Yale team excels at Putnam student mathematics competition

On a Saturday in December, Yale sophomore Richie Hsiung briefly laid down his fencing foil to deliver a piercing insight into probability density — and came up big for Yale mathematics.

Hsiung, along with teammates Deyuan Li and Andrew Milas, both Yale juniors, finished in the top five in the William Lowell Putnam Mathematical Competition, considered the preeminent mathematics contest for undergraduate college students in the U.S. and Canada.

It was Yale’s best performance in the Putnam exam since 1991.

“The Putnam is not something that you can really cram for in the week, month, or even year before,” said Hsiung, a computer science, mathematics, and philosophy major and member of Yale’s fencing team who had prior experience in math competitions. “Preparing really comes down to truly understanding mathematics and training your mathematical intuition to better see the crux of the problem.

“This intuition takes a very long time to develop.”

More than 3,400 students from 456 institutions took part in the six-hour competition, held at participating campuses across the U.S. and Canada, including Yale. During two three-hour sessions, students worked individually on a total of 12 challenging mathematical problems.

“I can assure you, this exam is really, really difficult,” said Wilhelm Schlag, the Phillips Professor and chair of the Department of Mathematics in Yale’s Faculty of Arts and Sciences. “It is perhaps the most famous university competition in mathematics. Hats off to these young people.”

Although the students work independently on the problems, the competition also recognizes the top-scoring “teams” — prizes are given to the students’ home mathematics departments based on the total score of their three highest scorers.

Hsiung drew on his high school experience with the USA Math Olympiad, the premier competition for math students in the country. Teammates Li and Milas, meanwhile, are computer science and mathematics majors in the School of Engineering & Applied Science (SEAS). The joint major in computer science and mathematics at SEAS is intended for students interested in computational mathematics.

“I’ve been extremely pleased with the broadening of participation in the Putnam exam in recent years at Yale,” said SEAS Dean Jeffrey Brock. “It’s helped build a more inclusive sense of community around mathematics, furthering the Department of Mathematics’ goals but also reinforcing that mathematical thinking and problem solving is truly cross-disciplinary. It’s great to see students at the interface of mathematics and engineering succeed here.”

Hsiung, who ended up among the top 20 in the individual rankings, said the Putnam competition requires innovative thinking and original solutions for the 12 problems that are presented. He credited Yihong Wu, a professor in Yale’s Department of Statistics and Data Science, for providing him with “solid intuition” — during a probability theory class at Yale — that he creatively applied to a Putnam problem by using the idea of probability density functions. (Probability density refers to the density of a continuous random variable.)

Here is a sample problem from the competition:

Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice’s goal is to maximize the number of uncovered squares when the game ends; Bob’s goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?

The correct answer is 290. But as Hsiung noted, reaching the correct answer is only one part of the test.

“Putnam problems require a series of difficult observations, or ‘clicks,’ en route to coming up with a creative solution,” he said. “And this need for originality is what separates these problems from what you’d see in a math class.”

Mihai Alboiu, a lecturer in the Department of Mathematics, organized Yale’s participation in the most recent Putnam event.

Schlag and Brock noted the importance of supporting Yale students from all academic majors who wish to compete in future Putnam exams.

“Yale has an abundance of gifted, well-rounded students, and in my own experience first-rate mathematical talent appears within every major,” Brock said. “I hope our Yale undergrads will continue to find the Putnam a source of community and challenge, and mostly of shared joy in mathematical problem solving.”