Pile Splitting Problem: Introducing Strong Induction
Bill Marion · Mathematical Association of America eBooks · 2009
In many textbooks in discrete mathematics there are numerous examples for teaching the Weak Form of the Principle of Mathematical Induction, but relatively few elementary problems for applying the Strong Form. What follows is a nice example to draw on when introducing the strong form. It can be presented as a classic puzzle, it has a number of variants and it is inherently recursive. By introducing the problem (Pile Splitting) as a puzzle, the instructor can engage the students in the process of finding a general solution. She can, then, raise the question as to how they can demonstrate that their conjecture is correct, and, thereby, motivate the need for strong induction. After an induction argument has been presented (the Worksheet includes one such proof), variants of the puzzle can be assigned for the students to work on in class or as a homework assignment. Notes for the instructor To give students practice in making conjectures about the solution to the puzzle, they should be asked to solve it themselves. One hands-on approach that works well is to provide each student with a sufficient number of beads or pennies for her to actively play the game enough times with different values of n so as to see a pattern emerge. Those students who correctly conjecture the general solution can assist the others. One way of reaching the conjecture is explained on the Solutions page.