strides with the cubical variety). There are some things that

do generalize considerably better. In fact I think the

generalizations are what some of us find so interesting about these n-

puzzles.

For example, I have a strong feeling that the 120-cell is solvable

using the same caging method I use to solve order-m n-cubes, with

only slight modifications to the algorithms. This 4-puzzle has even

parity, which increases my confidence further.

I agree that the "cube" isn’t a single cube, but in fact a composite

of many smaller pieces put together to make a cube. That’s why I

think saying "order-m" is so powerful and perfect. We have m pieces

extending out along each of the edges to make up the whole cubical

puzzle. That the individual cubies just happen to be cubes is a

coincidence. This can be seen in the magic dodecahedron/120-cell.

Some are rhombic, some are triangular, and some are pentagonal. I see

the opposing argument like "It’s not a beach, it’s trillions of

grains of sand."

I think we are generalizing by saying n-puzzle, when it would be just

as easy to say n-cube, which is the ultimate shape of these puzzles

(MC4D & MC5D). The frequency with which the generalized terminology

is required makes this point trivial. You can safely pull the meaning

of n-puzzle out of the context from which it’s taken.

Finally, while during a discussion of each class of n-puzzle, we may

not generalize between classes too frequently, if ever. However this

discussion itself shows the need for a terminology for the general n-

puzzles.

-Levi