“A Tile Game”:
Consider a square grid that is infinite in all directions with tiles placed on finitely many squares on the grid. Two grid squares are called adjacent if they share an edge. There are two types of legal moves: Move 1: If two tiles are on adjacent squares, then they can both be removed. Move 2: If a tile is on a square and all adjacent squares are unoccupied, then the tile can be removed with four new tiles then placed on the four adjacent squares. For which initial configuration is it possible to eliminate all tiles from the grid?
In a typical math class, you learn formulas and theorems, practice problems that are related to those specific topics, and then build on that to approach more complex formulas and theorems.
But what do you get when you are not given the formulas and theorems related to the math problem at hand? How do you know where to start with the problem or the tools needed to solve it?
That’s exactly what students in the North Carolina School of Science and Mathematics (NCSSM) Summer Research & Innovation Programs (SRIP) tackled in their Summer Research in Mathematics program.
Under the guidance of NCSSM Instructor of Mathematics Dr. Phillip Benge and Elon University Professor of Mathematics Dr. Todd Lee, students are given a choice of problems to tackle from mathematics journals.
The problem above, dubbed “Problem 12309: A Tile Game” by the “American Mathematical Monthly” journal, is one of the problems that a group of SRIP students took on this summer.
For three weeks, students work in groups to solve and write proofs of their solutions and submit them to the journals at the end of the course.
“Students try to reach into their bag of tools that they have developed in their lives so far and try to solve these problems,” said Benge. “The first step in a lot of math research is to build intuition. And then roll up your sleeves to get to work to see if your intuition is actually fruitful.”
Benge says that as in any research, part of the process is working through obstacles and overcoming frustration when ideas are not working out. “Part of what I do is help students manage that disappointment whenever an idea doesn’t work,” he said. “Hopefully, they grow through the process, and hopefully, they have a good experience.”
Rishyendra Medamanuri is a residential NCSSM student from Charlotte, North Carolina. He loved the independence that it took to think through the research problems.
“In a normal math class, the teacher gives you a topic, they give you questions, and you study it,” he said. “Here, we’re given a question, but there are probably many different ways to prove it. And the thing is, we had to figure out a way to do it ourselves….So it helped us grow, and it’s a lot different than a math class.”
Part of that growth comes from coping with failure, says Benge. “Working with failure is one of my hopes for the students,” he said. “To work with a group for a long period of time on maybe one or two problems. That’s a unique experience that they don’t get to encounter in their normal schoolwork.”
Arya Gupta, an online student from Charlotte, North Carolina, said while that failure and frustration were novel feelings, she ultimately came to love the process.
“I am not the most patient person,” she said. “It’s something new for me to just sit and patiently work on one thing. Because if it was just me, I would have given up 30 minutes in. But I spent two weeks on it, and I really liked that.”
For Medamanuri, that process turned into a new way to see the field of mathematics.
“If you just keep pushing through it, you’ll get used to it and it will be fun,” he said. “It allows you to learn more that isn’t exactly taught in the classroom. If you are really stuck on a problem, you kind of get a connection to the problem: you grow close to the problem and it has a meaning for you.”
Gupta stayed motivated because of her peers, and their shared love of doing math just for fun.
“Here, it’s like people are sitting and working on these problems because they liked it, and that was very cool to see,” she said. “The best part has been all the people I’ve met because I haven’t met a lot of people back at my school that are super into math.”
By the end of the program, every group came up with solutions and proofs that were submitted to the journals, so even while teaching the importance of perseverance and coping with failure, all of the students found success.
Gupta says if there’s one thing people considering Summer Research in Mathematics should know, is that they should apply and try it out.
“I don’t see research in math as something that a lot of people are ever going to experience in their life,” she said. “Even like me, I love math so much, and I have no idea if I’ll ever do math research again. I am really glad I got to experience it because I learned a lot of things.”
“My advice [for research] would be to be patient and hold out hope. You’ll get it eventually. And just enjoy it, because it’s really fun.”