Throughout his career, prolific mathematician Paul Erdős offered small rewards for hundreds of unsolved math problems.
Franklin & Marshall College senior Nart Shalqini is a few steps closer to cracking one. The Hackman Scholar’s research focuses on tiling a unit square.
“I liked how the problem combined geometry, number theory and graph theory in a beautiful way,” said Shalqini, who completed much of his research at home in Prizren, Kosovo, this fall.
There are many configurations in which one can pack a unit square with smaller squares, Shalqini explained.
“Since this question is broad, we focus on a more specific case where the small squares do not overlap and they cover the entire unit square. This case is called a tiling,” he said.
Shalqini tackled Erdős’ puzzle under the tutelage of Professor of Mathematics Iwan Praton. While studying in the Department of Mathematics’ common room, Praton’s paper on the topic caught Shalqini’s eye.
“There are many fascinating aspects to this question,” Praton said. “It is an example of a geometric optimization problem, so some computer scientists are interested in problems of this type to test out their algorithms. The earliest question of this sort started in the 1930s, so it has been unsolved for a long time.”
It’s a challenge tackled not necessarily for its real-world applications, but for exposure to a problem with an unknown answer.
“I loved [Praton’s] paper simply because it was very elegant,” Shalqini said. “I thought it would be a very good experience for me as an undergraduate researcher to try to tackle this problem to which I could have a meaningful contribution with my undergraduate background,” Shalqini said.
For two tile sizes, Shalqini and Praton were able to prove their conjectured optimal tiling, and for more than two tile sizes, they showed that the optimal tiling occurs only if the smallest tile is unique.
While they didn’t solve Erdős’ puzzle once and for all, the research exposed Shalqini to methods used by mathematicians that countered his thought process.
“It was another moment of realization on how a mathematician should always keep an open mind about their research, since they may not know what area of mathematics might be of use. Even the seemingly unrelated concepts may come in handy and lead to the crucial step of the solution,” he said.
"A mathematician should always keep an open mind about their research, since they may not know what area of mathematics might be of use. Even the seemingly unrelated concepts may come in handy and lead to the crucial step of the solution.”
Stories of the Spring Research Fair