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The games that Marquis Scholar Brian Kronenthal’07 (Yardley, Pa.) used during his honors thesis are easy enough to play. Developing a strategy to master them, however, is another issue.
Kronenthal, a mathematics major, researched combinatorial game theory, studying the strategies and mathematics of several two-player games.
“I explored surreal numbers, which are a representation of numbers that can be derived using set theory,” explains Kronenthal. “These surreal numbers can be used to analyze certain two-player games with equal information.”
Kronenthal’s honors thesis adviser was Derek Smith, associate professor of mathematics, who notes that Kronenthal was working to explain the “very subtle” mathematical theory of combinatorial games.
“There is a system of ‘numbers’ used to evaluate [these] games; a system that behaves a lot like the real numbers we all know, but that is much richer in structure,” he says. “Brian was attempting to explain various properties of this number system in a better manner than the standard sources on the subject.”
Smith explains that combinatorial game theory was developed in large part as an attempt to understand the game of Go. The endgame of Go, he notes, is often comprised of several small subgames in different areas of the board, with the values of these subgames essentially adding up to give the total value of the endgame.
“Brian’s thesis primarily studied the lesser-known game of Hackenbush, with positions that are analyzed and valued in a manner analogous to those in Go,” says Smith. “The exposition consists of two halves: one based on an inductively defined system of sets, and one based on certain combinatorial games.”
The fact that such complex thinking and analysis can be applied to such basic games made the thesis work more interesting for Kronenthal.
“This project was exciting for me because of the way it uses sophisticated mathematical concepts to analyze games that are very accessible, even to those without any mathematical background,” he says. “Thus, even though anyone could play these games, doing so with an optimal strategy requires this sophisticated approach.”
Kronenthal says that he received much of the support for his work from members of the mathematics department, particularly Smith.
“I was very pleased to be working with him,” says Kronenthal. “Derek was extremely helpful, understanding, and supportive throughout the process. His insights into certain aspects of the theory made my research much smoother. Lafayette is definitely an outstanding place to study mathematics, and has far exceeded any expectations I might have had in this regard. Not only is there a wide variety of courses offered, but the professors are all extremely accessible and eager to help students even outside of their office hours.”
Smith, who previously worked with the student on EXCEL Scholars research focusing on high-level math and problem-solving, says Kronenthal impressed him by being an “extremely disciplined and careful student of mathematics.”
“He is never content to understand half of a topic; he will work hard and follow through until all of his questions about the topic have been answered,” says Smith.
Kronenthal will begin doctoral studies in mathematics at the University of Delaware this fall.
“His thesis work will form a nice bridge between the expectations of undergraduate coursework and the independent scholarship required to excel in graduate school,” says Smith.
In April, Kronenthal and Lorenzo Traldi, Metzgar Professor of Mathematics, presented their EXCEL research on the prevalence of non-tying behavior in dice games at the 2007 American Mathematical Society’s Spring Eastern Section Meeting in Hoboken, N.J.
Kronenthal is a member of Phi Beta Kappa, America’s oldest and most respected honors organization, and mathematics honor society Pi Mu Epsilon. He also is a recipient of the Professor James P. Crawford Prize in Mathematics.
Chosen from among Lafayette’s most promising applicants, Marquis Scholars like Kronenthal receive a special academic scholarship and distinctive educational experiences and benefits. This includes a three-week, Lafayette-funded course abroad or in the United States during January’s interim session between semesters or the summer break. Marquis Scholars also participate in mentoring programs with Lafayette faculty and cultural activities in major cities and on campus.
Honors theses are among several major programs that have made Lafayette a national leader in undergraduate research. The College sends one of the largest contingents to the National Conference on Undergraduate Research each year; 21 students were accepted to present their research at this year’s conference.