Looking to keep your mind sharp and flexible? Try a math problem.

There is a beauty to math, when an equation's complex knot unravels to reveal a clear solution that, in retrospect, seems absurdly simple.

"The beauty comes with seeing the solution to a hard problem; everything clicks," reflected Binghamton University Associate Professor of Mathematics Alexander Borisov

Math students can rarely resist the lure of a tough problem, and the William Lowell Putnam Mathematical Competition presents some of the toughest. More than 3,800 students participated last December in Putnam, the preeminent undergraduate mathematics competition in the U.S. and Canada.

In the 2023 competition, Binghamton's top three scorers - Josiah Moltz, Levi Axelrod and Nikita Safronov - were ranked 40 among the 471 participating institutions; Moltz also received an honorable mention for being among the competitors ranked 26 to 100.

If you just go by the numbers, it looks as if Binghamton slid from 34th to 40th place from the previous year. But the stats don't tell the whole story: The 2022 team consisted of only five students, with three successfully solving problems, while a dozen joined in 2023 and 11 solved at least one problem.

The star participant in 2022, Ashton Keith, has since graduated and is now in a doctoral program at Purdue University. The 2023 team was younger; while Axelrod is a senior, Moltz and Safronov are first-year students, as are Aleksandr Aksenchuk, Alif Miah, Matt-Joshua Tan and Alexei Polonsky. Scorers also include sophomore Suhail Patel, juniors Jesse Hammer and Eric Wang, and senior Blanca Parker.

Associate Professor of Mathematics and Statistics Alexander Borisov hosts a problem-solving session on the evening of March 26, 2024, at his Whitney Hall office, where students participating in the William Lowell Putnam Mathematical Competition practice their skills. Pictured here, students Nikita Safronov, Blanca Parker and Josiah Moltz solve a problem. Image Credit: Jonathan Cohen. "I never really did competition math back in high school, so my involvement with the Putnam was like trying to ride a bicycle for the first time," said Miah, a dual major in mathematical sciences and physics.

Not coincidentally, Moltz, Safronov and Miah all attended Stuyvesant High School in New York City. Safronov and Moltz both took part in high school math competitions and came to Binghamton already knowledgeable - and excited - about Putnam.

"Putnam problems are all proof-based questions. This means that every problem must be answered with a mathematically based explanation demonstrating that your answer is correct," Moltz explained.

The contest is traditionally given on the first Saturday in December. It consists of two sections, with three hours allotted for each; each section includes six increasingly difficult questions.

Participating institutions proctor the competition for their own students and then submit all the work to the contest organizers for grading. At Binghamton, the Putnam contest is administered by Professor Marcin Mazur, who also runs the Problem of the Week series on top of his duties as the current chair of the Department of Mathematics and Statistics.

The problems are high-level pure math, as opposed to applied math, said Safronov, a statistics major from Brooklyn.

"They are incredibly challenging to the point where the median score is just a few points out of 120," he said.

The questions are intentionally challenging, but also designed in such a way that they don't require a large store of technical knowledge, Moltz added.

Detail of student Nikita Safronov's notes during a math problem-solving session on March 26. Image Credit: Jonathan Cohen. "This style of problem is very gratifying to think about because it forces you to use the bare minimum in very creative ways, and think precisely about the structure of the problem and what tools will be useful," he said.

Levi Axelrod, a senior mathematical sciences major from Connecticut, competed in Putnam every year since coming to Binghamton, even when the competition went online during the pandemic. He particularly enjoys problems that fall loosely into the field of combinatorics, which at its simplest is the study of counting things, he said.

"One problem that I answered correctly involved a set of rules for moving coins around in a grid of arbitrary size and a position for the set of coins to start in, and then trying to determine the number of positions possible for the set of coins to end up in," he explained. "Also related to this field are problems about the outcomes of games played by perfectly logical players, and I managed to solve a tricky problem of this type as well."

Competitions such as Putnam help students develop the confidence to tackle challenging problems, according to Borisov. The confidence, discipline and mental agility that undergraduates build through competitions like Putnam can later help them bridge the gap to graduate programs in the field.

To help students prepare and keep their math skills sharp, he offers informal Putnam problem-solving sessions every Monday and Tuesday from 6:30 to 7:30 p.m., although they often run late.

After all, there's nothing quite as appealing as a complex problem - if you're a math major.

"Most math we learn is quite simple to grasp since it's been studied meticulously already. On the other hand, the Putnam competition offers a glimpse into mathematics that is yet to be fully explored and understood, which is so exciting," Safronov said.

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