Quote:
Originally Posted by wik_chick88  how many distinct octahedra (all of the same size) are possible if each face of each octahedron is either red, blue or green? the octahedron is a regular solid with 6 vertices, 12 edges and 8 triangular faces.
im having trouble working out how many symmetries there are of an octahedron. 8 faces and 3 colours so different ways to colour the faces would be 8 choose 3 = 56. but obviously 2 ways might have the same physical result if the octahedron was rotated on a vertice, edge or face.
Please help!!!! |
wik_chick88,
As pointed out by The Perfect Hacker, the really slick way to solve this problem is to use Burnside's Lemma or, better yet, its cousin the Polya Enumeration Theorem, aka Polya's Theory of Counting. See
Pólya enumeration theorem - Wikipedia, the free encyclopedia.
However, that's not the only way to solve the problem. Consider the count of faces by color, written in the form #red + #blue: 8+0, 7+1, 6+2, 5+3, 4+4, 3+5, etc. There are only 5 cases to consider if you combine cases like 5+3 and 3+5 (meaning 5 red and 3 blue or 3 red and 5 blue faces). Then just work out the distinct possibilities; there aren't that many.