Let’s say you somehow manage to sleep through all of the National Collegiate Athletic Association’s March Madness and wake up in a stupor after all 64 college basketball teams in the main tournament have competed head-to-head and a champion was crowned. You stumble downstairs to find your friends’ brackets, in which they predicted which team would win each game, and a list of how many points each of your friends ended up scoring. Is it possible to figure out which team won each game? According to mathematicians, that depends on how many of these friends you have—and how each one filled out their brackets.
March Madness’s main tournament starts by pairing off 64 of the highest-level college basketball teams into 32 games. The 32 winning teams move on to the next round, in which they are paired into 16 games. That is followed by eight games, then four, then two and finally one championship game. Whether in national pools for cash rewards or just for fun among friends, millions of Americans each year fill out templates called “brackets” while they try to predict which team will win each game. For each of the 63 games, bracket makers have two choices for the winner, resulting in 263 possible bracket configurations. That’s more than nine quintillion! So it shouldn’t come as a surprise that a perfect bracket has never been verifiably achieved.
Instead of trying to craft the perfect bracket, Georgia State University mathematician Sam Spiro went in another direction. He asked: Given a collection of brackets scored based on how many games they correctly predicted, can I reconstruct how the tournament actually went?
On supporting science journalism
If you’re enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
Suppose someone got a perfect score on their bracket. Then their predictions were entirely correct, and you know precisely what went down during the tournament. But what if you only have a bracket from some poor, unfortunate soul who scored zero points? (A randomly chosen bracket has a 1⁄232, or one-in-more-than-two-billion, chance of scoring nothing. Pretty impressive!). That bracket tells you the winners of all the first-round games (the opposite of whatever your wonderfully inept friend selected), and you know who played in the second-round games but not much else.
Spiro was inspired to study this while writing a web novel about “a presidential fighting tournament.” Friends started sending him prediction brackets about which former U.S. presidents they thought would win out. “I realized that if I kept telling them how well they scored against each other, eventually I would be revealing the entire tournament…. So I just asked myself, ‘How much information am I giving? At what point, if they colluded, would this determine the whole thing?’” Spiro says. “And then I figured it would be better to pitch this as a March Madness thing instead of about my obscure presidents novel.”
From these silly beginnings came serious novel mathematical results. Spiro found that in any standard, single-elimination tournament with n teams, there exists a set of n⁄2 brackets that can be used to determine the actual results of the tournament games, no matter how the tournament goes. This means that for a 64-team tournament like March Madness, you could strategically choose 32 different brackets beforehand that, once scored, would reveal all the results to you. This is true for any scoring system that awards each game a positive number of points for a correct prediction, so long as you know the scoring system beforehand.
But what if your friends don’t comply with your specially chosen brackets? How many different brackets would you need to ensure you could determine the result of the tournament no matter what brackets are chosen? According to Spiro, this number is harder to pin down, but it’s “very, very large.” He determined a range that depends on how many teams are in the tournament. For March Madness, it’s somewhere between 8.9 quintillion and nine quintillion, which is pretty close to the total number of possible brackets. That’s around one billion times the number of people on Earth. So you’d better get to work at making some new extraterrestrial friends—or maybe you’d better just watch March Madness after all if you want to know how it plays out.
It’s Time to Stand Up for Science
If you enjoyed this article, I’d like to ask for your support. Scientific American has served as an advocate for science and industry for 180 years, and right now may be the most critical moment in that two-century history.
I’ve been a Scientific American subscriber since I was 12 years old, and it helped shape the way I look at the world. SciAm always educates and delights me, and inspires a sense of awe for our vast, beautiful universe. I hope it does that for you, too.
If you subscribe to Scientific American, you help ensure that our coverage is centered on meaningful research and discovery; that we have the resources to report on the decisions that threaten labs across the U.S.; and that we support both budding and working scientists at a time when the value of science itself too often goes unrecognized.
In return, you get essential news, captivating podcasts, brilliant infographics, can’t-miss newsletters, must-watch videos, challenging games, and the science world’s best writing and reporting. You can even gift someone a subscription.
There has never been a more important time for us to stand up and show why science matters. I hope you’ll support us in that mission.
