This week’s SUMS talk is a brief overview of impartial combinatorial two-player game theory. Abstract: Suppose you have a finite number of stones that have been divided into a number of piles. You and another player then take turns picking a pile and removing any positive number of stones from it. If the final stone is removed on your turn, you win. This game goes by the name of Nim, and a natural question to ask is what initial positions guarantee the first or second player wins. How does our answer change if we introduce or remove certain constraints on Nim, and can we apply this knowledge to any other two-player games? This talk will give a brief introduction to game theory for impartial combinatorial games like Nim, and discuss the number system that arises from position analysis of these games (known as nimbers).