Discrete Mathematics
# Combinatorial Games

The **nim** is a mathematical game of strategy where two players take turns removing objects from distinct heaps. A player must remove at least one object at his/her turn, and can remove as many as he/she wants as long as they come from a single heap. The player who removes the last object is the winner.

Seth and Gordon play nim with three heaps of ten coins each. If Seth goes first, who can guarantee a win?

The **nim** is a mathematical game of strategy where two players take turns removing objects from distinct heaps. A player must remove at least one object at his/her turn, and can remove as many as he/she wants as long as they come from a single heap. The player who removes the last object is the winner.

Seth and Gordon play nim with four heaps each containing 4, 5, 6, and 7 coins. If Seth goes first, who can guarantee a win?

The **nim** is a mathematical game of strategy where two players take turns removing objects from distinct heaps. A player must remove at least one object at his/her turn, and can remove as many as he/she wants as long as they come from a single heap. The player who removes the last object is the winner.

Seth and Gordon play nim with four heaps. If Seth goes first, which of the following is a possible 4-tuple of the heap sizes such that Gordon can guarantee a win?

**nim** is a mathematical game of strategy where two players take turns removing objects from distinct heaps. A player must remove at least one object at his/her turn, and can remove as many as he/she wants as long as they come from a single heap. The player who removes the last object is the winner.

Seth and Gordon play nim with two heaps of two coins each. If Seth goes first, who can guarantee a win?