Combinatorics

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.

A team of four prominent mathematicians, including two Fields medalists, proved a conjecture described as a “holy grail of additive combinatorics.”

Some strange mathematical sequences are always whole numbers — until they’re not. The puzzling patterns have revealed ties to graph theory and prime numbers, awing mathematicians.

A new proof breaks a decades-long drought of progress on the problem of estimating the size of triangles created by cramming points into a square.