Volume 96, No.1, January-February 2010

Go Figure
by Bridget Booher
Doing well in the William Lowell Putnam Mathematical Competition can bring a nimble thinker to the attention of the entire mathematics field. Doing poorly—well, that's to be expected.

If x: excerpt of problem B2 from 2007 competition.
If x: excerpt of problem B2 from 2007 competition.

The William Lowell Putnam Mathematical Competition may be the most intensely difficult intellectual contest ever designed for undergraduate college students. Time magazine has called the competition "a rite of passage for math cognoscenti … a coming-out party for the next generation of beautiful minds." Past participants include four who went on to win Nobel Prizes in physics and five who went on to earn the Fields Medal, the top honor for mathematicians under the age of forty and considered the equivalent of a Nobel Prize in math.

For the past seventy years, the Putnam—as it's known for short—has been held the first Saturday in December at colleges and universities throughout the U.S. and Canada. Doing well can bring a nimble thinker to the attention of the entire mathematics field or cement the reputation of a young prodigy. It bestows bragging rights on the winning institutions, helps consistently well-performing schools attract top high-school and faculty talent, and brings in a nice chunk of change for winning competitors and their institutions—as much as $2,500 and $25,000, respectively.

Doing poorly—well, that's to be expected. Out of a possible perfect score of 120—achieved only three times in the history of the contest—the median score for the past decade has been between one and two points.

Duke began fielding Putnam teams in 1977, managed to place in the top twenty-five in 1989, and finally broke through to the top-five winners' circle in 1990, with a second-place team win. Two years later, Jeff VanderKam '94 became the first Duke student to earn the designation of Putnam Fellow, one of the five top-scoring participants. But winning the Putnam continued to elude a department that was gaining visibility for its strong commitment to undergraduate education, among other assets.

By the early 1990s, Harvard University, where the Putnam originated, had racked up an eight-year winning streak. Even though Harvard had long dominated the first-place slot—and still has more first-place wins than any other institution—the streak was unprecedented in the history of the competition. Duke remained far behind Princeton University, the California Institute of Technology, and the Massachusetts Institute of Technology—all of which trailed Harvard—in  both individual and team performances.

And so, on Saturday, December 4, 1993, as thousands of students prepared to take the grueling six-hour test at their respective institutions, the Crimson powerhouse was the odds-on favorite to take home the title again.

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Do the math: weekly study session for Putnam competition candidates.
Jon Gardiner

Still, the Duke team had reason to be optimistic. In addition to VanderKam, Duke's team included Andrew Dittmer '98 and Craig Gentry '95. VanderKam had taken advanced-placement math in middle school and was awarded a college scholarship in eighth grade by placing in the top ten of all high-school students who participated in the North Carolina State High School Math Contest. He'd also won silver and gold medals at the International Math Olympiad while in high school. Not surprisingly, Dittmer arrived at Duke already planning to major in math and was tackling graduate-level work in his first semester.

Gentry, on the other hand, was a bit of a wild card: Although he excelled in his honors-level math courses, he had no international high-school competitions under his belt and, unlike VanderKam and Dittmer, had not been recruited by the math department. As one math professor noted at the time, "Craig came out of nowhere."

Despite Harvard's dominance, Dittmer recalls, he and his teammates were focused only on achieving personal bests, not on inter-institutional rivalries. "The fact that Harvard had won eight years in a row was not an intimidating force," he says. "Even if Harvard had won for the past forty-nine years, I would still have gone into the competition trying to do my best."

VanderKam recalls thinking afterward that some of the problems were "next to impossible for anyone."

Grading the thousands of tests generated during a single Putnam competition takes three to four months. Answers are in the form of proofs that illustrate the problem-solving process, and each test is reviewed by a panel of judges. In theory, anyone with a solid background in high-school mathematics can do well, but, in reality, participants should be well-versed in broader mathematical concepts (group theory, number theory, linear algebra). Creativity and ingenuity are essential—the problems are not of the textbook variety. Rigorous mathematical justification of each step in the argument is required for full credit.

Any number of students from an institution can take the test as individual participants, but the official team of three is selected by faculty members based on factors such as performance on past competitions. The winning teams are determined by adding up each individual member's ranking among the roughly 3,500 participants. If one member does poorly, the team's score and ranking will suffer. Determining the composition and collective strength of a team is an imprecise science; it's not uncommon for non-team members to outscore their team peers.

The Putnam consists of two three-hour sessions—six problems in the morning and six in the afternoon—with a break for lunch. Undergraduates selected by their institutions show up armed with pencils, scratch paper, and the intellectual capacity to tackle seemingly impenetrable problems such as this one from the 1993 contest:

The infinite sequence of 2's and 3's

2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 3, 3, 2,….

has the property that, if one forms a second sequence that records the number of 3's between successive 2's, the result is identical to the given sequence. Show that there exists a real number r such that, for an n, the nth term of the sequence is 2 if and only if n = 1+ [rm] for some nonnegative integer m. (Note: [x] denotes the largest integer less than or equal to x.)

In March 1994, the results of the text taken in December 1993 were announced: Duke had upset Harvard to claim the number-one spot. Team T-shirts with a Putnam problem on the front were printed up, and President Nannerl O. Keohane retired VanderKam's "jersey"—strictly speaking, a T-shirt—which is framed and hanging in the math department lounge. North Carolina Governor James B. Hunt wrote letters of congratulation.

"My initial reaction was that it was some hoax that some Harvard student had come up with," Dittmer told The Chronicle.

In fact, the Duke win was neither a hoax nor a fluke. Rather, it was a confluence of factors that included a departmental and institutional push begun several years earlier to recruit the nation's best high-school math students, students who had historically opted for—no surprise—Harvard, Stanford, MIT, and other top-ranked math schools.     

Phillip Griffiths was Duke's provost and James B. Duke Professor of mathematics from 1984 to 1991. An internationally renowned mathematician whose specialty is algebraic geometry, Griffiths became the director of the Institute for Advanced Study in Princeton after leaving Duke. He now serves on the Millennium Science Initiative project to strengthen science and technology endeavors in developing countries.

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