Combinatorics

Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future.

It was intuitive, even obvious. It was also wrong.

Born poor in colonial India and dead at 32, Ramanujan had fantastical, out-of-nowhere visions that continue to shape the field today.

A new proof marks the first progress in decades on a problem about how order emerges from disorder.

Mathematicians have illuminated what sets of points can look like if the distances between them are all whole numbers.

Recursion builds bridges between ideas from across different math classes and illustrates the power of creative mathematical thinking.

If Anna beats Benji in a game and Benji beats Carl, will Anna beat Carl?

Landmark results in Ramsey theory and a remarkably simple aperiodic tile capped a year of mathematical delight and discovery.

The Markov numbers reveal the secrets of irrational numbers and the patterns of the Fibonacci sequence. But there’s one question about them that has resisted proof for over a century.