March Mathness: fun facts on college basketball from the world of numbers

College Park, MD (March 18, 2004)--It's that crazy time of year again--the flurry of basketball activity known as "March Madness," in which U.S. college basketball teams compete to win the NCAA Division I Basketball Tournament. You can enjoy March Madness in new ways with some fun facts from the world of math:

How many games are there in March Madness? Count the Losers.

The number of games and teams gets dizzying after a while--63 seeded teams plus two "play in" teams that compete in an extra game to get the 64th spot.

But perhaps to make sure the insanity only lasts a month, March Madness is a single-elimination tournament-meaning that if a team loses a game, it's out of the tournament.

To figure out the number of games in the tournament, you can simply count the games in each round and add them up:

33 (includes the extra "play-in" game) + 16 + 8 + 4 + 2 + 1 = 64

But mathematicians suggest another way to do it: count the number of losing teams.

According to Mike Breen of the American Mathematical Society in Providence, RI, the single-elimination nature of the tournament means that there is one game for every losing team--something known as a "one-to-one correspondence" in math.

"Since only one team wins the entire tournament," Breen said, "sixty-four lose and so there are 64 games in the tournament."

What if March Madness were a double-elimination tournament, in which a team had to lose twice to be kicked out? Well, it would be a much longer tournament-128 or 129 games, depending on whether the eventual winner lost one game or none-and it would extend well into April. So March Madness would turn into April Absenteeism--since college basketball students who won too much might start missing too many classes.

How many ways can a bracket be filled out?

If you're in an office or online pool, you'd fill out a "bracket," a tournament chart that shows all the teams (see this year's example). In a bracket, you pick the teams that you think will win each game. You can fill out a second bracket if you want to make a second guess at the outcome of the tournament. Can you write down all possible outcomes in the tournament to be assured of winning the pool?

"Not unless you write very fast and get lots of help," said Breen.

Since there are 64 games and two possible outcomes for each game-a win and a loss-the number of possible outcomes for the tournament is a staggering 2 to the power of 64--2 multiplied by itself 64 times--or 18,446,744,073,709,551,616 if you want to spell it out.

With that many possibilities, every man, woman, child and baby on the planet could fill out 2.8 billion brackets--each of them completely unique--and still not exhaust the possibilities.

Want to guarantee winning your office pool? If you put in a dollar for each of those possibilities, that would be equal to paying off the U.S. National Debt (about $7 trillion as of March 2004) 26 million times.

Forget about winning for a moment. How about writing out all the possibilities? Even if you could write all your predictions for one bracket in one second, it would take over 500 billion years to write all the possibilities--much longer than the age of the universe. If you could get a billion close friends to help you and each of them could fill in one bracket per second, then you would be done in about 500 years--the 26th century, said Breen.

"It is doubtful that you can get the person who runs the office pool, the NCAA, CBS, and the players to wait that long," Breen said. "Better to take your chances and predict rather than trying to exhaust all possibilities (and your billion friends)."

How can you spot an upset in the tournament?

You don't have to be a college basketball expert to identify an upset--you can just use a little easy math.

At the beginning of the tournament the teams are bunched into four groups of 16. Teams in each group are given seeds, from one to 16, with 1 being the seed for the top-ranked team and 16 being the sixteenth-ranked team for each group.

How do you determine an upset? Simply add together the seed numbers for each game. In the full first round, the seed numbers in each game always add to 17. If the higher seeds win every game in the first round, the seeds in each game in the second round will add to 9…and so on.

If there were no upsets in the entire tournament, you have the following numbers:

"Unfortunately, there is no mathematical way to predict an upset before it happens," said Breen.

But after upsets happen, he pointed out you can spot them by noting the round of the game and adding up the sum of the seeds of the two opponents. When the sums are higher, that means there was an upset in the previous round or earlier.

More March Madness math

# # #

Bill Butterworth of Barat College of DePaul University, IL and Mark McFarling of the American Institute of Physics Statistical Research Center contributed to this report.