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And its Winning Strategy By Jason Jebbia In this presentation I will: Introduce the game play and rules of the game of Nim Demonstrate the winning strategy of Nim Discuss other variations of Nim including their winning strategies. Discuss the importance of Nim to other impartial games using the Sprague-Grundy Theorem Origins of the game are uncertain Charles Bouton, 1901 Coined the name “Nim” for the game Developed a complete theory of the game, including a winning strategy Game starts with a number of piles (or rows, columns, etc.) and a number of objects (pennies, peanuts, counters, etc.) in each pile. Example Two Player Game Players take turns On each turn, a player must choose one pile and remove at least one object from the chosen pile. Example Two ways to play: Normal Play: the player who removes the last object is the winner Misère Play: the player who removes the last object is the loser *For the normal play convention Convert the size of each pile into its binary notation Add the columns up independently modulo 2. The resulting value is called the Nim sum Example 7 = Binary 111 4 = 100 5 = 101 110 Nim Sum A winning position occurs when the Nim sum equal zero A losing position occurs when the Nim sum is greater than zero A player can always make a move from a losing position to a winning one. Once in a winning position, the next move will always result in a losing position Therefore, the optimal strategy is to always be converting losing positions to winning positions on each move. 7 4 5 = = = 111 100 101 110 001 100 101 000 = = = 1 4 5 111 010 101 000 = = = 7 2 5 111 100 011 000 = = = 7 4 3 Intuition If the Nim sum is greater than 0, then we know that at least one of the column sums is 1. 7 4 5 = = = 111 100 101 110 Intuition Thus, a player can convert a losing position into a winning one by changing the digits in the columns where the Nim sums are 1 starting with the left most column where the column sum is 1. 7 = 4 = 5 = 111 100 101 110 111 = 010 = 101 = 000 7 2 5 Intuition: If a player moves to a winning position, then the Nim sum equals 0. If this is not the final position (no objects left on the table), then every digit that is a 1 will have a corresponding 1 in a different row that will result in a column Nim sum of 0. 001 100 101 000 Intuition Thus, any move from a winning position will result in a losing position, then, since a player will change one, and only one, row of digits on a given move creating a new Nim sum. 1 4 5 = = = 001 100 101 000 Two Absolutes: 1. A player cannot move from a winning position back to a winning position (Nim Sum = 0) 2. Every losing position (Nim Sum > 0) has an allowable move to a winning position. Thus the winning strategy is to continue making moves converting losing positions to winning positions until the ultimate winning position is achieved (No more objects on the table). Of course, this assumes the other player does not know the winning strategy as well. If both players know the winning strategy and play without error, then the winner can be determined by the starting position of the game. If the game starts in a losing position, then the first player to move will ultimately win. If the game starts in winning position, then the second player to move will ultimately win. The player who removes the last object loses! Therefore, the goal is to leave the last object for your opponent to take Winning Strategy: Play exactly like you would in normal play until your opponent leaves one pile of size greater than one. At this point, reduce this pile to size 1 or 0, whichever leaves an odd number of piles with only one object. 100 010 001 111 = = = 4 2 1 011 010 001 000 = = = 3 2 1 001 010 001 010 = = = 1 2 1 001 001 001 = = = 1 1 1 000 001 001 = = = 0 1 1 000 001 000 = = = 0 1 0 100 010 001 111 = = = 4 2 1 011 010 001 000 = = = 3 2 1 010 010 001 001 = = = 2 2 1 001 001 000 000 = = = 2 2 0 001 010 000 011 = = = 1 2 0 010 000 000 = = = 1 0 0 Why this works Optimal play from normal Nim will never leave you with exactly one pile of size greater than one (because this leaves a losing position). Your opponent can’t move from two piles of size greater than one to no piles of size greater than one. So eventually your opponent must make a move that leaves only one pile of size greater than one. Same game as the normal version of Nim except one restriction. Restriction: An upper limit, k, is placed on the amount of objects that may be selected on a given turn. So a player may remove 1≤ m ≤ k objects on each turn. Once again, the winning strategy is very much like the normal game! Winning Strategy: Convert each pile into a “residue pile” by reducing each pile modulo (k + 1). Perform the same Nim Sum operation on these new piles Then, make the same moves to the actual piles that you would to make the residue piles into a Nim sum of 0. Example : k = 4 Pile Sizes 11 14 9 13 Residue Piles = 1 (mod 5) = 4 (mod 5) = 4 (mod 5) = 3 (mod 5) Binary 001 100 100 011 010 The Nim sum is not 0, so this is a losing position. Therefore, we would continue to use the same strategy as we would in normal play to make the Nim sum of the residue piles equal zero! Nim is an Impartial game. Impartial game: Combinatorial Game Two-players, Perfect Information, no moves are left to chance (so poker is not a combinatorial game) * Any play available to one player must be available to the other as well Examples: Hackenbush, Kayles, and Sprouts Partizan Game Combinatorial Game *Players have different sets of possible moves Examples: Chess, Checkers, and Go Sprague-Grundy Theorem: Every impartial game in its normal play convention is equivalent to some Nim pile using the SpragueGrundy Function Using this Sprague Grundy function, every impartial game gets assigned a Nim pile value that allows us to analyze the game. Discovered independently by R. P. Sprague (1935) and P. M. Grundy (1939) The Game of Nim The winning strategy of Nim Misère and Bounded Nim The Sprague-Grundy Theorem of Impartial Games To: Professor Tamas Lengyel Professor Ron Buckmire Professor Eric Sundberg [1] E. R. Berlekamp, J. H. Conway and R. K. Guy (1982) Winning Ways for your mathematical plays, vols. 1 and 2, Academic Press, New York. [2] C. L. Bouton (1902) Nim, a game with a complete mathematical theory, Ann. Math. 3, 35-39. [3] J. H. Conway (1976) On Numbers and Games, Academic Press, New York. [4] E. H. Moore (1910) A generalization of a game called nim, Ann. Math. 11, 93-94 [5] J. G. Baron (1974) The Game of Nim-A Heuristic Approach. Mathematics Magazine 47.1, 23-28. [6] B. L. Schwartz (1971) Extensions of NIM, Mathematics Magazine 44.5, 252-57. [7] G.H. Hardy and E.M. Wright (1960), An Introduction to the Theory of Numbers, 4th ed., Oxford, Clarendon Press, 117-120