One that I heard last week:
There is an 8x8 checkerboard in a room with a coin placed on each square. Each coin is either facing heads or tails up, and the face is determined randomly. Before you can inspect the board, a "master" comes in, picks a square of interest, and must make a manipulation to the board by flipping one of the 64 coins. He then exits the room. You are now allowed to enter, and must read out which square of interest the master chose by inspecting the state of the board. There is a strategy that is guaranteed to work for all possible configurations of the board.
Either I'm somehow missing some critical piece of this puzzle or your description is. The master could pick a square at random, leaving you with a board (and likelihood of success) that is still completely random.
Are you supposed to be collaborating with the "master"?