GoKawiilThe original version of this story appeared in Quanta Magazine.
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 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. But for the past two decades, no one has been able to advance any further.
Then last year, Matthieu Rosenfeld, a mathematician at the Laboratory of Computer Science, Robotics, and Microelectronics of Montpellier, settled the conjecture for eight runners. And within a few weeks, a second-year undergraduate at the University of Oxford named Tanupat (Paul) Trakulthongchai built on Rosenfeld’s ideas to prove it for nine and 10 runners.
The sudden progress has renewed interest in the problem. “It’s really a quantum leap,” said Beck, who was not involved in the work. Adding just one runner makes the task of proving the conjecture “exponentially harder,” he said. “Going from seven runners to now 10 runners is amazing.”
The Starting Dash
At first, the lonely runner problem had nothing to do with running.
Instead, mathematicians were interested in a seemingly unrelated problem: how to use fractions to approximate irrational numbers such as pi, a task that has a vast number of applications. In the 1960s, a graduate student named Jörg M. Wills conjectured that a century-old method for doing so is optimal—that there’s no way to improve it.
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