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

Go Figure

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.
January 31, 2010
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.

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.

David Kraines has worked hard to recruit talented math students and help them prepare for the Putnam.

David Kraines has worked hard to recruit talented math students and help them prepare for the Putnam. “For him it is a labor of love,” says the head of Duke’s math department. Jon Gardiner

"When I came to Duke in the mid-'80s, Duke's math department was already on a very good trajectory," says Griffiths, who admits that he himself "didn't do all that well" on the Putnam when he was at Wake Forest College, the undergraduate component of Wake Forest University. "We had an excellent chairman, Mike Reed. The Duke [Mathematical] Journal was one of the best. And the TIP program was bringing very bright high-school students from all around the country to Duke for the summer. Also, unlike many large research universities, Duke had a strong commitment to undergraduate education, including a faculty that sought out promising high-school students and convinced them to come to Duke."

Associate professor of mathematics David Kraines compares his department's strategy with that of another winning Duke program. "Mike Krzyzewski doesn't recruit players for his team by going around to dorms looking for tall students. With help from the admissions office, the president, the provost, and members of the math department faculty, we set about recruiting the top high-school math students to come to Duke." A few of the most exceptional of these were awarded merit scholarships.

William Lowell Putnam

William Lowell Putnam

The approach began paying off almost immediately. VanderKam applied to Stanford, Princeton, and Duke, but Duke emerged as his top choice because of faculty encouragement and accessibility. "I attended the TIP program the summer after my seventh- and eighth-grade years and had a blast," he says. "That's when I first started thinking about coming to Duke. Then, by the time I was a senior at the North Carolina School of Science and Math, I was visiting Duke's math and physics departments on a regular basis because I was doing an independent study project on topology."

Andrew Dittmer, the oldest of seven children, says Duke's "aggressive recruiting" and the offer of a full-tuition Angier B. Duke scholarship made his decision easy. Melanie Wood '03 was another highly sought-after star. At the age of four, she was solving linear equations in her head. She dominated middle- and high-school math competitions—including two silver medals in the International Math Olympiad—and was featured in Discover magazine ("The Girl Who Loved Math").

When considering where to go for college, Wood narrowed her selection to Duke and Harvard. She visited both campuses twice. Despite Harvard's track record for luring math scholars of her caliber, she disliked the "cold and competitive" nature of its department. Duke struck her as "a much friendlier place, where faculty were interested in working with undergraduates, and there was much more collaboration between undergraduates, graduate students, and faculty." She also appreciated the fact that Duke was purposefully recruiting students like her and recognized that she would be with like-minded peers who were valued for their talent and potential.

Wood was selected to represent Duke on two Putnam teams. In 1999, her freshman year, Wood and the team came in third, and her score put her in the top 2 percent of participants. In her senior year, Wood became the first American-born female to become a Putnam Fellow. She was also selected as the recipient of the Morgan Prize, established in 1995, awarded to an undergraduate student for outstanding research in mathematics, for her research on Belyi-extending maps and P-orderings—to date, she is the only woman to receive the award. 

 

Do the math: Rognlie,

Do the math: Matt Rognlie, top, leads weekly study session for Putnam competition candidates. Jon Gardiner

Since the 1993 team's first place win, the investment in undergraduate recruiting and mentorship has continued to pay off. Duke has gone on to win first place in the Putnam two more times, in 1996 and 2000. The university placed second once, in 1997, and third six times, in 1999 and from 2001 to 2005. With the exception of 1994, Duke has not dropped below the top ten.

Harold Layton Ph.D. '86, professor and chair of the mathematics department, is quick to note that the success did not happen overnight. "David Kraines deserves an enormous amount of credit for recruiting outstanding undergraduate math talent to Duke year after year and helping prepare students for the Putnam. For him it is a labor of love that has spanned decades."

As the caliber of students continues to climb, the math department has expanded its scope and offerings. In 2000, it began offering the PRUV Fellowship (the acronym stands for Practical Research for Undergraduates using VIGRE; VIGRE is a National Science Foundation initiative to encourage educational and research innovations in math and science). The competitive program provides summer stipends for math majors to pursue a research project leading to a senior thesis that qualifies for graduation with honors. Each summer since then, five to eight undergraduates have been paired with faculty mentors and given stipends for six weeks of intensive research.

Department chair Layton says that quantifying the effect that success in the Putnam competition has had on the math department can be tricky. "Only in specific cases can we point to a cause-and-effect relationship between our teams doing well on the Putnam and our attracting prospective students, but the record of Putnam successes do have a bearing on how Duke's math department is viewed," he says. "It helps attract very good students who come because they see other very good students coming here and doing well. Prospective graduate students perceive Duke as a stronger school because of the close interaction between undergraduates and faculty. And in the recruiting of faculty members to Duke, the opportunity to work closely with very talented undergraduate and graduate students is a significant and attractive component to the prospect of joining our department."

Indeed, faculty recruitment has brought a number of highly regarded math scholars to Duke. For example, Arlie Petters, the Benjamin Powell Professor of math, physics, and business administration, taught at MIT and Princeton before joining the Duke faculty in 2003. His research into gamma rays and black holes has earned him many distinctions, including designation by Queen Elizabeth of England as a Member of the Most Excellent Order of the British Empire.

And then there's Lenhard Ng, an assistant professor of math who grew up in Chapel Hill, where his father still teaches particle physics and cosmology at the University of North Carolina. A child prodigy who aced the math portion of the SAT at age ten, Ng graduated from Chapel Hill High School at the age of sixteen and enrolled at Harvard, receiving his degree summa cum laude in three years. He was a Putnam Fellow all three years, helping put Harvard's team back on top in 1994 and 1995.

From his sparsely decorated office on the second floor of the physics building, Ng conducts research in low-dimensional topology, symplectic and contact geometry, differential geometry, and mathematical physics—with particular interests in holomorphic curves, symplectic field theory, and knot theory.

"When I was applying to college, I wanted to get out of the area since I'd been here almost my entire life," he says. After earning a Ph.D. from MIT, he spent a year at the Institute of Advanced Study in Princeton, and was awarded a five-year postdoc fellowship by the American Institute of Mathematics. He was recruited by Duke in 2006 and says he welcomed the opportunity to join a department "that had made huge strides in the last ten years."

Ng and David Kraines serve as informal advisers to students enrolled in the weekly, half-credit problem-solving seminar that is unofficially known as the Putnam Club, since it serves as preparation leading up to the December competition. Kraines decides who will be on the team.

Oddly enough, Ng says he has mixed feelings about the Putnam. "I don't think it's the be-all and end-all of [gauging] mathematical capability," he says. "It's a bit of a speed competition, and because of the difficulty level of the problems, you have to balance creativity in solving them with ability to budget your time. But it can be a pretty good indicator of what a young mathematician could do if they decide to stay in the field."