Here's a fun problem. Polygons are chose to fill up a 360 degree angle (a point). Such as a triangle, two squares, and a hexagon. In how many ways can this be done when order doesn't matter?

You're right. It's a fun little problem. Presumably you mean equilateral polygons, or else the answer would be infinity.

I'm sure there's some really elegant way to do this using number theory, but it seems easy enough to just use brute force.

6 triangles
4 squares
3 hexagons
4 triangles + 1 hexagon
2 triangles + 2 hexagons
3 triangles + 2 squares
1 triangle + 2 squares + 1 hexagon
2 pentagons + 1 decagon
2 octagons + 1 square
1 triangle + 2 dodecagons
1 triangle + 1 octagon + 1 icosikaitetragon (24-sided)
1 nonagon + 1 triangle + 1 octakaidecagon (18-sided)
1 decagon + 1 triangle + 1 pentadecagon (15-sided)
1 square + 1 dodecagon + 1 hexagon
1 square + 1 dodecagon + 2 triangles
1 heptagon + 1 triangle + 1 tetracontakaidigon (42-sided)


So I guess that makes 16. Did I miss any?

Woops. I did miss one:

1 square + 1 pentagon + 1 icosagon (20-sided)

So the answer (I think) is 17 different ways.