Disclaimer: I'm not sure how math related this puzzle is (it could potentially be), but I thought it was an interesting puzzle, and I also don't know how to solve it so I wanted to see if anyone had any ideas.
You have a board divided in quarters and a coin is in each spot. You do not know whether each is facing heads or tails upwards. In each turn, you can choose flip any number of coins. Specify a sequence of turns that guarantees that at some point all coins will be facing the same direction.
Follow up: Between each of your turns, the board is rotated an arbitrary amount amount (90, 180, 270 degrees). Specify a sequence of moves that guarantees that at some point all coins will be facing the same direction.