Question

A square window is made up from 3x3 = 9 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 8 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 are there in all? For example, there is 1 "9-clear" window, there are 3 "1-red 8-clear" windows, and 8 "2-red 7-clear" windows...

Thanks

Answer 1

bump

Answer 2

I got an answer for you (just by considering all possible scenarios). Given that there are {0,1,2,3,4,5,6,7,8,9} red panes in the window, there are {1,3,7,17,23,23,17,7,3,1} different possibilities respectively. Further, there are then a total of 102 different possibilities in all.

Answer 3

for 2R/7C I get 8 possibilities not 7. Number the panes 1-9 and consider (15) (25) ; (12) (13) (16) (19) ; (24) (28) all 8 cases distinct.

Answer 4

You're right, I left out I believe case (28), since I wasn't actually writing out every single case and was just doing it mentally for the case of 2 reds. I actually wrote out some instances for the 3 and 4 red cases though, so I think those are correct. Still haven't thought of a good combinatorial way to do it..

Answer 5

On to the 4x4 window!
1x1: {1,1} = 2 types
2x2: {1,1,2,1,1} = 6 types
3x3: {1,3,8,17,23,23,17,8,3,1} = 104 types
4x4: {1,3,???