A famous math puzzle about how runners can avoid each other on a track has finally been solved for up to 12 people.
April 29, 2026
Original Paper
Eleven, twelve, and thirteen lonely runners
arXiv · 2604.23906
The Takeaway
The Lonely Runner Conjecture asks if every runner on a circular track will eventually be a certain distance away from everyone else. This deceptively simple problem has remained one of the most stubborn mysteries in number theory for decades. A new computer-assisted proof has finally extended the solution to cases with 10, 11, and 12 runners. It shows that as the number of runners or variables increases, the math becomes exponentially more complex. Pushing this boundary is a major milestone that helps mathematicians understand how patterns and gaps emerge in complex, repeating systems.
From the abstract
Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest integer. This statement is known as the Lonely Runner Conjecture. A computational method developed by Rosenfeld and the second author verified the conjecture for $k\leq9$. We further refine this method with new sieving techniques and employ a polynomial method argume