Abstract: The subject grew out of a math olympiad problem: The president and an assassin are in a room with reflecting walls. Assassin's bullets would hit the president even after any number of ricochets off the walls. Thus, the assassin has at his disposal infinitely many deadly possibilities. The presidential security detail can instantaneously position themselves at arbitrary spots in the room, and hence at the spots where they would intercept optimal subsets of deadly bullets. Can a finite number of security guards protect the prez? I will talk about mathematical ramifications of this olympiad question.