小蓝视频

What do you get when you put two curious 小蓝视频 math majors together with a classic board game? The result is new discoveries that could change how we think about strategy and problem-solving 鈥� all thanks to rainbow static Mastermind. 

If you鈥檝e 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鈥檚 code. None of the colors are repeated. The goal is to break the code with the shortest possible list of questions. 

For Riley Vavolizza 鈥�26, a mathematics and education studies major from Pleasantville, New York, and Rachel Xia 鈥�26, a mathematics major from Tianjin, China, the familiar childhood game has become the ultimate research 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鈥檚 Faculty Student Summer Research program. Scores of Skidmore students  鈥� more than 120 in 2025 鈥� 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鈥檚 not just about playing the game 鈥� it鈥檚 about pushing the limits of what is possible. 

 鈥淚t鈥檚 all about experimenting with different combinations, patterns, and new rules to discover better strategies,鈥� 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.

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.  

鈥淭hey鈥檙e so creative, and their input leads me to try new things,鈥� she says. 鈥淭heir curiosity has helped me get invested in projects that I never would have come up with on my own.鈥�  

Both students agree that the summer research experience has opened their eyes to what鈥檚 possible in math 鈥� and beyond. Xia says the challenge of proving something that seems obvious was surprisingly humbling.

"It taught me patience in problem-solving,鈥� 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.鈥�

鈥淚t鈥檚 made me more open-minded about pursuing research opportunities too,鈥�  she added. 

Their work is about more than just numbers on a page. It鈥檚 about learning how to ask better questions, thinking creatively, and building confidence to tackle problems 鈥� ideas that are the heart of a liberal arts education at Skidmore.  These skills matter whether you鈥檙e solving a board game, teaching high school students, or designing algorithms.  

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.