Message #264

From: Jay Berkenbilt <ejb@ql.org>
Subject: Re: [MC4D] 2^d and 3^d solve function
Date: Fri, 02 Jun 2006 19:35:03 -0400

Don Hatch <hatch@plunk.org> wrote:

> The crux of the matter is that
> in >=5 dimensions you can’t tell the difference
> between a cycle and an anti-cycle (of 3 stickers on a corner cubie);
> that is, one can be rotated to the other.
> So, to cycle 3 stickers a,b,c on a single corner cubie:
> 1. cycle a,b,c and anti-cycle 3 stickers on some other cubie
> 2. twist a face containing a,b,c and not the other cubie so that:
> a goes to a
> b goes to c
> c goes to b
> d goes to e
> e goes to d
> (where d,e are two more stickers on the same corner cubie
> as a,b,c – this is where d>=5 is needed)
> (you can’t just swap b and c without swapping some other d and e,
> since that would turn the cubie inside out)
> 3. undo 1.
> 4. undo 2.
> The result is that a,b,c got forward-cycled twice and nothing else
> on the puzzle was touched– i.e. they got backward-cycled once.
> So do all of the above in reverse to cycle a,b,c forward, as desired :-)

In 1987, I won the first prize in math at the Prince Georges County,
MD regional science fair with basically this concept: a general
solution to an n-dimensional Rubik’s cube with n pieces per side could
always be constructed by moving a set of pieces to an isolated face,
performing a sequence of moves that altered only those pieces on that
face, rotating the face, and performing the same sequence of moves in
reverse. The above certainly adds some interesting additional
insights! In any case, I’ve never met a puzzle of this ilk that
couldn’t be solved using that approach. The three-dimensional
hand-held puzzle "Square 1" actually has a twist (so to speak) that
separates it from all the other puzzles. Since I realized this
approach, I’ve never met a Rubik’s cube style puzzle that I could
solve within an hour or two of first picking it up except for Square 1
which took me several days of an hour here and an hour there before I
realized what the difference was. Once I figured it out, it only
required adding one trick to the bag of tricks to be able to solve it.

It’s particularly interesting, but not that surprising, that as the
number of dimensions increase, the difference between a cycle and an
anti-cycle disappears. I’ll have to chew on that for a while.

–Jay