Some Basic Mathematical Exercises

This chapter begins with drill exercises—those used to drill students until they acquire the necessary skills to move on. Drill exercises can range from the mundane, through word problems designed either to illustrate the situations in which the skills are useful or to provide some entertainment in acquiring the skills. A number of examples are offered, the section on such exercises ending with one of the most fruitful drill exercises ever proposed: Fibonacci’s Rabbit Problem. The second section of this chapter discusses challenge exercises by presenting three problems from an early 20th century mathematical competition along with their solutions. The bulk of the chapter is the discussion of exploratory exercises in which some new concept or tool is introduced and one is encouraged to explore, spotting regularities, formulating conjectures, and proving them. The two main examples offered are the Fibonacci numbers arising from the aforementioned Rabbit Problem and the Tower of Hanoi. The Fibonacci numbers have long been a staple of recreational mathematics because they have so many readily discovered properties, many of which are easily proven. The Tower of Hanoi, also originating in recreational mathematics, has become a standard example, this time in computer science, of solving a problem easily using recursion and not so obviously easily directly. The study of these topics is mildly extensive.