The Mathematics of Ultimate Frisbee

Ultimate frisbee is a sport of choices. Who do you pass the disc to? Who do you defend? What is the optimal way to score — and win? While making a choice in ultimate frisbee might seem entirely subjective, a number of mathematical principles, including probability, counting, and angles, influence the game. 

To begin an ultimate frisbee contest, the captains of both teams meet for a disc toss to determine the starting position. The captains typically do a double disc flip: a captain from each team flips a disc in the air to see where it lands. Prior to the toss, each team selects “odds” or “evens.” The term “odds” means that both flipped discs land in opposing directions, and the term “evens” means that either both discs land facing up or both discs land facing down. The winner of the toss gets to select which team starts on offense and which side of the field they want to begin on. Arguably, the winner of the toss should want to start on offense in order to give the team the opportunity to score on their first position and take an early lead before the opponent has the opportunity to score. The double disc flip can, thus, put the team who won the toss in a favorable position right at the outset of the game. 

Four different possibilities can happen from the double disc flip: both discs land facing up, both discs land facing down, one disc lands facing up while the other disc lands facing down, and the opposite of the former. “Evens” wins if either of the first two possibilities happens, and “odds” wins if either of the last two possibilities happens. Since there are four total possibilities, and each of the two teams has two ways to win, this is fair, right?

No, it is not. The double-disc flip is actually not fair! There is a slightly higher probability that a frisbee lands facing up than facing down, which increases the probability of having two frisbees land facing up, increasing the probability of “evens” winning. While the difference is not too large, 50.32% for “evens” and 49.68% for “odds,” there is a slight advantage for choosing “evens” in a double disc flip to win the toss. “Evens” could ultimately lead to having the first possession of the disc and, therefore, the opportunity to score first.

Once the game begins, there are still many choices to make on the field — especially for the team that lost the toss and ends up on defense. One of them is choosing a defensive configuration. Many teams elect to use “person defense,” where each of the seven players is responsible for defending one of the seven players on the opposing team. I am currently taking MATH 39 (Discrete Mathematics with an Introduction to Proof), and we spent an entire one-third of the course on counting. It wasn’t until a few weeks ago in class that I truly realized how many ways “person defense” can be set up. Typically on the field, we choose which player from the other team we want to defend before the next point starts, but sometimes, we switch to “person defense” while we are actively playing and have very little time to decide. Since there are seven players on a team, there are 7! ways to decide who defends who. But what does 7! even mean?

Typically, the fastest person on the field gets to choose who they want to defend first. That person has seven possible players to choose from. The next fastest defender then has six options to choose from: they can choose any of the remaining six players that the first defender did not choose. The next player then has five options, the next player has four, and this continues until the last player to find their defensive matchup has only one option to choose from. Using multiplication, there are 7 x 6 x 5 x 4 x 3 x 2 x 1 (or 7!) possible ways to choose who to defend, which totals to 5040 possibilities. When I’m on the field, my teammates are often yelling, “Asha, find your person!” while I’m looking for the last available player to defend; since there are so many possible ways to set up a “person defense,” this takes time! Sometimes, we plan ahead who we are going to defend before we take the field, but sometimes, someone forgets and ends up defending the wrong person on the other team. This throws off the person who was originally going to defend that player, and potentially the rest of the team. In MATH 39, we learned a formula that calculates derangements, which would be the probability that no player ends up with the player they were originally going to defend, which happens if a mistake is made. If we calculate D7 = 7!(1-11!+12!-13!+14!-15!+16!-17!), there are 1854 ways that everyone could end up defending the wrong player, and out of all 5040 possibilities, there is a chance this could happen 36.79% of the time. 

There are also choices regarding which teammate to throw the disc to, which can also be counted. An offensive player with a disc has six teammates they can throw to, but not every teammate may be able to break away from their defender to receive a disc. If an offensive player does receive a throw, they also have six options to throw the disc to. However, they might look at different possibilities than the person who threw the disc to them. They may, instead, throw the disc further down the field. If a disc is thrown down the field in three throws, with the first thrower having six options, the second thrower having six options, and the third thrower having six options, there are a total of 63, or 216, possible exchanges that can happen over the course of three throws. In reality, there may not be as many options depending on where the disc is on the field and how good the defense is, but in a sport that feels like there are so few options, there are actually so many. 

Another important mathematical aspect of frisbee is angles. We often practice breaking away from our defenders while on offense to receive a throw, and most ways do not involve running in straight vertical or horizontal lines. Instead, we often run at an angle. Defenders usually prevent offensive players from making a pass along a straight line. Players have to run along acute angles when running close to a thrower to receive the disc, or along obtuse angles when running deep to receive the disc down the field. As an ultimate frisbee player, I often don’t think about which angles I am taking to get in position to receive the disc, but naturally, as I try to break away from my defender, I end up choosing to make an angled run to receive the disc deep from the thrower. 

Like many other players, I don’t think about math while I’m on the field. I just let it happen and hope that I end up finding a player to defend, but the more I think about it, the better player I have become. There are so many possibilities that can happen in a game, but as each choice gets made, we have to immediately be ready for the next choice with little time to think about the mathematics even though it surrounds us. 

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