Newsgroups: rec.puzzles
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: 19 Sep 94 21:36:40 GMT
Subject: Re: 4-D puzzles

bco...@acsu.buffalo.edu (Bram Cohen) writes:

> Here are two puzzles which were in an old Martin Gardner column:

The first, at least, was reprinted in Gardner's _Mathematical Circus_,
and is touched on in Rudy Rucker's _The Fourth Dimension_.

> (1) The surface of a cube can be 'unfolded' in the following way:
>         _
>       _|_|_
>      |_|_|_|
>        |_|
>        |_|
> There are other ways as well. How many ways can a 4-D cube be
> 'unfolded' as a bunch of connected cubes in 3-space?

I wrote about this on sci.math in 1992 and again earlier this month.
There are 261 octocubes you can get from unfolding the surface of a
tesseract, or 355 if you count reflections of the chiral ones.  I am
quite certain of this, having it both in manual analysis and from a
program to generate them.

Then I found a paper by Peter Turney in J Rec Math V17(1), 1984, in
which he shows the problem to be equivalent to counting the number of
8-vertex graphs that consist of a tree together with a differently-
colored pairing of the 8 vertices, disjoint from the tree (the tree
represents face adjacency in the unfolding, and the pairing represents
opposite pairs of faces in of the solid.)  This would plainly extend to
the problem of unfolding the surface of a 5-dimensional hypercube into
a dekatesseract, but I don't know if it's been done.  Also, I don't
know how to figure out which ones are chiral.

> Martin Gardner commented that he received about 10 responses to
> each of these.  Unfortunately, he got as many different answers as
> responses! For that reason, please ... specify what the unfolded
> patterns are for (1).

I could give you twenty pages of diagrams, but how would that help?
If you can't solve the problem, how could you verify exhaustiveness
and nonduplication in a list?  I suggest you see Turney's proof,
instead.

There's an interesting ObPuzzle here.  Notice that when you unfold a
non-convex polyhedron and flatten it out, the surface may overlap
itself.  For instance, the following depicts one form of the unfolded
surface of the L-tricube:

              +--+-----+-----+
              |  |     |     |
  +--+     +--+--+-----+-----+
  |  |     |XX|  |
  |  +--+--+--+  |
  |     |  |     |
  +-----+--+-----+

We know that doesn't happen for the tetrahedron, cube, or octahedron.
Prove it doesn't happen for the dodecahedron or icosahedron.  It
doesn't happen for the tesseract, either: Can you prove that unfolding
the N-dimensional cube never overlaps?  What about the other regular
N-dimensional polytopes?

Dan Hoey
Hoey@AIC.NRL.Navy.Mil