15 251 Great Theoretical Ideas in Computer Science 15 251 Game Playing for Computer Scientists Combinatorial Games Lecture 3 September 2 2008 A Take Away Game 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 21 chips Player that removes the last chip wins Which player would you rather be 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 Mis re Normal Play Rule The last player to move wins Mis re 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 This rules out games like poker No hidden moves This rules out games like battleship No draws in a finite number of moves This rules out tic tac toe P Positions and N Positions P Position Positions that are winning for the Previous player the player who just moved N Position Positions that are winning for the Next player the player who is about to move 0 4 8 12 16 are Ppositions if a player moves to that position they can win the game 21 chips is an N position 21 chips What s a P Position Positions that are winning for the Previous player the player who just moved That means For any move that N makes There exists a move for P such that For any move that N makes There exists a move for P such that There exists a move for P such that There are no possible moves for N P positions and N positions can be defined recursively by the following 1 All terminal positions are P positions 2 From every N position there is at least one move to a P position 3 From every P position every move is to an N position Chomp Two player game where each move consists of taking a square and removing it and all squares to the right and above Player who takes position 0 0 loses Show That This is a P position N Positions Show That This is an N position P position Let s Play I m player I No matter what you do I can mirror it Mirroring is an extremely important strategy in combinatorial games Theorem Player I can win in any square starting position of Chomp Proof The winning strategy for player I is to chomp on 1 1 leaving only an L shaped position Then for any move that Player II takes Player I can simply mirror it on the flip side of the L What about rectangular boards Theorem Player I can win in any rectangular starting position Proof Look at this first move If this is a P position then player 1 wins Otherwise there exists a P position that can be obtained from this position And player I could have just taken that move originally Move the Token Two player game where each move consists of taking the token and moving it either downwards or to the left but not both Player who makes the last move to 0 0 wins The Game of Nim Two players take turns moving x y z Winner is the last player to remove chips A move consists of selecting a pile and removing chips from it you can take as many as you want but you have to at least take one In one move you cannot remove chips from more than one pile Analyzing Simple Positions x y z We use x y z to denote this position 0 0 0 is a P position One Pile Nim What happens in positions of the form x 0 0 The first player can just take the entire pile so x 0 0 is an N position Two Pile Nim Seen this before It s the Move the Token game Two Pile Nim P positions are those for which the two piles have an equal number of chips If it is the opponent s turn to move from such a position he must change to a position in which the two piles have different number of chips From a position with an unequal number of chips you can easily go to one with an equal number of chips perhaps the terminal position 3 Pile Nim Two players take turns moving x y z Winner is the last player to remove chips Nim Sum The nim sum of two non negative integers is their addition without carry in base 2 We will use to denote the nim sum 2 3 10 2 11 2 01 2 1 5 3 101 2 011 2 110 2 6 7 4 111 2 100 2 011 2 3 is associative a b c a b c is commutative a b b a For any non negative integer x x x 0 Cancellation Property Holds If x y x z Then x x y x x z So y z Bouton s Theorem A position x y z in Nim is a P position if and only if x y z 0 Proof Let Z denote the set of Nim positions with nim sum zero Let NZ denote the set of Nim positions with non zero nim sum We prove the theorem by proving that Z and NZ satisfy the three conditions of Ppositions and N positions 1 All terminal positions are in Z The only terminal position is 0 0 0 2 From each position in NZ there is a move to a position in Z 001010001 001010001 100010111 100010111 111010000 101000110 010010110 000000000 Look at leftmost column with an odd of 1s Rig any of the numbers with a 1 in that column so that everything adds up to zero 3 Every move from a position in Z is to a position in NZ If x y z is in Z and x is changed to x x then we cannot have x y z 0 x y z Because then x x k Pile Nim Combinatorial Games …
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