Math major masterminds crack the code of a classic board game
What do you get when you put two curious èƵ math majors together with a classic board game? The result is new discoveries that could change how we think about strategy and problem-solving — all thanks to rainbow static Mastermind.
If you’ve never played Mastermind, here are the basics: One player creates a secret code using colored pegs, and the other tries to guess it using logic, pattern recognition, and plenty of creativity.
In the rainbow static variation of the game, the codebreaker must supply a list of questions to determine their opponent’s code. None of the colors are repeated. The goal is to break the code with the shortest possible list of questions.
For Riley Vavolizza ’26, a mathematics and education studies major from Pleasantville, New York, and Rachel Xia ’26, a mathematics major from Tianjin, China, the familiar childhood game has become the ultimate research puzzle.
Associate Professor Kirsten Hogenson uses math to show a new way to solve the classic peg-and-color puzzle.
The two Skidmore students are spending five weeks over the summer working with Associate Professor Kirsten Hogenson in the Department of Mathematics and Statistics as part of Skidmore’s Faculty Student Summer Research program. Scores of Skidmore students — more than 120 in 2025 — from fields as disparate as philosophy and neuroscience participate each summer.
Together, Hogenson, Vavolizza, and Xia have been diving into strategies for cracking codes in the shortest number of guesses. It’s not just about playing the game — it’s about pushing the limits of what is possible.
“It’s all about experimenting with different combinations, patterns, and new rules to discover better strategies,” Vavolizza explains.
One of their coolest discoveries? They determined a shorter winning question list than previously known for when a player has three pegs and multiple colors. In math speak, they found a new upper bound (a value greater than or equal to all other values in a given set) for gr(3,k), where 3 is the number of pegs and k represents the number of colors.
They verified that their list will always win the game and that it's the shortest list when the number of colors is small. They're now working on a proof that will show they have the best possible list for larger values of k.
Riley Vavolizza ’26 (left), who works with Associate Professor Kristen Hogenson, is one of more than 120 students to participate in Faculty Student Summer Research with Skidmore faculty this summer.
Professor Hogenson, a graph theorist who studies networks and their hidden connections, says working with students like Vavolizza and Xia is her favorite part of the job.
“They’re so creative, and their input leads me to try new things,” she says. “Their curiosity has helped me get invested in projects that I never would have come up with on my own.”
Both students agree that the summer research experience has opened their eyes to what’s possible in math — and beyond. Xia says the challenge of proving something that seems obvious was surprisingly humbling.
"It taught me patience in problem-solving,” Xia says.
For Vavolizza, the project connects perfectly to future goals.
I plan on teaching math after Skidmore, and this experience has been a great way to expand my interest in math and engage in creative problem-solving.”
“It’s made me more open-minded about pursuing research opportunities too,” she added.
Their work is about more than just numbers on a page. It’s about learning how to ask better questions, thinking creatively, and building confidence to tackle problems — ideas that are the heart of a liberal arts education at Skidmore. These skills matter whether you’re solving a board game, teaching high school students, or designing algorithms.
Associate Professor Kirsten Hogenson works with math majors Rachel Xia ’26 and Riley Vavolizza ’26 during their summer research on Rainbow Static Mastermind.
Their experience also highlights the unique opportunities Skidmore students have to pursue rigorous and creative research alongside leading faculty experts.
As Vavolizza and Xia look ahead to new challenges, they are carrying the confidence to tackle real-world questions with them and integrating Creative Thought Matters in their fields and futures.