The Power of Creative Mathematical Thinking

Math That Connects Where We’re Going to Where We’ve Been: Recursion Builds Bridges Between Ideas From Across Different Math Classes.

Illustration by Robert Neubecker for Quanta Magazine

Patrick Honner in Quanta: Say you’re at a party with nine other people and everyone shakes everyone else’s hand exactly once. How many handshakes take place?

This is the “handshake problem,” and it’s one of my favorites. As a math teacher, I love it because there are so many different ways you can arrive at the solution, and the diversity and interconnectedness of those strategies beautifully illustrate the power of creative thinking in math.

One solution goes like this: Start with each person shaking every other person’s hand. Ten people, with nine handshakes each, produce 9 × 10 = 90 total handshakes. But this counts every handshake twice — once from each shaker’s perspective — so the actual number of handshakes is 902=45. A simple and lovely counting argument for the win!

There’s also a completely different way to solve the problem. Imagine that the guests arrive one at a time, and when they get there, they shake hands with everyone present. The first person has no hands to shake, so in a one-person party there are zero total handshakes. Now the second person arrives and shakes hands with the first person. This adds one handshake to the total, so in a two-person party, there are 0 + 1 = 1 total handshakes. When the third person arrives and shakes hands with the first two guests, this adds two handshakes to the total. The fourth person’s arrival adds three handshakes to the total, and so on.

This strategy models the sequence of handshakes recursively, meaning that each term in the sequence is defined relative to those that come before it. You’re probably familiar with the Fibonacci sequence, the most famous recursive sequence of all. It starts out 1, 1, 2, 3, 5, 8, 13, 21, and continues on with each subsequent term equal to the sum of the previous two.

More here.