Original version of this story appeared inside Quanta Journal.
Visualize the incredible workout: A group of runners begin running around a circular track, with each runner maintaining a unique and consistent pace. Will every runner end up “lonely,” or far from everyone else, at least once, regardless of their speed?
Mathematicians think that the answer is yes.
The “lone runner” problem may seem simple and trivial, but it presents itself in many ways in mathematics. It is similar to questions in number theory, geometry, graph theory, and more—about when it is possible to find a clear line of sight in a field of obstacles, or where billiard balls can be moved on a table, or how to plan a network. “It’s very multifaceted. It touches on many different areas of mathematics,” he said Matthias Beck of San Francisco State University.
With only two or three runners, proof of concept is basic. Mathematicians proved it for four runners in the 1970s, and by 2007, they had found it. until seven. But for the last two decades, no one has been able to move forward.
Then last year, Matthieu Rosenfelda mathematician at the Computer Science, Robotics and Microelectronics Laboratory of Montpellier, solved the concept of eight runners. And within a few weeks, a second-year student at the University of Oxford called Tanupat (Paul) Trakulthongchai is built on Rosenfeld’s ideas to prove it nine and ten runners.
The sudden development has increased interest in the problem. “It’s really a big scale,” said Beck, who was not involved in the work. Adding just one runner makes the task of proving the concept “more difficult,” he said. “To go from seven runners to now 10 runners is amazing.”
Starting Dash
At first, the lone runner problem had nothing to do with running.
Instead, mathematicians became interested in a seemingly unrelated problem: how to use fractions to approximate irrational numbers like pi, a function that has a large number of applications. In the 1960s, a graduate student named Jörg M. Wills the assumption that century way of doing it it is better—that there is no way to improve it.
In 1998, a group of mathematicians rewrite the concept in the language of flight. Say N runners start from the same point on a circular track of length 1 unit, and each runs at a different constant speed. Wills’ hypothesis is equivalent to saying that every runner will end up alone at some point, regardless of the speeds of the other runners. More precisely, each runner will at some point find himself at a distance of at least 1/N from any other athlete.
When Wills saw the lone runner paper, he emailed one of the authors, Luis Goddyn of Simon Fraser University, to congratulate him on “this beautiful and poetic name.” (Goddyn’s response: “Oh, you’re still alive.”)
Mathematicians also showed that the lone runner problem is similar to another question. Imagine an endless sheet of graph paper. In the center of each grid, place a small square. Then start at one of the corners of the grid and draw a straight line. (A line can point in any direction except vertically or horizontally.) How big can the smaller squares be before the line hits one?
As versions of the lone runner problem proliferated throughout mathematics, interest in the question grew. Mathematicians proved different cases of the concept using completely different methods. Sometimes they relied on tools from number theory; sometimes they turned to geometry or graph theory.
