1. How many different tetrahedrons can be produced by coloring each face a solid color and using n different colors? (Two tetrahedrons are the same if they can be turned and placed side by side so that corresponding sides match in color.)

2. How many cubes with n colors?

Figuring out Polya's Theory of Counting is a worthwhile project that is still on my "to do" list. It should be considered "advanced" combinatorics because it goes quite a way beyond elementary counting techniques. For example, it is the very last topic covered in the Schaum Outline on Combinatorics by Balakrishnan that is probably available at your local Border's or Barnes and Noble, or from amazon.com etc. An introductory college text that covers it is Introductory Combinatorics by Brualdi (last chapter of 2nd ed.) -- he in fact does the cube problem as an example and gives (p. 587 of 2nd ed.)

(1/24)(k^6+3k^4+12k^3+8k^2)

as the answer, where k is the number of colors.

Edited on September 26, 2004, 8:30 pm

Posted by Richard on 2004-09-26 20:20:36