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Luke Fagley ’22 (left) and Andrea Potylycki ’21 (right) showcase their S.U.R.F. project with Assistant Professor of Mathematics Matt Prudente, Ph.D. (center)

by Kristin Boyd Edwards

As a spring altered by COVID-19 folded into summer, senior Andrea Potylycki and junior Luke Fagley moved in sync, steadily applying the game theories they learned during Dr. Matt Prudente’s mathematics courses to crack the code in their ambitious Student Undergraduate Research Fellows (S.U.R.F.) project— Two-Player Graph Pebbling.

Prudente hand-picked the math majors for the collaborative project. The pair knew expectations were high, as their research would build upon studies previously conducted by other researchers, including Prudente’s doctoral dissertation from 2015. If they succeeded, they could make a name for themselves in the math world. If not, it was back to the drawing board.

So even while limited by pandemic restrictions, they spent much of their summer days conducting solo research and tag-teaming via Facetime on their iPhones to discuss how the graphs they implemented impacted algorithms, and, ultimately, the outcome of the pebbling game.

The chesslike game features two players, a Mover and a Defender. The Mover uses pebbling moves, while the Defender uses pebbling moves to ensure that a specified vertex (or space) remains pebble free. Unlike Chess, the two-player pebbling game always yields a winner, eliminating the possibility of a draw. For this to occur, each player must take their turn and, if Mover pebbles from u to v, then the Defender cannot pebble from v to u in the same round.

Their goal was to determine all of the ways a player could win the game.

“Game theory applications is a popular field of study in mathematics with wide interests, whether it be chess, sports or games of chance,” said Prudente. “For this project, we answered some conjectures that previous discussions and papers have spurred.”

The research was exhilarating and, at times, frustrating, especially when Potylycki and Fagley said they were exhausted or found their ideas weren’t clicking. “Sometimes I was nervous because we wanted to figure this out,” says Fagley, a member of the university’s golf team who jokes he would often think of potential scenarios in between swings. “This research was always on my mind.”

Throughout the summer, they made progress and grew more confident until they reached their goal. “That’s when it helped the most to have someone working with you on it,” Potylycki said. “We would write up proofs, which solidify what you’re doing is correct and acceptable in the math community.”

Despite their success, the pair continues to play it cool, Potylycki said with a laugh, but admit they’re excited about the next steps. Potylycki will present the groundbreaking pebbling research at the 2021 Joint Mathematics Conference, the country’s largest mathematics conference. The research is also being prepared for submission to an academic journal for possible publication. Overall, the project “showed them how modern mathematics research is completed,” Dr. Prudente added.

The duo have continued their work together, recently becoming O’Pake Institute Fellows. They are utilizing their skillsets in mathematics and business to help Southeastern Pennsylvania entrepreneurs and businesses solve various problems. “It’s a really great way to help businesses, but we are also getting our foot in the door,” Fagley said. “We’re networking and gaining experience.”

Moving forward, Fagley, a business minor, is determining his honors thesis topic and would like to one day become the head pro at a golf course. Potylycki, a researcher at heart, is considering grad school opportunities and would like to pursue her Ph.D. eventually.

“I’m thankful for all of the experiences that ����ֱ�� gave me,” Potylycki said. “It was really fun to work with Luke and produce original research. That’s not something I thought I would be able to do as an undergrad. We got to prove something new that no one has ever done before.”