The Math Exam That Stumps Literal Geniuses

This right here is the Putnam, one of the hardest math exams in the world. The median score on this exam is a 2 out of 120, and the average score? An 8. The Putnam is a math competition that 4,000 college students from across the US and Canada take each year. They have 6 hours to answer 12 problems in categories such as calculus, number theory, and linear algebra. The exam is known for its deceivingly hard questions. Although some seem straightforward, most of them require an exceptional degree of intuition and creativity. The top five individual scorers are known as the Putnam Fellows, a lifelong title in the math community, and the school with the number one team wins $25,000. In recent years, the school that has dominated the rankings is none other than my alma mater, MIT. Not only did they place first every year from 2024 to 2021, but it also had all five Putnam fellows. So who are these students crushing the Putnam? How do they train? And most importantly, what makes the exam so difficult? Right now we're in Building 2, which is home to the MIT Math Department. We're soon about to head up to meet with Dr. Henry Cohn, a mathematician specializing in discrete math, coding theory, and combinatorics. In other words, he's brilliant, which makes him the perfect fit for running a course known as 18A34, or the Putnam Seminar. This course trains first-year undergrads for the Putnam competition, but it's not open to everyone. This course is only intended for those with previous competition experience, and we're about to get an inside look. So, okay, what we're going to be talking about today is number theory. And this is something that doesn't come up that much on the exam. You don't need to have taken a course in number theory in order to do these problems. As Professor Cohn delivers today's lesson, I see students scribbling notes, collaborating on problems, and asking each other questions. Meanwhile, I'm just in the back wondering what all the squiggles mean. But the students in this room know a thing or two about tricky math problems. So I did like the AMC 8. I did USAMO as well. And then also some smaller math competitions like statewide on the side. I did EGMO which is the European Girls Math Olympiad and also MEMO which is the Middle European Mathematical Olympiad. I did the IMO for two years and now it's like the natural college version of everything that I've done before. And then if you don't mind me asking, how did you perform in the IMO? I got two gold medals. These are freshmen taking the Putnam for the first time tomorrow. Yet they don't seem that nervous at all. Maybe it's because of all the insane practice they've gotten through the seminar. So it's a bunch of problems, like usually 40 or something. So we solve like maybe five of those on our own, and then we write up solutions for a few of them, and then on Wednesdays we have an hour where we each present a problem that we really liked. But what does a Putnam problem actually look like? The thing is they're hard if you sort of take a look at the Putnam and say, oh yeah, I can solve all 12 of these problems quickly. You're deluding yourself. And... If your first attempt ends up in failure, it doesn't mean you're not cut out to be a mathematician. I mean, it's a skill that you develop over time. Unlike many math competitions which test what you know, the Putnam tests how you think. I did a lot of math competitions in high school as well, but I think the Putnam is just different. Like, you have to think very deeply. Students get six hours to answer 12 questions worth 10 points each. The questions span topics such as algebra, number theory, combinatorics, geometry, and real analysis. Simple stuff, you know? The exam is split across two sessions, session A and session B, with questions A6 and B6 being notoriously difficult. The tricky part about the exam is that students can't brute force these problems. Each one requires a creative, almost unexpected insight. I was on the committee that writes the problems from 2014 to 2016. Part of the thing is, you want the problems to be different from what people have seen before. In 2022, I solved problem B5, which is a problem about combinatorics and probability. My solution involved some integration, so it's very unexpected and I'm glad I was able to pull that off. This is Mark, a TA for the Putnam seminar. The Putnam problem he mentioned is this one right here. Pause to take a look. Wow, you paused for that long? Well, I don't blame you. So let's break down what this question asks in simpler terms. Let's say you have a handful of unfair coins. They're weighted so they mostly land on tails and rarely land on heads. The twist is that you can write any number you want on these coins. A five on the first, a negative ten on the second, a thousand on the third, and so on. You then flip all the coins at once and add up all the numbers on the coins that come up heads. The question here is, how rare does heads have to be so that no matter what numbers you write on the coins, the most likely final score Now, if you're stumped, don't worry, because roughly 15 out of 3,400 test takers received full points on this question. The fact that Mark was able to discover a calculus-based approach to this problem is incredibly impressive, considering the official solution has zero calculus. I think the most important thing is to look for small bits of the problems. For example, if a problem has small cases you can play with, you can just try to play with. If the problem asks you for the answer, try to guess what the answer looks like before attempting to prove it. That would be helpful. Persistence is really definitely necessary because a lot of these problems, like the Putnam style problems, you'll try lots of different techniques and none of them work. And maybe it's like the fifth strategy you try is the one that works. So it's important to just keep trying strategies and not getting discouraged because a lot of these problems the solution kind of hinges on making some clever connection between one topic and another that doesn't initially look related but also it happens very often that i can kind of like dig into my mind and find something that's similar and just have a general idea and which direction to start what to try because we have a lot of tools so sometimes it reminds me of one of them and i just tried And once you see like one path, You get this like jolt of like understanding the whole thing and that feels really nice. Speaking of having the right tools in your arsenal, let me tell you about the sponsor of this video, Manus AI. 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Check out Manus AI through the link in my description. The more I speak with the students, the more I realize the way they approach the Putnam problems is how I approach problems in middle school algebra and high school calculus and in college level discrete math. Although it feels like they're using wizardry to tackle these problems, It comes down to incessant practice and an intuition that they build over time. It's not like people are sort of born with mathematical techniques baked into their brains. If you've never thought about math before, you're going to be in over your head, not because you're not talented or don't have a lot of potential, but just because you've got to get used to these ideas and how they fit together and get comfortable manipulating these sorts of abstract objects. In order to build up Your intuition, you need to practice a lot and persist through your practice problems, and that eventually builds your intuition, which you can apply to like more problems. So it's kind of a loop. And that loop leads to these results right here. This sheet of paper is the Putnam announcement of winners for 2024. And what inspired this entire video? Actually, it wasn't just this. I was also inspired by the announcement of winners for 2023, 2022, 2021. I think you get the point. This announcement reveals the highest-scoring individuals and teams, with teams consisting of each school's top three scorers. The top five individuals are known as the Putnam Fellows and receive $2,500 each, and the institution with the top team receives $25,000. Okay, and get this, MIT's team has not only come in first for the past five years, but it has also had all five Putnam Fellows, one of the most honorable distinctions in the math community. Students would be lucky to receive this distinction once, but three current MIT students have received it three times. In 2024, MIT also had the top scoring woman who finished in the top 25 and earned the $1,000 Elizabeth Lowell Putnam Prize. And if all of this isn't crazy enough, That same year, 69 MIT students scored within the top 100 among nearly 4,000 contestants. In 2022, I got top 25, and in 2023 I did not as well, so I was only in the top 100. And in 2024, I was a bit lucky and got top 15. To be in the top 15 in 2024, you had to score a 71 or higher out of 120. And the highest score that year was a 90 with the median being a 2. Mind you, the students taking this exam are often math majors brave enough to sign up in the first place. I think just participating is a huge accomplishment in and of itself. But what makes MIT particularly special? Lots of universities have seminars and it's not like MIT has some sort of secret sauce that makes the MIT seminar different or better than other ones. I think partly it's the students that come to MIT and partly also the synergy between students. If you're in an environment where lots of other people are doing something, you're encouraged to do it yourself. I did a Math Olympiad before. I was from Thailand, so I represented Thailand in the International Math Olympiad. And then I come to MIT, and I think Putnam is a nice continuation. I started the AMC 8 in fourth grade. And after that, I've just been always on the math competition track. So it was kind of natural for me, like, because I'm from Hungary, so national competitions kind of, like, grew into, like, international olympiads as I was in school. But to quote Ruchira Agarwal, president of the Global Talent Fund, recruiting the most talented students is only the beginning. A top-tier university education with excellent professors, supportive mentors, and an engaging peer community is key to unlocking their full potential. Partly it is a celebration of mathematical problems that it's a chance for thousands of people to sit down in December and think about some really fun problems and enjoy the experience. Does doing well in the Putnam have its material benefits? Of course. You get the title, the resume boost, and even a cash prize. But these aren't the primary incentives that draw students to the competition. With math competitions, I think the main thing is to try to find joy in them. Honestly, I just think it would be fun. It's fun math problems, and a lot of them have really cool solutions. So it's just an excuse to spend a day doing fun, interesting math problems. It makes me really sad when I see people who feel very stressed out about this feeling like, Their intellectual worth is being judged here, or worse yet, their value as a human being, which is ridiculous. I'm not super stressed because I view it as an opportunity to do math and discuss with other people. There were a lot of people at MIT who were taking part in them. It's a really large community, so it's fun to be engaged and be a part of it. Instead, the important thing is to enjoy the beauty of mathematics, the beauty of surprises. The way puzzles get under your skin, and doing well on contests has its rewards. It can get you recognition or prizes. But on the other hand, it's a pretty low hourly rate of return for the effort of actually learning mathematics. That basically, if you want to do well on contests because you want to be a winner, there are probably easier ways to be a winner in life. And then, If it leads to rewards and recognition, that's great. And if it doesn't, at least it's a springboard to success doing or using mathematics in ways that go well beyond contests.