Can you 3-color the cells of a 9x9 grid (connected toroidally) so that each possible coloration of a 2x2 subsquare occurs exactly once?

Can you 4-color a 16x16 grid so that each possible 2x2 subsquare occurs exactly once?

Can you 2-color a 256x256 grid so that each possible 4x4 subsquare occurs exactly once?

Can you 4-color a 512x512 grid so that each possible 3x3 subsquare occurs exactly once?

I've made some progress on these and will post my results later. If someone has a link for information on these sorts of problems, I'd like to see it. I've come up with them as an independant reinviention (they're simple enough in concept to undoubtedly have been thought of before).