## Math team takes national title

by Monte Basgall

For the second time in four years, a Duke University team placed first in the William Lowell Putnam Mathematical Competition, a grueling six-hour-long test that for math students is the equivalent of an NCAA basketball tournament.

By amassing the highest cumulative ranking, Duke sophomore Andrew Dittmer of Eureka, Mo., junior Noam Shazeer of Swampscott, Mass., and senior Robert Schneck of Charlotte beat out the perennial favorite, Harvard, which this year came in third behind second-place Princeton. Schneck recently won a Churchill Award for a scholarship to Cambridge University.

Of the 2,407 contestants from 405 institutions who took the test, less than half scored more than 2 points out of a possible total of 120. By contrast, the Duke team's high scorer, Shazeer, received 68 points, while Dittmer's total was 62 and Schneck's was 60.

"Many schools have won the Putnam one time and then never again," said David Kraines, a Duke associate professor of mathematics and the team's coach. "Duke is on the road to a very solid tradition. In large part because of our previous Putnam victory, more students who are extremely talented in mathematics are eager to come to Duke."

Under the Putnam rules, each university designates three students as its official team members but can also enter other students in the competition. That means that the competition's highest scorers may not necessarily be members of the winning team.

Shazeer placed among the top 10 in the nation in the latest competition, while Dittmer ranked in the next five with Schneck a few places behind that. The cumulative ranking of this team was higher than that of Duke's winning 1993 Putnam team.

"What I found most satisfying was the breadth in addition to the depth of our performance," Kraines added. "Of the 15 Duke students who took the latest test, we had seven scoring within the top 200 nationwide - by far the most we've ever had."

The competition was held on Dec. 7, but the results arrived while the Duke campus was on spring break. Before they returned to class, though, team members got the word by phone and e-mail.

"I knew we all did really well so I expected we'd be in the top three," said Shazeer. "But I'm a little surprised that we won."

"I personally didn't think my performance is what it could have been," said Dittmer, also a member of Duke's winning 1993 team. "I kind of ran out of steam in the second part."

"It's a very long test," added Schneck. "You come in Saturday morning and you're there for most of the day. Definitely at the end you're glad for it to be over. But during a competition like this, you really get into a groove sort of analogous to a runner's high. It just carries you with it."

For winning the Putnam competition, Duke's mathematics department will receive $7,500, which Kraines said helps pay for students travel to national Mathematical Society meetings. Each team member also will receive $500. In addition, Shazeer won another $500 and Dittmer another $250 for their high contest rankings.

For Schneck, the Putnam competition win comes on the heels of notification that he has won a Winston Churchill Foundation Scholarship for a year of graduate studies at Cambridge University in England.

Here are four sample questions from the 57th annual competition, named for William Lowell Putnam, a member of the Harvard class of 1882 whose widow initiated a trust fund to support the competition.

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Find the least number A such that for any two squares of combined area 1, a rectangle of area A exists such that the two squares can be packed into that rectangle (without the interiors of the squares overlapping). You may assume that the sides of the squares will be parallel to the squares of the rectangle.

Let C1 and C2 be circles whose centers are 10 units apart and whose radii are 1 and 3. Find, with proof, the locus of all points M for which there exist points X on C1 and Y on C2 such that M is the midpoint of the line segment XY.

Suppose that each of twenty students has made the choice of anywhere from zero to six courses from a total of six courses offered. Prove, or disprove: There are five students and two courses such that all five have chosen both courses or all five have chosen neither.

Define a selfish set to be a set which has its own cardinality (number of elements) as an element. Find, with proof, the number of subsets of {1,2,...,n} which are minimal selfish sets, that is, selfish sets none of whose proper subsets is selfish.

"Putnam competition questions may be attacked in different ways," Kraines said. "Often it's the case that students will come up with solutions much more elegant than the proposers realized were possible."