How can you quickly figure out how many games are needed to determine the winner of a tournament? What are the odds of predicting the winner of each and every game? And how does this explain why your NCAA tournament bracket was so wrong? Keep on reading to find out.
It's that time of year again—when people all across the nation who don't really know or care much about college basketball watch a lot of college basketball. So if you, like millions of other people, have spent a significant number of hours over the past few weeks watching the NCAA tournament, you're in good company.
Because it's March. Wherein there is madness. But is there math in this March Madness? Are there perhaps numerical fun facts waiting to be discovered? Indeed, there are lots of them! And today we're going to talk about three of my favorite—including one that will help you understand why that bracket you spent so much time toiling over turned out so terribly.
Before we get into the nitty-gritty details of the NCAA basketball tournament, I'd like to take a few minutes to talk about tournaments in general. In particular, I'd like to talk about how single-elimination or knockout style tournaments are set up. Because there's some pretty cool math within their structure.
To begin with, barring some fancy finagling with byes or the like, every single-elimination tournament—whether it be the NCAA tournament, the knock-out stage of the World Cup, a major tennis championship, or whatever—starts with an even number of teams. Which makes sense because every team must have another team to play in the first round.
But not only are there an even number of teams, the number of teams must actually be a power of 2 (again, barring some more of that fancy finagling … which, we'll soon see, the NCAA tournament actually contains a bit of). If the tournament takes, say, 4 rounds to determine a winner, we know that there must be 24=16 competitors—since 16 teams in the 1st round beget 8 teams in the 2nd (quarter-final) round which beget 4 teams in the 3rd (semi-final) round which beget 2 teams in the 4th (and final) round determining the winner.
How to Quickly Calculate the Games in a Tournament
This leads us to the question that will give rise to our first numerical fun fact for today: How many games are required to determine the NCAA tournament winner? There are (at least) two ways to think about this: the hard way and the easy way.
Let's start with the hard way. As we've seen, once we know the number of teams in a tournament, we can fairly simply figure out the number of rounds it'll take to determine the winner. Namely, since the number of teams in a tournament must be equal to 2 raised to the power of the number of rounds played, we know that, for example, a 64 team tournament like the NCAA tournament must have 6 rounds—since 26=64 teams. (I know, technically the NCAA tournament starts with 68 teams, but 4 of the low-seeded teams are quickly whittled down in a sort of pre-tournament tournament … leaving us with 64 teams in the main bracket.)