Suppose you are a point in a square room. The walls of the room are mirrors, and there is a man with a laser gun standing somewhere else in the room. The man is also a point, and both of your positions are fixed points, that is, neither of you can move. A beam from the laser will bounce off of the walls of the room at an angle equal to its angle of incidence. To protect yourself, you are allowed to place any number of bodyguards (possibly infinitely many) at any points in the room. The laser beam will stop if it hits a bodyguard.
Is there an arrangement of finitely many bodyguards that completely protects you from being shot? If so, what is the largest number of bodyguards you need to protect yourself no matter how you and the gunman are placed in the room?
I encountered this problem a while back, and haven't been able to solve it. An initial idea is to tile the plane with 'rooms' and translate your image into each square, but only draw the gunman in one of the tiles. Then all possible fatal shots are represented by a segment from the shooter to one of your images. Somehow, we must place bodyguards along these segments, and cover each one.
A simple generalization of this problem is to begin with a room of a different shape (triangle, hexagon, circle, etc.). I imagine these problems could get quite tough, for instance, my approach above would only work with shapes that tile the plane.