167shw1 - Game Theory Steven Heilman Please provide...

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Game Theory Steven Heilman Please provide complete and well-written solutions to the following exercises. Due April 5, in the discussion section. Homework 1 Exercise 1. Let n be a positive integer. Consider the game Chomp played on an n × n board. Explicitly describe the winning strategy for the first player. (Hint: the first move should remove the square which is diagonally adjacent to the lower left corner.) Exercise 2. Compute the following nim-sums: 3 4, 5 9. Then, let a, b, c be nonnegative integers. Prove that a a = 0 and ( a b ) 0 = a b . Exercise 3. Consider the nim position (9 , 10 , 11 , 12). Which player has a winning strategy from this position, the next player or the previous player? Describe a winning first move. Exercise 4. Consider the game of Chomp played on a board of size 2 × ∞ . Recall that a typical Chomp game board is n × m , so that the board has n rows and m columns. We can label the rows as { 1 , 2 , . . . , n } and we can label the columns as { 1 , 2 , . . . , m } , where n, m are positive integers. On a 2 × ∞

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• Fall '08
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