Consider two games:

• (For two players) Start with several piles of pebbles. On her turn, a player removes any number of pebbles from any one pile. Whoever removes the last pebble wins. (This is a very famous game, called Nim.)
• (For N players) Each player is given a black or white hat, randomly and with equal probability. Each player can see his teammates' hats but not his own. At a signal from the judge, all the players simultaneously guess their own hat colors; they are also allowed to pass and guess nothing. If there is at least one correct guess and no incorrect guesses among the players, the whole team gets a prize. Is there a strategy the players can decide on in advance to maximize their chances of winning? (Try it with N=3.)

The strategies for these games are closely related to each other and to the theory of error-correcting codes. We will see how the first game can be used to derive an optimal strategy for the second one, and then discuss how game strategies in general can lead to good codes.