Cutting a Cube

Date: 07/05/99 at 08:18:37
From: Denis Borris
Subject: Cutting a cube

Make every (flat) slice through a cube that goes through exactly three 
of the cube's corners -- no more and no less. When you are done, take 
the whole thing apart and count the pieces. How many are there?

I tried hard, even on a cube of cheese, but couldn't hold it together 
long enough. I can see that each face of the cube will end up with a 
corner-to-corner "X" from the cuts, and that the middle piece will be 
a polyhedron.

Thank you very much for your help.

Date: 07/06/99 at 09:10:53
From: Doctor Rick
Subject: Re: Cutting a cube

Hi, Denis.

Problems like this are a real challenge for 3-dimensional 
visualization. I have paper models of a cube, a tetrahedron, and an 
octahedron on my desk just for such problems. I cut them from index 
cards or business cards and taped them together.

You have made a good start: you see the X's on the faces of the cube. 
How about trying to take it apart mentally from the outside? You can 
see 4 pieces on each face of the cube. Focus on one of these pieces. 
You see a second face of this piece; it has 2 other faces inside the 
cube, made by the cuts. How many pieces of this shape are there? 
(Hint: one per edge.)

Now look at the piece that is cut off from the cube by a single cut. 
What shape is it? When the pieces from the previous paragraph are 
removed from this piece, you will have something left. All of its 
vertices are on the surface of the cube. It is a regular polyhedron - 
can you name its shape? How many of this shape are there in the cube? 
(Hint: come up with your own hint similar to the previous one.)

Now for that inner polyhedron. It too has all of its vertices on the 
outside of the cube. Maybe you already see this; if not, do whatever 
you have to do to convince yourself. Then find the vertices and count 
them. Visualize the edges and faces that connect the vertices. It is 
another regular polyhedron. What is it?

If you need a review of the names of the shapes, see our Dr. Math FAQ 
on regular polyhedra:

http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html   

Your question didn't say anything about identifying the shapes, but I 
think that is the most interesting part of the problem.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/