Suppose that N runners are running laps on a circular track. Each runner has a constant speed that is different from all the others. A runner is lonely whenever there is no other runner closer than 1/N of a lap. Is it true that every runner will be lonely at some time? Jörg Wills conjectured in 1965 that the answer is always “yes”. This problem is not yet solved for more than seven runners. Yet its solution would have consequences in computational geometry, diophantine approximation and graph coloring.