I have a set of N identical Platonic (regular) solids, each painted with a different color on each face--the same set of colors for each of them. If I choose at random one face of each of the solids, the probability that all the chosen faces are of different colors is 2/3 the probability that there are three of one color and the rest all different colors.

How many of these solids do I have, and how many faces are on each?

I don't have time to give the derivation but for N polyhedra with X sides the formula for the ratio of the probabilities boils down to:

(X+2-N)(X+1-N)/(C(N,N-3))

A search among the platonic solids X={4,6,8,12,20} to see which comes out to 2/3
yeilds X=20, N=11

So we have eleven icosahedra.

Posted by Jer on 2011-11-21 16:13:44