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Combinatorial Games

Whether you’re a master of games or just playing around, learn how combinatorial ideas can be used to analyze and solve games such as Nim.

Concept Quizzes

Winning Positions

         

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?

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 two heaps of two coins each. If Seth goes first, who can guarantee a win?

Goofy and Roxanne are playing a game with 31 stones. On each turn, a player can remove one, two, three, or four stones (he/she has to remove at least one on his/her turn). The person who picks the last stone loses. If Goofy goes first, who can guarantee a win?

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