Nim: Serious Math With a Simple Game

Children will practice inductive thinking as they find winning strategies for 22 variations on the game of NIM. All they need is this book and a pile of paperclips, buttons, checkers, or whatever is handy.Students begin by learning the five rules of the Basic Game of1. There is 1 pile of objects.2. There are 13 objects.3. There are 2 players.4. Players take turns removing either 1 or 2 objects from the pile.5. Whoever takes the last object loses.As students play, the book’s thoughtful questions lead them to discern the game’s winning strategy.Students then consider variations. They may play with 15, 20, 30, or more objects in the pile. They may allow players to “Take 1, 2, or 3” or “Take 1, 2, 3, or 4” objects from the pile. They may experience the power of generalization when they formulate the winning strategy for all games of the type “Take 1, 2, . . ., max.” They may play games with two or more piles. Advanced students may even change the number of players.Winning strategies involve patterns and sequences. Young students may describe a sequence of winning numbers as “starting at one and counting up by threes.” Older students may use algebra to describe “numbers of the form 3n+1,” or use modular arithmetic to refer to “numbers equivalent to 1 in modulo 3.” Advanced students analyzing a 3-player game may use probability to determine which moves are more likely to win.The book provides attractive game rules to give to students, leading questions for teachers to ask, and complete descriptions of strategies. These games are enjoyed by a wide range of students.

Christopher Freeman holds a bachelor’s degree in math and an master’s degree in math education from the University of Chicago. He teaches math to grades 6–12 at the University of Chicago Laboratory Schools. Freeman also teaches math enrichment classes in the Worlds of Wisdom and Wonder and Project programs for gifted children in the Chicago area, sponsored by the Center for Gifted at National-Louis University. His books are the fruits of curricula he has developed for gifted children in these programs and in the regular classroom.

All of Freeman’s activities involve students in inductive thinking. Students are presented with an intriguing situation or set of special cases, and they formulate conjectures about the fundamental mathematical properties that govern them. Students in Freeman’s classes practice inductive thinking when they find winning strategies for math games, formulate conjectures about the structure of many-pointed stars, or figure out which polygons can fit together to form polyhedra—and why.

Freeman is a regular presenter at the annual conventions of the National Association for Gifted Children. He contributed a chapter on math curriculum in the NAGC publication Designing and Developing Programs for Gifted Students, edited by Joan Franklin Smutny. He has published three books with Prufrock Press, Nim: Variations and Strategies, Drawing Stars and Building Polyhedra, and Compass Constructions.