Let’s say you somehow manage to sleep through all of the National Collegiate Athletic Association’s March Madness and wake up in a daze after all 64 major league college basketball teams have competed against each other and a champion has been crowned. You stumble down to find your friends’ brackets, where 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 find out which team won each match? According to mathematicians, it depends on how many of these friends you have – and how each filled out their parentheses.
March Madness’ main tournament kicks off by pairing 64 of college basketball’s highest-level teams in 32 games. The 32 winning teams advance to the next round, where they are paired in 16 matches. Then follows eight games, then four, then two and finally a championship game. Whether in national pools for prize money or just for fun among friends, millions of Americans fill out templates called “bracket” each year as they try to predict which team will win each game. For each of the 63 games, bracket makers have two picks for the winner, resulting in the 2nd63 possible bracket configurations. That’s more than nine quintillion! So it should come as no surprise that a perfect bracket has never been confirmed achieved.
Instead of trying to create the perfect bracket, Georgia State University mathematician Sam Spiro went in a different direction. He asked: Given a collection of brackets scored based on how many matches they predicted correctly, can I reconstruct how the tournament actually went?
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Suppose someone got a perfect score on their bracket. Then their predictions were absolutely correct and you know exactly what went down during the tournament. But what if you only have one bracket from a poor, unlucky soul who scored zero points? (A randomly chosen bracket has a 1⁄232or one-in-more-than-two-billion, chance of not scoring anything. Quite impressive!). That bracket tells you the winners of all the first-round games (the opposite of whatever your wonderfully inept friend picked), and you know who played in the second-round games, but not much else.
Spiro was inspired to study this while writing an online novel about “a presidential match tournament.” Friends began sending him predictions about which former US presidents they thought would win. “I realized that if I kept telling them how well they scored against each other, I would eventually give away the whole tournament… So I asked myself, ‘How much information am I giving? At what point, if they cooperated, would this decide the whole thing?'” says Spiro. “And then I thought it would be better to say this as a March Madness thing rather than about my obscure presidential novel.”
From these silly beginnings came serious new mathematical results. Spiro found that in any standard, single-elimination tournament with n layer, there is a set of n⁄2 brackets that can be used to determine the actual results of the tournament games, regardless of how the tournament goes. This means that for a 64-team tournament like March Madness, you can strategically select 32 different brackets in advance that, once scored, will reveal all the results to you. This applies to any scoring system that gives each bet a positive number of points for a correct prediction, as long as you know the scoring system beforehand.
But what if your friends don’t adhere to your specially chosen brackets? How many different brackets do you need to ensure that you can determine the outcome of the tournament regardless of which brackets are selected? According to Spiro, this number is more difficult to determine, but it is “very, very large”. He determined a range that depends on how many teams are in the tournament. For March Madness, that’s somewhere between 8.9 quintillion and nine quintillion, which is pretty close to the total number of possible brackets. That’s about one billion times the number of people on Earth. So you should start working on making some new alien friends — or maybe you’d rather just watch March Madness if you want to know how it plays out.
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