Newsgroups: rec.puzzles
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: 19 Sep 94 21:15:28 GMT
Subject: Painted cubes

I don't see this in the archive, though I'm pretty sure I've seen it
here before.  Looking through _Lewis Carroll's Games and Puzzles_, I
saw it again:

  Given six pots of paint and a supply of wooden cubes, paint each
  side of a cube a different color.  How many ways can this be done,
  up to rotation?

The problem also asks that we develop a method of coding and recording
the different cubes.

The counting puzzle is very easy, but I won't spoil it, except to say
that it ends up as a number that is immediately recognizable as the
number of a certain combinatorial object constructed from the six
colors.  The interesting question, for which I have not found an
answer, is to provide a natural (or at least simply described)
one-to-one correspondence between the combinatorial objects and the
colorings.  Most satisfying would be a rule for placing the cubes so
we could in some sense see the combinatoric.

As another puzzle, we can ask into what shape of box we can pack our
set of colored cubes with like colors abutting.  What if we require
the colors on the top face of the box to match the bottom?  What if we
require all the opposite pairs of faces to match?

Dan Hoey
Hoey@AIC.NRL.Navy.Mil