Great Theoretical Ideas In Computer Science V Adamchik D Sleator Lecture 10 CS 15 251 Feb 11 2010 Spring 2010 Carnegie Mellon University Mathematical Games Plan Introduction to Impartial Combinatorial Games References Game Theory by T Ferguson Download from http www math ucla edu tom Game T heory Contents html Related courses 15 859 21 801 Mathematical Games Look for it in Spring 11 A Take Away Game Two Players 1 and 2 A move consists of removing one two or three chips from the pile Players alternate moves with Player 1 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 21 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 1 can win 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 A set of positions Two players Rules specify for each player and for each position which moves to other positions are legal moves The players alternate moving A terminal position in one in which there are no moves The game ends when a player has no moves The game must end 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 state This rules out games like battleship No draws in a finite number of moves This rules out tic tac toe Impartial Versus Partizan A combinatorial game is impartial if the same set of moves is available to both players in any position A combinatorial game is partizan if the move sets may differ for the two players In this lecture we ll study impartial games P Positions and N Positions P Position Positions that are winning for the Previous player the player who just moved Sometimes called LOSING positions N Position Positions that are winning for the Next player the player who is about to move Sometimes called WINNING positions 0 4 8 12 16 are Ppositions if a player moves to that position they can win the game 21 chips 21 chips is an Nposition 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 Ppositions normal winning condition 2 From every N position there is at least one move to a Pposition 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 position and above Player who takes 1 1 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 1 No matter what you do I can mirror it Mirroring is an extremely important strategy in combinatorial games Theorem Player 1 can win in any square starting position of Chomp Proof The winning strategy for player 1 is to chomp on 2 2 leaving only an L shaped position Then for any move that Player 2 takes Player 1 can simply mirror it on the flip side of the L Theorem Every rectangle is a Nposition Proof Consider this position This is either a P or an N position If it s a Pposition then the original rectangle was N If it s an N position then there exists a move from it to a P position X But by the geometry of the situation X is also available as a move from the starting rectangle It follows that the original rectangle is an Nposition The Game of Nim Two players take turns moving Winner is the last player to remove chips x y z 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 We use x y z to denote this position x y z 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 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 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 P positions 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 010010110 101000110 000000000 Look at leftmost column with an odd of 1s Rig any of the …
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