# Faculty Research: What’s the Big Idea? (part 3)

*Public debate over the place of religion in society; a puzzle in a magazine; headlines about cultures clashing. Sometimes it’s such fragments of daily life that capture the imagination of a scholar and fuel the desire to understand an idea or an event. Academic research grows from just that source: the curiosity to pursue a topic to its roots, to refine a theory with students and colleagues, and to share discoveries with the wider world. Dartmouth faculty member Peter Winkler shares the big ideas that drive his work as a teacher-scholar. (Preston H. Kelsey Professor of Religion Susan Ackerman ’80 is profiled in the first installment of this story, published November 30, 2010, and Paul D. Paganucci Professor of Italian Language and Literature Graziella Parati was featured in the second, published December 2, 2010.)*

**Puzzling it out**

Some mathematicians will explore the outer reaches of their field, but Professor of Mathematics and Computer Science Peter Winkler has a different approach: he looks for elementary things that mathematicians do not yet know how to do.

“Sometimes simple problems intrigue your intuition,” says Winkler, who is also the Albert Bradley Third Century Professor in the Sciences. “And you may need to find a whole new mathematical way to study a problem, because the most far-reaching work in mathematics is reaching far in the wrong direction.”

One recent example in Winkler’s work is an ancient problem called “overhang”: how far a stack of bricks can be made to lean over a table’s edge. Winkler says that people have explored this problem since the 19th century, and many thought they knew the solution. “But it turned out the answer was quite different from what people thought—you can actually make bricks hang out much farther than people thought you could.”

In his research, Winkler focuses on phase transitions, changes that occur when an aspect of a problem is slowly altered and the problem suddenly shifts in character. “An example would be if you have puzzle,” Winkler says, “and you slowly change some parameter that’s associated with this puzzle, and all of a sudden it changes from being very easy to very difficult to solve. And we’d like to understand how and when this happens.”

Winkler also works on random walk, a process that begins at some point in space, where each step’s characteristic is determined by chance. “Random walks can be taken by physical things, by people, or by tiny particles,” Winkler explains, “but they also come up a lot in abstract settings, where you just have some random process and it behaves like a random walk. In fact, random walks actually come up in the brick overhang problem, even though there doesn’t seem to be anything random about it.”

Winkler has acquired a certain amount of fame for topics outside the usual mathematical research, including mathematical puzzles. His work on puzzles includes two written collections, a column for the professional magazine Communications of the ACM, and talks on the subject to groups of students and even Wall Street investment bankers. In 2006, he appeared with New York Times’ puzzlemaster Will Shortz and Sudoku founder Wayne Gould on New Hampshire Public Radio’s *The Exchange*. Winkler collects and designs mechanical puzzles and organized Dartmouth Mechanical Puzzles Day in February 2008, where he brought some of the world’s best puzzle designers, makers, and collectors to campus.

Winkler also encourages a spirit of play. His ragtime piano compositions were entered at Artworks, Dartmouth’s employee arts show; he is the faculty advisor for the Dartmouth Bridge Club; and one of his favorite courses is Math 5, “Combinatorial Games.”

“I had a lot of people who were taking the course just because they love playing games—including a bunch of football players. Every time we met, I would hand out boards, explain several new games, and everybody would pair up and play. And then for the second half of class we would study these games mathematically and see who should have won. The students had a lot of fun with that.”

It is highly probable that Winkler did, too.