Sphere packing

Maryna Viazovska is a Ukrainian mathematician, born in Kyiv. She studied in Ukraine and Germany and earned her PhD at the University of Bonn in 2013. She is a professor at EPFL in Switzerland.

In 2022, she became the second woman to receive a Fields Medal, one of the major awards in mathematics.

Her work asks how tightly identical spheres can be packed without overlapping. In 2016, she proved the best possible packing in eight dimensions; with collaborators, she also solved the problem in 24 dimensions. Finding an arrangement and proving that nothing can beat it are different challenges. Try a smaller version below: fit as many identical disks as you can into three shapes. This game uses flat disks and finite regions to introduce the idea of packing.

Graphs and their structure

Maria Chudnovsky is a professor of mathematics at Princeton University. She studied at the Technion in Israel and earned her PhD at Princeton in 2003. She received a MacArthur Fellowship in 2012.

She works in graph theory: the study of vertices and the edges joining them. Graphs can represent friendships, transport networks or tasks that cannot be scheduled at the same time.

With Neil Robertson, Paul Seymour and Robin Thomas, she proved the Strong Perfect Graph Theorem, settling a conjecture proposed by Claude Berge in 1961. Their proof was announced in 2002 and published in 2006.

The theorem gives a way to tell whether a graph is perfect by looking for particular cycles. Try it below: move the vertices, mark an odd hole, and explore the complement.

Words, games and mathematics

Edward Doolittle is Kanyen’kehake (Mohawk), from Six Nations in southern Ontario. He is an associate professor of mathematics at First Nations University of Canada. He earned his PhD at the University of Toronto in 1997, specializing in partial differential equations.

His work includes Indigenous mathematics and mathematics education. The Canadian Mathematical Society recognized his contributions to mathematics education with the 2023 Adrien Pouliot Award.

With Zack Wolske, he has explored word versions of Spot-It, a card game in which every pair of cards shares exactly one symbol. Finding a whole deck whose symbols also make real words creates a mathematical design problem. His July 2026 presentation included work with English, French and Plains Cree dictionaries.

Try a small English example below. Match letters, complete a deck, then see whether you can add one more card. This example was built for this activity using the same mathematical idea.

The mathematics of parking cars

Pamela E. Harris is a Mexican-American mathematician. She moved from Mexico to Milwaukee and was the first person in her family to attend university.

Her research includes combinatorics: the study of discrete structures and the ways they can be arranged and counted. One topic is parking functions, which describe when cars following a simple parking rule can all find a space.

Try the three parking games below. Each has three different queues. Predict who parks, repair a queue by changing one car, then repair queues that need changes to several cars.

Pursuit games on graphs

Anthony Bonato is a Canadian mathematician at Toronto Metropolitan University. He is openly gay and has helped organize events that support LGBTQ+ people in mathematics.

He studies graphs and networks, including Cops and Robbers: a game in which players move between connected vertices. A central question is how many cops are needed to guarantee a capture, however the robber plays.

Choose a side below, or play with another person. The computer can play either side. Both sides can see every piece.

Building signals from simple waves

Lawrence W. Baggett is a mathematician at the University of Colorado Boulder. Blind since childhood, he has worked in harmonic analysis, wavelet theory and related areas of mathematics.

Harmonic analysis studies ways to understand functions and signals through simpler components. A sound can contain several frequencies at once; the strengths of those components help determine what it sounds like.

Reconstruct a sound below. Adjust the strengths of four simple tones. You can play by listening, comparing waveforms, or using both.

Billiards, surfaces and dynamics

Maryam Mirzakhani (1977–2017) was an Iranian mathematician. In 2014 she became the first woman to receive a Fields Medal, recognized for her work on the geometry and dynamics of surfaces.

Her work with collaborators also connects to the paths of billiard balls in polygonal tables. A ball moving in straight lines and reflecting from walls leads to surprisingly rich mathematical questions.

Aim a ball at a target below. Start with rectangles, then try an L-shaped table. Unfold a shot into reflected copies of the table to see the geometry behind the bounces.

When the order matters

Chelsea Walton is a Black American mathematician and a professor at Rice University. She grew up in Detroit, attended public schools and enjoyed patterns and logic puzzles from a young age.

Her research is in noncommutative algebra, including the symmetries of algebraic structures. “Noncommutative” means that changing the order of two operations can change the result.

Try this idea with labelled shapes. Start with a square: turn it, reflect it and compare what happens when you change the order. Then work out how to undo a sequence, finishing with a pentagon.