Question

A square window is made up from 4x4 = 16 square subpanes that are tinted either clear or red. (Since the frame is symmetric it can be rotated or flipped before mounting it.) For example if the window-maker is delivering orders for windows that have 1 red and 15 clear panes, they only need to manufacture 3 different window types: {red in corner; red in side; red in central}.

How many distinct window types have 7 red and 9 clear panes?

How many distinct window types are there in all?

Thanks!

Answer 1

I'm still working on the first part of the problem, since I haven't yet found the pattern (since 1, 3, 21 for the first three cases of 0, 1, and 2 reds is not a familiar sequence to me). However, the answer to the second question is (I believe) 8192. Ignoring rotations and reflections, you clearly have 2^16 possibilities. Reflections and rotations of a square form the dihedral group of size 8 (if you haven't studied any abstract algebra, just convince yourself that there are 8 distinct ways to renumber the vertices of a square using only reflections and rotations), so we just divide 2^16 by 8 to give 2^13 = 8192 distinct window types.

I'll let you know if I think of a nice way to solve the first part.

Answer 2

this assumes that all the transformations of every window under the dihedral group yield distinct configurations, but unfortunately some windows are already symmetric (the completely red window always goes to itself under rotations and reflections) and don't have 8 independent versions

consider for example the 2x2 window example: there are {1,1,2,1,1} = 6 distinct red/clear window types, which is not a factor of 2^4 = 16 windows before symmetry...

Answer 3

Ah, true, true, I'll keep working on it.