MATH
167exam1soln

# 167exam1soln - 167 Midterm 1 Solutions1 1 Question 1(i Give...

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167 Midterm 1 Solutions 1 1. Question 1 (i) Give an example of a game mentioned in class, or the notes, or any course textbook, such that: the first player has a winning strategy. (You only need to mention the game; you do not need to prove anything.) Solution . Hex, Connect Four, and Chomp on a finite rectangle of size larger than 1 × 1 are all valid answers. (ii) Give an example of a game mentioned in class, or the notes, or any course textbook, such that: the second player has a winning strategy. (You only need to mention the game; you do not need to prove anything.) Solution . Chomp played on a 2 × ∞ board, or on an n × ∞ board for n > 2. (iii) Give an example of a game mentioned in class, or the notes, or any course textbook, such that: both players have a strategy guaranteeing at least a draw. (You only need to mention the game; you do not need to prove anything.) Solution . Tic-tac-toe and Checkers are valid answers. 2. Question 2 Consider the game of Nim, where the game starts with four piles of chips. These piles have 1 , 5 , 3 and 15 chips, respectively. Which player has a winning strategy from this position, the first player, or the second? Describe a winning first move.

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