NCAA Tourney and Math

In the past 10-15 years the NCAA Basketball tournament has become a popular item of interest to students, particularly when their favorite team is in the tournament. I found that even kids who weren't normally interested in basketball were talking about it and some of them even would join their classmates (or parents)in picking the outcomes of the games. I wanted to take advantage of this natural interest and come up with some math questions related to the whole thing.

What I ended up doing was pretty simple but turned out to work very well and certainly was interesting to the students. I made a transparency of the full set of brackets with all the teams listed. I showed that to the students and explained the basic idea that a team continued to play as long as they won. (The NCAA actually invites 65 teams into the tournament. Two teams play an early "play-in" game and that narrows the field to 64 teams.) I ask the students to tell me how many total games will then be played, after the "play-in" game, in the tournament to determine a winner. It's the type of question that most, if not all, kids don't already know. Neither do many adults.

When I first thought of this question, I solved it by doing what most people do when they hear this question. I looked at the bracket sheet and counted the first round games -quickly realizing that there had to be 32 games in the first round because there are 64 teams (after the "play-in" game). The next round would be 16 games (because there were 32 winners n the first round), then the third round would have 8 games and the fourth round (regional final) would have four games. Those four games produce the "Final Four." Of course then you have two games in the national semi-finals and then the final game for the championship.

32+16+8+4+2+1 = 63. (I prefer not to count or sometimes even mention the "play-in" game because it isn't usually shown in the brackets that kids see. That way they can count all of the games one by one if they don't see any faster way.) When I first did this question, many years ago and saw the answer "63" I quickly realized that I had not used the most efficient method. When you have 64 teams in a single elimination tournament, 63 teams have to be eliminated in order for the champion to be determined. Each game eliminates one team so 63 games need to be played to eliminate 63 teams. An even better way to look at it is that all teams have to be eliminated except one, so 64-1=63.

Most kids and adults can easily figure out by the original counting method, that the tournament with 64 teams requires 63 games to determine the winner. They usually don't notice or think about the shortcut. Therefore when I present the original question I also present an additional question. Almost every year there are people who say that all of the NCAA (Division 1) teams should participate, regardless of record, just like they do in high school. In a given year I wouldn't always know exactly how many teams were in Division 1 so I would tell the students to assume that there would be 200 teams and to figure out how many games would need to be played. Assuming they didn't know the shortcut mentioned above, they would normally try to solve it just like they had the original question. With 200 teams there would be 100 games, then the second round would have 50 games, the third round 25 games and then they would get stuck. Very few, if any, students or adults, would be able to figure it out from there. What do you do with an odd number of teams?

So the next day when I asked them how they solved the original 64 team question (after they passed their papers in) most would have the correct answer but would not have seen the shortcut. I can't remember any of my 8th graders getting the 200 team question right because they always got stuck with the odd number of teams. At this point, if I had the time, I could suggest two problem solving strategies - solving a simpler problem and finding a pattern along with making a list.

I start with a tournament that has only one team which I know is ridiculous but it serves the strategy. Obviously with only one team, zero games need be played. If there are two teams, you need one game. When you get to three teams, you show them that the only way to do it is to have two teams play and the winner of that game plays the other team for the championship. So three teams requires two games. Four teams (like the "Final Four") needs three games. As this list is being made some kids will see the pattern that every time you add a team you add a game. That is true but if that is all they see then a tournament with 200 teams is going to require making a long list. Eventually somebody usually does see the most efficient method - just subtract one from the number of teams. They can then instantly determine that if the tournament had 200 teams that 199 games would be needed. If the tournament had 35 teams, then 34 games are played.

If you have any other additional time (highly unlikely) you can add in the concept of powers of two. When you have a single elimination tournament, you have to have enough "play-in" games to cut the field down to a number of teams that is a power of two. So a tournament with 35 teams would require three "play-in" games to cut the field to 32 teams. In fact you could ask the students at the beginning why the NCAA field (after the play-in game)is 64 teams. Why not 60 teams or 80 teams or 50 teams?

This was a great problem solving situation that took advantage of the natural interest that students and adults have in the NCAA tournament. One of the things I used to always say to my students is that, "A good mathematician is a lazy one." This particular situation is a perfect example of that.