By the Numbers

“Let’s play a game. We start with a pile of 21 stones and take turns removing stones. On each turn, we can remove one or two stones. The goal is to take the last stone. I will go second. Can you beat me?”

So begins the introduction to Combinatorial Puzzles and Games, a hands-on course during which students play and discuss the mathematical implications of games such as the one described.

Offered last spring by the Osher Lifelong Learning Institute (OLLI), the course was taught by Aaron Li, a third-year PhD in mathematics candidate in the University of Minnesota (UMN) College of Science and Engineering.

Li explains that while the games are seemingly simple, they all have rich connections to topics such as combinatorics, game theory, and number theory. This from a budding researcher who, alongside Professor Jasmine Foo, is exploring how to use probability and phylogenetic trees to better understand the behavior of evolutionary processes like cancer or infectious diseases.

But he insists no previous math experience was required. “My focus was on creating activities that were inquiry-based and accessible to all backgrounds.” The goal: to develop students’ problem-solving skills.

A Fortuitous Match

Combinatorial Puzzles and Games was offered through the OLLI Scholars program, which has a rich history of selecting UMN graduates and postdoctoral researchers to teach courses for the Institute during the academic year. Twenty scholars are chosen annually based on the quality of their proposals. The 20 go on to hone their teaching skills by developing curricula and leading courses in their respective areas of expertise.

It’s a win-win situation. The scholars get valuable teaching experience and stipends of $1,000, which are funded, in part, by donor contributions. OLLI members, most of whom are 50-plus years of age and retired from myriad professions, are enthusiastic students who are endlessly curious and eager to learn.

Intergenerational. Reciprocal. It's a fortuitous match and, year after year, both scholars and members express gratitude for learning from and contributing to the growth of one another.

On the whole, members are extremely supportive and enjoy contributing to the growth and experience of early career professionals. Intergenerational. Reciprocal. It's a fortuitous match and, year after year, both scholars and members express gratitude for learning from and contributing to the growth of one another.

“It was really amazing to be able to design a course of my own and work with a bunch of really engaged, enthusiastic students,” says Li. “I think what stands out most about the experience was getting to know the students and helping them discover or rediscover their interest in math. One activity involved counting in binary, and a student told me she was familiar with binary because she had built a computer back in middle school but hadn't really worked with math for decades. It was great to be able to make that connection. Another student who was a retired math professor told me that my course inspired them to propose their own OLLI course... It meant a lot to me to be able to share the fun side of math with others.”

Extending that fun and intergenerational learning still further, Li’s hope is that his students mastered “some of these games well enough to play them with a niece/nephew or grandchild.”

In the Classroom

A Portland, Oregon native, Li graduated from Carleton College in 2020, just prior to entering the University’s PhD program. In addition to his research, he is a teaching assistant in the math department and would one day like to work at a “small, teaching-focused college or university… somewhere that I can design and teach courses that interest me and work with enthusiastic and engaged students.”

He was looking for such an experience when he heard about the OLLI Scholars program from past Scholars. Li decided to apply. It was the spring of 2021, the COVID-19 vaccine was in its infancy, and OLLI courses were still being taught completely online. Still, the application gave him a choice of teaching venue: online or, in hopes of future COVID-mitigating advances, in person. Comfortable teaching in either venue, Li knew his proposed course would only be successful if offered in person.

“I designed the course to involve small group collaboration and a lot of hands-on interaction with games. Zoom breakout rooms are not a terrible facsimile for something like this, but I thought being able to physically move the game pieces and talk to each other would help the learning process a great deal.”

By the time Li taught his course, the pandemic landscape had changed, allowing him to teach in a classroom where he found the discussions and activities to be as lively as he had hoped.

“The students did not hesitate to collaborate, ask questions, and bravely try to work things out,” he says. “I was also surprised by how diverse the backgrounds of the students were… [they] ranged from people with PhDs in math to people who had not done any math since taking calculus 40+ years ago.”

The Fun Side of Math

In addition to his research and undergraduate teaching, and hobbies such as rock climbing, cubing, and soccer, Li is involved in numerous activities, including supporting students in the Undergraduate Math Club and serving as a (now past) deputy treasurer of the University’s chapter of the Society for Industrial and Applied Mathematics.

Whatever the activity, Li’s passion for both math and teaching are evident, and he doesn't let a learning opportunity go to waste.

“Designing and leading a course like this was an extremely valuable experience for me. I had never done anything like it before and I learned a lot about how to work with students with different learning styles and adjust the material of the course to fit the pace and interests of the students.”

He has led sessions on Eulerian cycles, coding theory, and graph colorings for the University’s Mathematics Center for Educational Programs, which offers Saturday Morning Enrichment classes for 5th–7th grade students. The classes “are an opportunity to show young people higher-level math problems in a fun and accessible way,” he says.

The experience helped Li to better understand pedagogy. “The lesson plans… taught me a lot about how to make advanced math understandable and interesting to students without any formal background.”

Li also is an organizer for the UMN graduate-student-run Directed Reading Program, which gives undergraduates the opportunity to work with graduate-level math students on independent reading projects. 

Because of this work, Li was selected to be a fall 2022 mentor for the Spelman-Morehouse Directed Reading Program, which pairs students at Spelman and Morehouse Colleges with graduate students to work on virtual, semester-long reading projects. Li and his mentee explored The Mathematics of Elections and Voting by W.D. Wallis.

“I think the Directed Reading Program is a great way to work one-on-one with undergraduate students on a topic that may not be part of their normal curriculum. I am very grateful that the program has let me work with several bright and passionate young math students.”

21 Stones Revisited

Wondering how the game of 21 stones works?

The "solution" is that the second player can guarantee a win. (Li admits to being “a little sneaky” by saying he would go second.) If the first player takes one stone, the second player should take two. If the first player takes two stones, the second player should take one. This way, the stones are removed in groups of three (two, then one or one, then two). 

The pile will therefore shrink from 21 to 18 to 15 and so on, until only three stones remain, and this will happen when it is the first player's turn. No matter what they do, they cannot take all three of the remaining stones, leaving the second player to take whatever is left—and win! 

And while this is the trick to the game, Li suggests that someone who is thinking like a mathematician might wonder: How does this change if you start with 22 stones? How about 23? What if you can take more than two stones in one turn? What if there are three players? 

“Math is just as much about asking questions as it is about answering them,” he says.

OLLI Takeaways

When asked what one thing from his OLLI Scholar experience he will most use in his professional life, Li says, “Designing and leading a course like this was an extremely valuable experience for me. I had never done anything like it before and I learned a lot about how to work with students with different learning styles and adjust the material of the course to fit the pace and interests of the students.”

And in his personal life? “On a personal level, I think the program was a great way to make some connections with the broader community in the Twin Cities. As a graduate student at the University, the vast majority of my day-to-day interactions are with other people affiliated with the University in some way. It was great to be able to get to know some people outside of that bubble.”

Li's advice for future OLLI Scholars: "Be flexible and be ready to work with a variety of backgrounds."

Would you like to join the nearly 300 UMN students who have participated in the OLLI Scholars program? Now's your chance! Applications for the 2023–24 academic year will be accepted through December 31, 2023. Here's how to get started: OLLI Scholar Overview and Guidelines.

You can support UMN graduates and post-doctoral researchers through the OLLI Scholars Program Fund.