How to find your one true love using mathematics
A guide on using mathematics in our romantic lives.

Every cisgender man in the University of Toronto Mississauga (UTM) has around 68 potential cisgender female dating partners on campus, while every cisgender woman has 54 potential cisgender dating partners. How do I know this?

I used a trick called Fermi estimation to break down the question of “how many potential partners can a cisgender UTM student have” into tinier estimates. The errors in rounding up or down for each estimate cancel each other out, resulting in a fairly close estimate to the actual result.

This technique has been used to estimate the number of intelligent lifeforms in the universe. Mathematician Peter Brackus also used it to calculate how many potential partners he could have in his paper “Why I don’t have a girlfriend?” We will use British mathematician Hannah Fry’s take on Brakus’s method in her book Love and Mathematics.

We start by calculating the number of students at UTM. Based on this source, there are about 17,000 students, with 50 per cent identifying as female, 40 per cent as male, and 10 per cent as other identity groups.

We also need to consider the likelihood that the other person would find us attractive, that we would find them attractive, and how well we would get along. For simplicity, let’s assume there is a 20 per cent chance for each.

Using these tinier estimates we can come up with a formula for total number of partners: (number of students in UTM) * (percentage of students that are the opposite gender) * (percentage of students that we find attractive) * (percentage of students that find us attractive) * (percentage of students we get along with).

Hence there are 17,000 * 0.5 * 0.2 * 0.2 * 0.2 = 68 potential female partners for cis-gendered men (we multiply all the probability estimates together with the total UTM population), and 17000 * 0.4 * 0.2 * 0.2 * 0.2 = 54 potential male for cis-gendered women.

These values can change depending on other factors like how large your search pool is—beyond UTM or more within—or your flexibility in dating other genders.

This is one way of using mathematics to answer the questions of love. In the spirit of Valentine’s Day, let’s look at the different takeaways and approaches to love that math gives us.

When do you find your One True Love?

The media has hopelessly romanticized the concept of one true love (OTL). At the same time, most of us have been disillusioned by the idea of love at first sight. So the question is—how many “sights” does it take to find your OTL? The definitive, mathematical answer is 37 per cent.

This number comes from the “optimal stopping” problem. Let’s say you have to decide to date one person from a total of N people. You can only see one person at a time, and once you reject someone, you can’t choose them again. So, how do you choose the best partner out of all N partners?

The optimal mathematical solution is explained in Algorithms to Live By, by Brian Christian and Tom Griffiths: for the first 37 per cent of candidates, simply observe each candidate, and don’t pick any of them. Then, as you go through the remaining candidates, choose the first one who was better than the initial 37 per cent of candidates you observed.

By following this algorithm, you have a 37 per cent chance of getting the best partner. This might seem low, but think about approaching this with no algorithm and choosing a partner at random. You would only have a 1/N chance to get the best option. As N increases, your opportunity to get the best match dwindles. However, by following the “look then leap” approach, even if N increases, your chance to get the best candidate stays the same. It’s particularly effective the higher N becomes.

But in the dating world, N actually won’t increase that much. In fact, most people will only date two to three partners— within this range, the 37 per cent rule isn’t useful. However, with one small change, we can still make it applicable to love.

Rather than applying optimal stopping over the number of potential partners, we can apply it to the time over which we search for partners. In Algorithms to Live By, we can see Michael Trick, now Senior Associate Dean of Operations Research at Carnegie Mellon, utilized this method as a graduate student to see when he should marry someone.

Assuming he would search for someone to marry from the age of 21 to 40, by the 37 per cent rule, after the age of 26, Trick should marry the next person who is a better match for him than all the people he dated before. In fact, he found someone like that and proposed on the spot. He got rejected.

The optimal stopping problem we’ve discussed thus far doesn’t consider how our partner of interest can reject our proposal. If you recalculate the math with the possibility of rejection, you find that the new optimal place to stop observing is 25 per cent. After meeting the first quarter of potential partners, you should choose the next best person. This will give you a 25 per cent chance to meet our OTL, which is still better than going in blind.

How do you win in a confession?

How do you know when the right time to confess is? This is a question that frequently occupies the mind of lovestruck young adults. There are even anime that revolve around this question; the most notable example is Kaguya-sama: Love is War, where the two protagonists try to “win” at a confession by making the other person confess to them first.

While each situation is different, mathematics does offer an answer in how to reach the most successful outcome possible in a confession through the stable marriage problem.

Let’s say we have two sets of M men and women. Each man has a list of rankings for every woman in terms of what he likes about her as a romantic partner. Similarly, each woman has a list of rankings for every man. The stable marriage problem asks us to pair all men and women together in a way such that no man or woman wants to switch their partners. In the context of the problem, a man or woman would only want to switch partners to someone who ranks higher on their personal ranking scale— and only if that someone prefers them over his/her own partner.

The optimal solution to this problem exists, and it is called the Gale-Shapely algorithm. The algorithm works like this: make one group the proposers, and the other group the receivers. To break cliché, let’s have the group of women propose. 

Every woman proposes to their top male preference. Then, each man who got a proposal tentatively accepts their best female proposal. The women who got rejected propose to their second choices, and the men select their new best female proposal. This process continues until all people are matched up.

In the end, no one will want to switch their partners. No woman will want to switch their male partners, because they were already rejected by all the men higher on their list. No man will want to switch because they have already accepted the best proposal they received. 

The Gale-Shapley algorithm reveals something else: the proposers are better off than the receivers. In the case above, each woman gets the best partner they possibly can, while the men have to make do with the best offer they received.

So, Kaguya-sama is definitely wrong—people who wait to get a proposal are the ones more likely to lose. If you want to be in the best possible relationship for yourself, then you have to seize the day and confess—the only possible way to win.

Mathematics is an excellent way to make sense of human behaviour. This article is only a tip of the iceberg on how to view love and humans with math. If you enjoy this kind of thinking, I suggest reading Love and Mathematics by Hannah Fry.

If you want to go beyond love and look at math in daily life, then Algorithms to Live By would be the best choice for you.