Newsgroups: sci.math, rec.puzzles
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 20 Dec 1994 16:11:25 GMT
Subject: Re: Cube inside other cube

d...@sjfc.edu (Dan Cass) writes (excerpts in reverse order):

> what does "ObPuzzle" mean...

Obligatory puzzle.  Some of us consider it de rigueur.  On newsgroup
X, you will often see ObX as a sop to being on-topic; on rec.puzzles,
we seem to be pretty well confined to on-topic posts and spam.

> [Q: what is the general solution of 1/a + 1/b + 1/c = 0 in integers?]
> Is that related to Egyptian fractions, or a simple case of it?

As p...@eurocontrol.fr (Philip Gibbs) pointed out, the general
solution is

  a = tu(u+v),   b = tv(u+v),  c = -tuv

It's easy to prove that this is the general form, and that any
solution with c < 0 < b <= a has a unique such parameterization
with (t,u,v)=1, 0 < v <= u, 0 < t.  "Egyptian fractions" are just
reciprocals of integers (possibly together with 2/3), which these
certainly are.

> I lost the post,...

davi...@spcvxb.spc.edu (Dave Davis) asked:

> One square nests neatly in another, one vertex of the inner on each
> side of the outer. But with cubes it's not so neat....  find the
> maximum number of points on the surface of a cube, at most one per
> surface, such the vertices which complete it to a cube lie entirely
> within.

Your idea, to find

> three vectors (a,b,c), (b,c,a), (c,a,b) which were mutually
> orthogonal, and then use these three, along with (0,0,0) and the
> four sums of 2 or 3 of them, as the vertices.

is very nice.  It amounts to taking a copy of the original cube,
rotating it about the x=y=z axis, then shrinking it to fit inside the
original.

With the above parameterization, the ratio of the edge of the
internal cube to that of the external cube is

sqrt(a^2+b^2+c^2)/(a-c) = (u^2 + u v + v^2)/(u^2 + 2 u v)

This approaches 1 as v/u approaches 0 or 1.  Its infimum of sqrt(3/4)
is approached as v/u approaches sqrt(3/4)-1/2.

ObPuzzle]:  Find the smallest integer cube-in-an-integer cube touching
six sides and less than 2/3 of the volume of the larger cube.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil