Take a perfect cube. While keeping the cube intact and all the pieces together, make as many planar slices as possible through exactly three vertices.

After doing so, separate the resulting pieces. What shapes result and how many are there of each and in total?

To what do these numbers correspond?

Note: Please do NOT punch the problem into a 3D graphics program and then rush to post the solution here so you can be first. This remains my most enjoyable solution to date because I attempted it without pen and paper, computer or polyhedron.

This type of problem is my favorite, and I simply can't let it go without giving it a try!

There are so many methods to do it, but I decided to try a cross-section approach.  If the cube is a unit cube, the 5 square cross-sections are at heights 0, .25, .5, .75, and 1.

 ________    ________    ________
|\      /|  | /\ 1/\2|  |   /\  2|
| \ 1  / |  |/  \/ 3\|  |  /  \  |
|  \  /  |  |\  /\  /|  | /    \ |
|   \/   |  | \/  \/ |  |/      \|
|   /\   |  | /\ 4/\ |  |\  4   /|
|  /  \  |  |/  \/  \|  | \    / |
| /    \ |  |\  /\  /|  |  \  /  |
|/______\|  |_\/__\/_|  |___\/___|

Note: 4th and 5th are same as
2nd and 1st respectively.

Based on that, there are 12 tetrahedrons corresponding to the 12 edges.  Each has five sqrt(1/2) edges and one 1 edge.  Labels 1 and 2 are examples of these.

There are 8 equilateral tetrahedrons corresponding to the 8 corners.  Each has side length of sqrt(1/2).  Label 3 is an example.

There is 1 octahedron corresponding to the 1 cube.  It has side length of sqrt(1/2).  It is labeled 4.

In all, 21 solids.

Edited on August 10, 2005, 4:40 am

Edited on August 10, 2005, 4:44 am

Posted by Tristan on 2005-08-10 04:39:15