A straightforward conjecture about runners moving around a track turns out to be equivalent to many complex mathematical questions. Three new proofs mark the first significant progress on the problem in decades.

Picture a bizarre training exercise: A group of runners starts jogging around a circular track, with each runner maintaining a unique, constant pace. Will every runner end up “lonely,” or relatively far from everyone else, at least once, no matter their speeds?

Mathematicians conjecture that the answer is yes.

The “lonely runner” problem might seem simple and inconsequential, but it crops up in many guises throughout math. It’s equivalent to questions in number theory, geometry, graph theory, and more — about when it’s possible to get a clear line of sight in a field of obstacles, or where billiard balls might move on a table, or how to organize a network. “It has so many facets. It touches so many different mathematical fields,” said Matthias Beck(opens a new tab) of San Francisco State University.

For just two or three runners, the conjecture’s proof is elementary. Mathematicians proved it for four runners in the 1970s, and by 2007, they’d gotten as far as seven(opens a new tab). But for the past two decades, no one has been able to advance any further.

Then last year, Matthieu Rosenfeld(opens a new tab), a mathematician at the Laboratory of Computer Science, Robotics, and Microelectronics of Montpellier, settled the conjecture for eight runners(opens a new tab). And within a few weeks, a second-year undergraduate at the University of Oxford named Tanupat (Paul) Trakulthongchai(opens a new tab) built on Rosenfeld’s ideas to prove it for nine and 10(opens a new tab) runners.

...read more at quantamagazine.org