Download Presentation
## The Game Of Nim

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**The Game Of Nim**And its Winning Strategy By Jason Jebbia**The Game of Nim**• 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**History of Nim**• 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**Rules of Nim**• Game starts with a number of piles (or rows, columns, etc.) and a number of objects (pennies, peanuts, counters, etc.) in each pile. • Example**Rules of Nim**• 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**Rules of Nim**• 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**Winning Strategy**• *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**Winning Strategy**• Example Binary • 7 = 1 1 1 • 4 = 1 0 0 • 5 = 1 0 1 1 1 0 Nim Sum**Winning/Losing Positions**• 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.**Winning Strategy**• 7 = 1 1 1 • 4 = 1 0 0 • 5 = 1 0 1 1 1 0 0 0 1 = 1 1 0 0 = 4 1 0 1 = 5 0 0 0 1 1 1 = 7 0 1 0 = 2 1 0 1 = 5 0 0 0 1 1 1 = 7 1 0 0 = 4 0 1 1 = 3 0 0 0**Winning Strategy**• Intuition • If the Nim sum is greater than 0, then we know that at least one of the column sums is 1. 7 = 1 1 1 4 = 1 0 0 5 = 1 0 1 1 1 0**Winning Strategy**• 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. = 1 1 1 = 1 0 0 = 1 0 1 1 1 0 1 1 1 = 7 0 1 0 = 2 1 0 1 = 5 0 0 0**Winning Strategy**• 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. 0 0 1 1 0 0 1 0 1 0 0 0**Winning Strategy**• 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 = 0 0 1 4 = 1 0 0 5 = 1 0 1 0 0 0**Winning Strategy**• 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).**Winning Strategy**• 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.**Misère Nim**• 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.**Misère Nim**1 0 0 = 4 0 1 1 = 3 0 1 0 = 2 0 1 0 = 2 0 0 1 = 1 0 0 1 = 1 1 1 1 0 0 0 0 0 1 = 1 0 0 1 = 1 0 1 0 = 2 0 0 1 = 1 0 0 1 = 1 0 0 1 = 1 0 1 0 0 0 0 = 0 0 0 0 = 0 0 0 1 = 1 0 0 1 = 1 0 0 1 = 1 0 0 0 = 0**Misère Nim**1 0 0 = 4 0 1 1 = 3 0 1 0 = 2 0 1 0 = 2 0 0 1 = 1 0 0 1 = 1 1 1 1 0 0 0 0 1 0 = 2 0 0 1 = 2 0 1 0 = 2 0 0 1 = 2 0 0 1 = 1 0 0 0 = 0 0 0 1 0 0 0 0 0 1 = 1 0 1 0 = 1 0 1 0 = 2 0 0 0 = 0 0 0 0 = 0 0 0 0 = 0 0 1 1**Misère Nim**• 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.**Bounded Nim**• 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!**Bounded Nim**• 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.**Bounded Nim**• Example : k = 4 Pile SizesResidue PilesBinary 11 = 1 (mod 5) 0 0 1 14 = 4 (mod 5) 1 0 0 9 = 4 (mod 5) 1 0 0 13 = 3 (mod 5) 0 1 1 0 1 0 • 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!**Importance of Nim**• 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**Importance of Nim**• Sprague-Grundy Theorem: • Every impartial game in its normal play convention is equivalent to some Nim pile using the Sprague-Grundy 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)**What have we learned?**• The Game of Nim • The winning strategy of Nim • Misère and Bounded Nim • The Sprague-Grundy Theorem of Impartial Games**Special Thanks**• To: • Professor TamasLengyel • Professor Ron Buckmire • Professor Eric Sundberg**References**• [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