stat155hw1soln - identified with a heap size and the...

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Stat 155 Fall 2009: Solutions to Homework 1 (was due September 10, 2009) 1. Chomp the square at (3,1), gobbling 9 pieces: z - z P 2. Using the binary expressions of the heap sizes, 9 = 1001 , 10 = 1010 , 11 = 1011 , 12 = 1100 , and so the nim-sum of the heap sizes is: 9 10 11 12 = 4 . Thus, the position is in N and a win for the next player. Since 4 = 0100, the winning move would be to remove 4 chips from the heap of 12, so the nim-sum would reduce to 0. The new position would be (9 , 10 , 11 , 8). 3. The game of Nimble is just the game of Nim in disguise, if each coin is
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Unformatted text preview: identified with a heap size, and the left-most slot is numbered 0. So a game with a single coin is single-heap nim, and if there are two coins on the same slot, this means that the corresponding nim game has two heaps of the same size, etc. If a coin moves to the left-most slot, that is equivalent to reducing that heap to 0. Now Bouton’s solution can be applied to Nimble, and the starting P-positions are those in which the nim-sum of the initial coin positions is 0. 1...
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