The Math Exam That Stumps Literal Geniuses

0:00 This right here is the Putinham, one of the hardest math exams in the world.

0:03 The median score on this exam is a two

0:06 out of 120 and the average score an eight.

0:10 The Putnham is a math competition that 4,000 college

0:13 students from across the US and Canada take each year.

0:15 They have 6 hours to answer 12 problems in categories such as calculus,

0:19 number theory, and linear algebra.

0:21 The exam is known for its deceivingly hard questions.

0:24 Although some seem straightforward,

0:26 most of them require an exceptional degree of intuition and creativity.

0:30 The top five individual scorers are known as the Putinham Fellows,

0:33 a lifelong title in the math community.

0:36 And the school with the number one team wins $25,000.

0:39 In recent years, the school that has dominated

0:41 the rankings is none other than my alma mater, MIT.

0:45 Not only did they place first every year from 2024 to 2021,

0:49 but it also had all five Putnham fellows.

0:52 So, who are these students crushing the Putinham?

0:54 How do they train?

0:56 And most importantly, what makes the exam so difficult?

0:59 Right now, we're in building 2, which is home to the MIT math department.

1:03 We're soon about to head up to meet with Dr.

1:04 Henry Conn, a mathematician specializing in discrete math,

1:08 coding theory, and combinatorics.

1:10 In other words, he's brilliant, which makes him the perfect fit for running

1:13 a course known as 18A34 or the Putinham Seminar.

1:17 This course trains firstear undergrads for the platinum competition,

1:20 but it's not open to everyone.

1:21 This course is only intended for those with previous

1:24 competition experience and we're about to get an inside look.

1:27 So, okay, what we're going to be talking about today is number theory,

1:31 and this is something that doesn't come up that much on the exam.

1:34 You don't need to have taken a course

1:36 in number theory in order to do these problems.

1:39 As Professor Cohen delivers today's lesson, I see students scribbling notes,

1:43 collaborating on problems, and asking each other questions.

1:46 Meanwhile, I'm just in the back wondering what all the squiggles mean.

1:48 But the students in this room know a thing or two about tricky math problems.

1:53 So I did like the AMC Amy.

1:55 I did USAMO as well and then also

1:58 some smaller math competitions like statewide on the side.

2:00 I did EGMO which is the European girls math olympiad

2:04 and also MIMO which is the middle European mathematical olympiad.

2:08 I did the IMO for 2 years and now it's

2:12 like the natural college version of everything that I've done before.

2:16 And then if you don't mind me asking, how did he perform in the IMO?

2:20 I got two gold medals.

2:22 These are freshmen taking the Putinham for the first time tomorrow.

2:25 Yet, they don't seem that nervous at all.

2:27 Maybe it's because of all the insane

2:29 practice they've gotten through the seminar.

2:31 So, it's a bunch of problems, like usually 40 or something.

2:33 So, we solve like maybe five of those on our own,

2:36 and then we write up solutions for a few of them, and then on Wednesdays,

2:39 we have an hour where we each present a problem that we really liked.

2:42 But what does a Putin problem actually look like?

2:44 The thing is, they're hard.

2:46 If you sort of take a look at the platinum and say, "Oh,

2:48 yeah, I can solve all 12 of these problems quickly, you're deluding yourself.

2:53 And if your first attempt ends up in failure,

2:57 it doesn't mean you're not cut out to be a mathematician.

3:00 I mean, it's a skill that you develop over time."

3:03 Unlike many math competitions which test what you know,

3:05 the Putinham tests how you think.

3:08 I did a lot of math competitions in high school as well,

3:11 but I think the Putinham is just different.

3:13 like you have to think very deeply for them.

3:16 Students get six hours to answer 12 questions worth 10 points each.

3:20 The questions span topics such as algebra,

3:22 number theory, combinatorics, geometry, and real analysis.

3:26 Simple stuff, you know.

3:27 The exam is split across two sessions,

3:29 session A and session B with questions A6 and B6 being notoriously difficult.

3:35 The tricky part about the exam is

3:37 that students can't brute force these problems.

3:39 Each one requires a creative almost unexpected insight.

3:43 I was on the committee that writes the problems from 2014 to 2016.

3:48 Part of the thing is you want the problems

3:51 to be different from what people have seen before.

3:55 In 2022, um I saw problem B5,

3:58 which is like a problem about like combinatorics and probability.

4:01 My solution involved like some like integration.

4:04 So it's like very unexpected and I I'm glad I was able to pull that off.

4:08 This is Mark, a TA for the Putinham seminar.

4:11 The Putinham problem he mentioned is this one right here.

4:13 Pause to take a look.

4:15 Wow, you paused for that long.

4:16 Well, I don't blame you.

4:18 So, let's break down what this question asks in simpler terms.

4:21 Let's say you have a handful of unfair coins.

4:23 They're weighted so they mostly land on tails and rarely land on heads.

4:27 The twist is that you can write any number you want on these coins.

4:30 A five on the first, a -10 on the second, a th00and on the third, and so on.

4:35 You then flip all the coins at once and add

4:38 up all the numbers on the coins that come up heads.

4:40 The question here is how rare does heads have to be

4:44 so that no matter what numbers you write on the coins,

4:47 the most likely final score is zero.

4:50 Now, if you're stumped, don't worry because roughly 15 out of 3,400

4:54 test takers received full points on this question.

4:57 The fact that Mark was able to discover a calculus-based approach

5:00 to this problem is incredibly impressive

5:03 considering the official solution has zero calculus.

5:06 I think the most important thing is that like

5:08 look for like small bits of the problems.

5:11 For example, like if a problem has small cases you

5:13 can play with, you can just like try to play with.

5:15 If the problem ask you for the answer,

5:17 try to guess what the answer look like before attempting to prove it.

5:20 That will be like helpful.

5:22 Persistence is really definitely necessary because a lot of these problems

5:25 like the Putinham style problems you'll try lots of different techniques

5:28 and like none of them work and maybe it's like you

5:31 know the fifth strategy you try is the one that works.

5:33 So it's important to just keep trying strategies

5:36 and not getting discouraged because a lot of these problems

5:39 the solution kind of hinges on making some clever connection

5:42 between one topic and another that doesn't initially look related.

5:45 But also it happens very often that I can kind of like dig into my mind

5:50 and find something that's similar and just have

5:52 a general idea and which direction to start,

5:55 what to try cuz we have a lot of tools.

5:57 So sometimes it reminds me of one of them and I just try it.

6:00 And once you see like one path, you get this like jolt of like understanding

6:05 the whole thing and that feels really nice.

6:09 Speaking of having the right tools in your arsenal,

6:11 let me tell you about the sponsor of this video, Manis AI.

6:14 Manis is the world's first general AI agent.

6:17 And I want to be clear about what that means.

6:19 It's not a chatbot.

6:20 It doesn't just give you instructions.

6:22 It actually does the work.

6:24 You give it a goal and it autonomously plans your steps,

6:27 organizes your tasks, tracks the deadlines, and hands you a finished output.

6:31 So, watch this.

6:32 I'm going to type in I'm a college student juggling five classes,

6:36 a research project, and three club commitments.

6:38 Build me a dashboard to track everything.

6:41 And just like that, Manis gets to work.

6:43 It's not asking a dozen follow-up questions.

6:46 It's planning, building,

6:47 and within a few minutes gets back to me with a fully functional dashboard.

6:51 This has deadlines, reminders, and everything in one place.

6:54 Now, that's the difference between a tool that just talks and one that delivers.

6:58 I've been genuinely impressed by how far it goes without me having to guide it.

7:02 It's the kind of tool I could have used in college, especially around finals.

7:05 Check out Manisai through the link in my description.

7:08 The more I speak with the students,

7:10 the more I realize the way they approach the platinum problems is how I approach

7:14 problems in middle school algebra and high

7:16 school calculus and in college level discrete math.

7:18 Although it feels like they're using wizardry to tackle these problems,

7:22 it comes down to incessant practice and an intuition that they build over time.

7:27 It's not like people are sort of born

7:29 with mathematical techniques baked into their brains.

7:32 If you've never thought about math before, you're going to be in over your head.

7:37 Not because you're not talented or don't have a lot of potential,

7:40 but just because you've got to get used to these ideas and how

7:44 they fit together and get comfortable

7:45 manipulating these sorts of abstract objects.

7:48 In order to build up your intuition,

7:50 you need to practice a lot and persist through your practice problems

7:56 and that eventually builds your intuition which

7:58 you can apply to like more problems.

8:00 So, it's kind of a loop and that loop leads to these results right here.

8:03 This sheet of paper is the Putinham announcement

8:05 of winners for 2024 and what inspired this entire video.

8:10 Actually, it wasn't just this.

8:12 I was also inspired by the announcement of winners for 2023, 2022, 2021.

8:17 I think you get the point.

8:19 This announcement reveals the highest scoring individuals and teams

8:22 with teams consisting of each school's top three scorers.

8:26 The top five individuals are known as the Putnham Fellows and receive 2500 each.

8:31 And the institution with the top team receives $25,000.

8:36 Okay?

8:36 And get this, MIT's team has not only come in first for the past 5 years,

8:41 but it has also had all five Putnham fellows,

8:45 one of the most honorable distinctions in the math community.

8:48 students would be lucky to receive this distinction once,

8:50 but three current MIT students have received it three times.

8:55 In 2024, MIT also had the top scoring woman who finished

8:59 in the top 25 and earned the $1,000 Elizabeth Lel Putnham Prize.

9:04 And if all of this isn't crazy enough, that same year,

9:07 69 MIT students scored within the top 100 among nearly 4,000 contestants.

9:13 In 2022, I got top 25.

9:15 And in 2020, I did not as well.

9:17 So, I only in top 100.

9:19 And then 2024, I was a bit lucky.

9:21 I got top 15.

9:22 To be in the top 15 in 2024, you had to score a 71 or higher out of 120.

9:28 And the highest score that year was a 90 with the median being a two.

9:33 Mind you, the students taking this exam are often math

9:35 majors brave enough to sign up in the first place.

9:38 I think just participating is a huge accomplishment in and of itself.

9:42 But what makes MIT particularly special?

9:45 Lots of universities have seminars and it's not like MIT has some sort

9:49 of secret sauce that makes the MIT seminar different or better than other ones.

9:55 I think partly it's the students that come

9:57 to MIT and partly also the synergy between students.

10:01 If you're in an environment where lots of other people are doing something,

10:05 you're encouraged to do it yourself.

10:07 I did Matt Matt olympiad before like I was

10:11 from Thailand so I represented Thailand in the international

10:14 matt Olympiad and then yeah I come to MIT

10:17 and I think Patnham is like a nice continuation.

10:20 I started the EMC8 in fourth grade and after

10:24 that I've just been always on the math competition track.

10:29 So it was kind of natural for me like cuz I'm from Hungary so

10:32 national competitions kind of like grew into like

10:34 international Olympiads as I was in school.

10:36 But to quote Richer Agarwal, president of the Global Talent Fund,

10:40 recruiting the most talented students is only the beginning.

10:43 A top tier university education with excellent professors, supportive mentors,

10:48 and an engaging peer community is key to unlocking their full potential.

10:52 I think partly of it as a celebration

10:55 of mathematical problems that it's a chance for thousands

11:00 of people to sit down in December and think

11:02 about some really fun problems and enjoy the experience.

11:07 Does doing well in the Putinham have its material benefits?

11:10 Of course, you get the title, the resume boost, and even a cash prize,

11:14 but these aren't the primary incentives that draw students to the competition.

11:17 With math competitions, I think the main thing is to try to find joy in them.

11:23 Honestly, I just think it would be fun.

11:24 Like, it's fun math problems and a lot of them have like really cool solutions.

11:28 So, it's just an excuse to spend a day doing fun, interesting math problems.

11:33 It makes me really sad when I see people who feel very

11:37 stressed out about this, feeling like

11:40 their intellectual worth is being judged here,

11:43 or worse yet, their value as a human being, which is ridiculous.

11:47 I'm not like super stressed cuz I view it as like

11:50 a opportunity to like do math and discuss with other people.

11:54 There were like a lot of people at MIT who were taking platinum.

11:56 That's like really large community.

11:58 So, it's fun to like be engaged and being part of it.

12:01 Instead, the important thing is to enjoy the beauty of mathematics,

12:06 the beauty of surprises, the way puzzles get under your skin.

12:11 And doing well on contests has its rewards.

12:14 It can get you recognition or prizes, but on the other hand,

12:18 it's a pretty low hourly rate of return

12:21 for the effort of actually learning mathematics

12:24 that basically if you want to do well

12:27 on contests because you want to be a winner,

12:30 there are probably easier ways to be a winner in life.

12:33 And then if it leads to rewards and recognition, that's great.

12:38 And if it doesn't, at least it's a springboard to success

12:42 doing or using mathematics in ways that go well beyond contests.