How many people need to be in a room to guarantee that there is a large probability that a pair of them have the same birthday? The number of people is surprisingly small, a fact that is sometimes called the ”birthday paradox” because it contradicts our intuition. The simple mathematics behind the birthday paradox leads to ”birthday attacks” on cryptographic protocols that can be remarkably effective if the hidden keys are chosen poorly. The birthday problem becomes significantly more compli- cated if we change the question slightly: How many people need to be in a room to guarantee that there is a large probability that a pair of friends have the same birthday? This leads to interesting problems about graphs and colorings.