lecture05 - 15-251 Great Theoretical Ideas in Computer...

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1 15-251 Great Theoretical Ideas in Computer Science 15-251 Game Playing for Computer Scientists Combinatorial Games Lecture 5 (September 13, 2011) 21 chips Two Players: I and II A move consists of removing one, two, or three chips from the pile Players alternate moves, with Player I starting Player that removes the last chip wins A Take-Away Game Which player would you rather be?
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2 Try Small Examples! If there are 1, 2, or 3 only, player who moves next wins If there are 4 chips left, player who moves next must leave 1, 2 or 3 chips, and his opponent will win With 5, 6 or 7 chips left, the player who moves next can win by leaving 4 chips 0, 4, 8, 12, 16, … are target positions; if a player moves to that position, they can win the game Therefore, with 21 chips, Player I can win! 21 chips What if the last player to move loses? If there is 1 chip, the player who moves next loses If there are 2,3, or 4 chips left, the player who moves next can win by leaving only 1 In this case, 1, 5, 9, 13, … are a win for the second player Combinatorial Games There are two players There is a finite set of possible positions The rules of the game specify for both players and each position which moves to other positions are legal moves The players alternate moving The game ends in a finite number of moves (no draws!) Normal Versus Misère Normal Play Rule: The last player to move wins Misère Play Rule: The last player to move loses A Terminal Position is one where neither player can move anymore What is Omitted No random moves No hidden moves No draws in a finite number of moves (This rules out games like poker) (This rules out games like battleship) (This rules out tic-tac-toe)
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lecture05 - 15-251 Great Theoretical Ideas in Computer...

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