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The problem set
Taking the hardest undergraduate math test around
Most Wednesdays this past fall, if you happened to wander past Carney 309 late in the afternoon, as the daylight dwindled, you would have seen a small group of students and faculty staring at a blackboard strewn with equations. Chances are you would have found them sitting in a companionable silence, each lost in a private fog of thought—which is to say, you would have seen the Boston College Problem Group in action, preparing for the toughest and most prestigious undergraduate mathematical contest in the world.
It’s called the Putnam Competition. Endowed by Elizabeth Lowell Putnam in 1927, in honor of her late husband, William, a lawyer, the competition was launched in its present form in 1938. It’s been an annual event ever since, designed “to stimulate a healthy rivalry in mathematical studies in the colleges and universities of the United States and Canada,” according to the Mathematical Association of America, which has run the competition since its inception. Each year some 5,000 students take part, sitting in early December for an exam administered simultaneously on campuses all over North America.
It’s a legendarily difficult competition. The winners have almost always been students from Harvard, MIT, and a few other math powerhouses. Two of them, including theoretical physicist Richard Feynman, have gone on to win Nobel Prizes. Students enter the competition as individuals, but each institution can also nominate a team of three to represent it. Prizes are awarded to individuals and to teams, whose individual scores are added together. The exam itself takes all day. Students have three hours in the morning to solve six problems, and three hours in the afternoon to solve six more. Each is worth 10 points, for a total of 120. And here’s a sobering fact: Each year, when the results of the exam are ranked from best to worst, invariably the median score is . . . zero.
Let that sink in for a minute. Each year 5,000 of the very best collegiate mathematicians in the United States and Canada take the Putnam exam, and at least half of them don’t score a single point.
Max Jackson ’14, from Granby, Connecticut, is a two-time member of that club. Max took the test as a sophomore and a junior, and each time scored a zero. Aware that this would be his last chance to compete, he carved time out of a busy pre-med schedule to attend as many Problem Group sessions as he could. Not that he harbored illusions about becoming a contender. Just scoring points, any points, would represent a triumph. “It’s a personal challenge, like when I ran the marathon,” he said after a November practice session. “These problems are some of the hardest problems I’ve ever seen, that most people will ever see in their mathematical careers. I’m not trying to win. I’m just fighting to get into that top 50 percent.”
Throughout the fall semester, along with a handful of other Problem Group regulars, Max did battle on Wednesdays with sample Putnam questions (see box). At one session, thinking he’d figured something out, he got to his feet, mumbled something about slope as a constant, and shuffled to the front of the class, where he began transferring notes to the blackboard. He started out confidently enough, but within a minute or so he had to stop. He stepped back and reviewed what he’d written on the board. Something wasn’t right.
After a while, Maksym Fedorchuk, one of two young assistant professors who helped run the practice sessions, spoke up. “Max,” he said, “are you trying to get your hands on a second derivative?”
“I’m trying to get . . .” Max’s sentence trailed off into thought, and then he tapped his chalk on an equation. “I think this is where I messed up,” he said.
Radu Cebanu, the other young professor, nudged Max toward a solution. “The problem,” he reminded him, “only says that slope is periodic.”
This helped Max, who brightened and said, “Then I guess you’d have to go down to the second derivative to get that.”
“Right,” said William Keane, the assistant chair of the mathematics department. “I think the second derivative is the way to go.”
Short, bearded, and a model of soft-spoken professorial geniality, Keane was sitting off to the side of the blackboard. Above him, on the wall, hung a framed Renaissance portrait of Pythagoras, who looked a little vexed himself by the problem on the board. Keane has taught at Boston College for 36 years, has served as the University’s Putnam liaison for 16 of them, and took the test himself as a college student, all of which means he knows better than anybody the odds the students are up against. “Winning the Putnam isn’t realistic,” he said when the exam was just a few weeks away. “But preparing for it for months really improves your mathematical acuity. These aren’t the kinds of problems you look at and say, ‘I know how to do that.’ They almost always require you to think across boundaries, to search for connections and patterns. That’s the fun of it: when you see patterns emerge and then can leap from them to others. That’s beautiful mathematics.”
The six students who turned up at Carney 309 on the morning of Saturday, December 7, to take the Putnam exam weren’t really concerned with doing beautiful math. They just wanted to score some points, elegantly or not. The official team consisted of Stephanie Ger ’14, Tianyu Xiang ’15, and Jing Xu ’16; they were joined by Max Jackson, Zachary Skarka ’15, and Christopher Coscia ’15, who were competing individually.
Max’s strategy—everybody’s strategy, really—was simple: Scan all six problems, locate one or two that seem doable, and then spend the full three hours attacking just those.
The first session wrapped up at 1:00, at which point the whole group headed out for a lunch at Union Street, in Newton Centre—a math-department tradition on Putnam exam day. College-football games flickered on flat-screen TVs around the room, but the group ignored them entirely, instead chattering abstrusely about the problems they’d just been grappling with. The mood was surprisingly lively and upbeat. “They all thought they got at least one,” Professor Keane later reported. “But I’ve learned to wait and see.”
After lunch, the group headed back to Carney 309, and at 3:00 on the dot launched into the afternoon’s problems. Three hours later, Professor Keane collected the exams, and it was all over. Some of the students began checking their mobile devices, to consult the solutions already being posted online after the exam by test-takers around the country. But not Max. Poking around online, he decided, might make him feel worse rather than better about how he’d done. All things considered, he felt pretty good, so he figured he’d just wait for the official results, which will be announced in the spring. “I definitely have a shot,” he said. “I suppose we’ll see in March.”
Toby Lester is a writer in the Boston area.
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