March 23, 2011 Volume 4 Issue 5
. . . an e-conversation with the Faculty of Gordon College . . .
By Richard Stout
It’s time for “March Madness,” that round of college basketball games that never seems to end, but (thankfully) signals the end of winter and the coming of spring. For faithful fans—like me—it can also be an emotional roller coaster where our team plays an inspired game one night, but loses the next on a last-second basket. And the drama is heightened by the fact that if a team loses, it goes home. No second chances. A season has ended.
But even amidst all the emotion of the sport and the drama of the competition, something else emerges: an interesting mathematical problem. At the heart of the tournament is a basic question upon which the madness depends: How many games must be played to determine a winner?
The answer, of course, is related to the number of teams in the tournament. The larger the field (of teams), the more games needed. But what, exactly, is the relationship between the number of games and the number of teams? For this year’s 68 teams, how many games will it take to determine a champion?
One way to solve the problem is to draw a schematic diagram showing how teams move through the tournament from round to round, and then count the number of games. That works, but to find the answer for another tournament we need a new diagram.
Or we can take a more analytical approach and consider how many games are needed in each round. This method would be easier to use if there were 64 teams instead of 68. For instance, with 64 teams we’d need 32 games in the first round, then 16, and so on. Different calculations are needed to account for 68 teams, a method that might not be easy to do—which is part of the challenge, and the fun. And what if a tournament had, say, 138 teams?
Or suppose we generalize the original problem to the following, which seems even more difficult: In a single-elimination tournament with “N” teams, how many games are needed to determine a winner? Although the original problem can be solved with drawings or specific calculations, this more general problem, needs an entirely different approach. It isn’t possible just to make calculations or follow a prescribed procedure. A solution requires a different level of analysis and understanding.
Unfortunately, many people will stop there because they see mathematics as a jungle of unintelligible rules and confusing procedures. But keep pushing and you’ll find mathematics becomes more than just useful laws and procedures. Press on and it reveals an aesthetic quality.
In fact, doing mathematics provides an opportunity to appreciate that beauty. Some see mathematical beauty in the process of uncovering simplicity within a logical framework. Others find it in the way patterns produce enlightenment and understanding. Or, some find beauty from the satisfaction achieved after understanding a concept at a level where you really “know” it is correct.
What does this have to do with basketball tournaments? As a wise professor pointed out to me years ago, in order to produce a winner, it helps first to realize that every team but one must lose a game. If we started with “N” teams then “N minus 1” teams must lose. In order for that to happen, there must be “N minus 1” games. A tournament with 68 teams requires 67 games, and one that starts with 138 teams requires 137 games. It’s that simple. And this solution to the general problem, using such a simple argument, becomes a thing of beauty.
Knowing as much mathematics as possible and being able to apply that knowledge to solve practical problems is crucial in today’s complex world. Still, it’s not enough. As an educator I hope others will also see the beauty in what we are doing. When I ask students why a particular statement is true, I sometimes stop and write “BMIB”—Because Mathematics is Beautiful—simply to emphasize that there is an aesthetic quality in valid mathematics.
Consequently, we make calculations or construct arguments that help us understand how a basketball tournament works. But we’re doing more than that. We’re also trying to understand something that lies beyond our sensory world, to access an abstract but ordered universe that many mathematicians believe has an existence of its own. And the opportunity to glimpse that universe can be a beautiful experience indeed, a little like watching the last three point shot swish to win the game.
Richard Stout is professor of mathematics at Gordon College in Wenham, MA. He and his wife live in Ipswich, MA.