In the first lesson of this unit, students learn how to represent numbers in other bases and convert to and from decimal representations. In the second lesson, students learn the most basic of Nim games, Static Nim.
Individual Lessons
Lesson 1 - Number Representations
Students learn about the repeated subtraction and repeated division methods for converting a decimal number
N to a numeral in base
b, provided
b is an integer other than ‑1, 0, or 1. Students also learn about the Fibonacci representation, which is a method for representing a numeral as a sum of Fibonacci numbers. The Fibonacci representation will be useful in later lessons in this unit when exploring Nim games.
Lesson 2 - Static Nim
Static Nim is a one-pile game between two players. In this game, the maximum number of tokens that can be removed on each turn remains constant throughout the game. In this lesson, students will learn to represent the positions as the vertices of a directed graph and the moves as the edges of the graph. Also, they will learn that solving a game means finding a partition of the vertices into two sets such that three important properties are satisfied.
Lesson 3 - Optimal Strategies
In static nim, the set of possible move sizes remains the same during the play of the game. In various versions of dynamic nim, the rules are such that the maximum number of counters that can be removed on each turn changes as the game is played. This maximum can depend on the current size of the pile, the number of counters removed on the previous play, or the move number of the game. In this lesson, students will explore the second type, where each move determines the maximum move size of the next move.
