1. What are Combinatorial Games?

Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie,tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. And so is Go. The shorter terminology game, games is used below to designate combinatorial games.

2. Why are Games Intriguing and Tempting?

Amusing oneself with games may sound like a frivolous occupation. But the fact is that the bulk of interesting and natural mathematical problems that are hardest in complexity classes beyond NP, such as Pspace, Exptime and Expspace, are two-player games; occasionally even one-player games (puzzles) or even zero-player games (Conway's “Life“). Some of the reasons for the high complexity of two-player games are outlined in the next section. Before that we note that in addition to a natural appeal of the subject, there are applications or connections to various areas, including complexity, logic, graph and matroid theory, networks, error-correcting codes, surreal numbers, on-line algorithms and biology.

But when the chips are down, it is this “natural appeal” that compels both amateurs and professionals to become addicted to the subject. What is the essence of this appeal? Perhaps the urge to play games is rooted in our primal beastly instincts; the desire to corner, torture, or at least dominate our peers. An intellectually refined version of these dark desires, well hidden under the façade of a passion for local, national or international tournaments or for scientific research, is the consuming strive “to beat them all”, to be more clever than the most clever, in short — to create the tools to Math-master them all in hot combinatorial corafeat! Reaching this goal is particularly satisfying and sweet in the context of combinatorial games, in view of their inherent high complexity.