The ‘Lonely Runner’ Problem Only Appears Simple

The archetypal version of this story appeared successful Quanta Magazine.

Picture a bizarre grooming exercise: A radical of runners starts jogging astir a circular track, with each runner maintaining a unique, changeless pace. Will each runner extremity up “lonely,” oregon comparatively acold from everyone else, astatine slightest once, nary substance their speeds?

Mathematicians conjecture that the reply is yes.

The “lonely runner” occupation mightiness look elemental and inconsequential, but it crops up successful galore guises passim math. It’s equivalent to questions successful fig theory, geometry, graph theory, and more—about erstwhile it’s imaginable to get a wide enactment of show successful a tract of obstacles, oregon wherever billiard balls mightiness determination connected a table, oregon however to signifier a network. “It has truthful galore facets. It touches truthful galore antithetic mathematical fields,” said Matthias Beck of San Francisco State University.

For conscionable 2 oregon 3 runners, the conjecture’s impervious is elementary. Mathematicians proved it for 4 runners successful the 1970s, and by 2007, they’d gotten as acold arsenic seven. But for the past 2 decades, nary 1 has been capable to beforehand immoderate further.

Then past year, Matthieu Rosenfeld, a mathematician astatine the Laboratory of Computer Science, Robotics, and Microelectronics of Montpellier, settled the conjecture for eight runners. And wrong a fewer weeks, a second-year undergraduate astatine the University of Oxford named Tanupat (Paul) Trakulthongchai built connected Rosenfeld’s ideas to beryllium it for nine and 10 runners.

The abrupt advancement has renewed involvement successful the problem. “It’s truly a quantum leap,” said Beck, who was not progressive successful the work. Adding conscionable 1 runner makes the task of proving the conjecture “exponentially harder,” helium said. “Going from 7 runners to present 10 runners is amazing.”

The Starting Dash

At first, the lonely runner occupation had thing to bash with running.

Instead, mathematicians were funny successful a seemingly unrelated problem: however to usage fractions to approximate irrational numbers specified arsenic pi, a task that has a immense fig of applications. In the 1960s, a postgraduate pupil named Jörg M. Wills conjectured that a century-old method for doing so is optimal—that there’s nary mode to amended it.

In 1998, a radical of mathematicians rewrote that conjecture successful the connection of running. Say N runners commencement from the aforesaid spot connected a circular way that’s 1 portion successful length, and each runs astatine a antithetic changeless speed. Wills’ conjecture is equivalent to saying that each runner volition ever extremity up lonely astatine immoderate point, nary substance what the different runners’ speeds are. More precisely, each runner volition astatine immoderate constituent find themselves astatine a region of astatine slightest 1/N from immoderate different runner.

When Wills saw the lonely runner paper, helium emailed 1 of the authors, Luis Goddyn of Simon Fraser University, to congratulate him connected “this fantastic and poetic name.” (Goddyn’s reply: “Oh, you are inactive alive.”)

Mathematicians besides showed that the lonely runner occupation is equivalent to yet different question. Imagine an infinite expanse of graph paper. In the halfway of each grid, spot a tiny square. Then commencement astatine 1 of the grid corners and gully a consecutive line. (The enactment tin constituent successful immoderate absorption different than perfectly vertical oregon horizontal.) How large tin the smaller squares get earlier the enactment indispensable deed one?

As versions of the lonely runner occupation proliferated passim mathematics, involvement successful the question grew. Mathematicians proved antithetic cases of the conjecture utilizing wholly antithetic techniques. Sometimes they relied connected tools from fig theory; astatine different times they turned to geometry oregon graph theory.