For most people, the image of runners circling a track may suggest competition, but for Tanupat Trakulthongchai, a second-year mathematics undergraduate at St John’s College at Oxford University, it became the doorway into one of mathematics’ most deceptively simple unsolved problems: the Lonely Runner Conjecture.
The problem, first posed nearly six decades ago, sounds almost like a children’s riddle.
The conjecture says that, at some point, every runner will become “lonely” — a distance far enough away from all the others on the track. Simple enough to picture, yet difficult enough to have challenged generations of mathematicians.
Tanupat Trakulthongchai (Photo: Evan Nedyalkov)
Tanupat explains it plainly: “If you have n runners … starting from the same point and running on a circular track with different speeds … at one point each runner would be lonely.”
The track is assigned a length of 1, and a runner is considered “lonely” when they are at least 1/n of the track away from all other runners. The lonely moment does not have to happen at the same time for everyone. Each runner only needs to have their own moment of separation.
That combination — easy to state, hard to prove — was exactly what drew Tanupat in.
“I was very surprised when I learned that this is a conjecture,” he said. “It seemed very obvious … something very simple to prove. But it turns out that we still haven’t been able to prove that.”
Compared with many advanced mathematical problems, the Lonely Runner Conjecture does not require pages of technical language to understand. Yet underneath its seemingly benign surface lies a complicated structure involving number theory, geometry and combinatorics.
Finite verification process
Tanupat’s work builds on the method of the French mathematician Matthieu Rosenfeld, whose paper reduced parts of the problem into a finite verification process, in this case for eight runners. In other words, instead of proving the conjecture all at once in pure abstraction, mathematicians could narrow it down to a large but checkable number of cases.
The difficulty, however, was that the number of cases remained enormous.
“What Rosenfeld did was … turn the problem into a practical finite checking problem,” Tanupat explained. “But that still is going to take a lot of time.”
Tanupat’s contribution was to streamline that process using combinatorics — the branch of mathematics concerned with counting, structure and arrangements.
By doing precomputation beforehand, he was able to reduce the main computational bottleneck. “It introduced an extra step, but the extra step reduced the work in the main step,” he said.
Initially, Tanupat’s work advanced the conjecture up to 10 runners. But the story did not stop there. In a later preprint with another Thai co-author, he said they had extended the result up to 13 runners.
Being young, and Thai, in such a specialised field also requires a particular mindset. Tanupat described research as a balance between humility and confidence. On one hand, he said, progress is impossible without consulting existing literature. On the other hand, “you have to have a certain sort of arrogance” — the belief that perhaps you can see something others have missed.
Real-world applications?
For readers wondering whether this mathematical loneliness has any real-world use, Tanupat is careful not to oversell it. “There is no application directly from this conjecture,” he said. But the broader field it belongs to, Diophantine approximation, has important links in how we represent numbers in computers.
The conjecture also has a surprisingly literal connection to running. Tanupat noted that many mathematicians who work on the problem seem to mention their personal experience with running in some form, despite there being no mathematical reason for the link.
He himself used to be a gifted sprinter in primary school. “I was a very good runner,” he said, laughing that he did not train seriously and “fell down over the years”.
As for the future, Tanupat believes that extending the current computational approach can only go so far. As the number of runners increases, the cases multiply. “You only have a finite amount of time. Eventually this computational approach reaches its limit.”
The full conjecture remains unresolved, but Tanupat’s work has pushed its boundary forward. For a problem built around distance, his story shows how mathematics runs through a connection of old ideas, new methods and a young Thai student willing to chase a mystery still running after six decades.
