What are all the ways that 30-, 60-, and 90-degree angles can be arranged about the origin in the plane? Let two ways be equivalent if one can be carried into the other by either a rotation or a flip of the plane -- i.e., by an element of the group O(2). This appears to be a messy counting job but certainly doable by hand because of its small scale. As a first step, I found there are 19 partitions of 12 using only 1's, 2's, and 3's (appended). Can anyone suggest a smart way to go about this, or is there no good way to shorten the task? --Dan ----------------------- 3+3+3+3 2+2+2+3+3 2+2+2+2+2+2 1+2+3+3+3 1+2+2+2+2+3 1+1+2+2+3+3 1+1+2+2+2+2+2 1+1+1+3+3+3 1+1+1+2+2+2+3 1+1+1+1+2+3+3 1+1+1+1+2+2+2+2 1+1+1+1+1+2+2+3 1+1+1+1+1+1+3+3 1+1+1+1+1+1+2+2+2 1+1+1+1+1+1+1+2+3 1+1+1+1+1+1+1+1+2+2 1+1+1+1+1+1+1+1+1+3 1+1+1+1+1+1+1+1+1+1+2 1+1+1+1+1+1+1+1+1+1+1+1 ----------------------- ________________________________________________________________________________________ It goes without saying that .