ma017-2010-hw3 - (4 1994 girls are seated around a table...

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MA017 2010 HOMEWORK 3 Due 2009-10-19 Tue. (1) Start with two piles of chips of sizes x , y . Two players alternately take any number of chips from one pile or the same number of chips from each pile, and the winner is the one who takes the last chip. Assuming both players play perfectly, who wins? (2) Start with p = 1. Two players alternately multiply p by one of the numbers 2 , 3 ,..., 9, and the winner is the one who first reaches p 10 6 . Who wins? (3) Two players alternately place move a knight on a 1994 × 1994 chessboard; the firstst player to move chooses its initial position. The first player makes only horizontal moves ( x,y ) 7→ ( x ± 2 ,y ± 1), while the second makes only vertical moves ( x,y ) 7→ ( x ± 1 ,y ± 2); neither is allowed to visit a square that has already been visited. The loser is the one who cannot move. Prove that the first player can win.
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Unformatted text preview: (4) 1994 girls are seated around a table. Initially, one girl has n tokens. Each turn, some girl with at least two tokens passes one token to each of her neighbors. The game ends when every girl has at most one token. Show that if n < 1994, the game must terminate, while if n = 1994, the game cannot terminate. (5) A game starts with four heaps of beans, containing 3 , 4 , 5 and 6 beans. Two players move alternately. A move consists of taking either (a) one bean from a heap, provided at least two beans are left behind in that heap, or (b) a complete heap of two or three beans. The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy. 1...
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