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/
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