#lonely runner

[electromirror]

Ditian - Lonely Runner *clip

TECH HOUSE

2015-01-12

[curiosamathematica]

The Lonely Runner conjecture

Consider some, say k, runners on a circular track of unit length. They start on the same position, and run at constant but pairwise distinct speeds. A runner is called lonely if he is at distance of at least 1/k from every other runner.

The lonely runner conjecture states that each runner is lonely at some time. Oddly, the conjecture has only been solved up to 7 runners! Perhaps running is a more social activity in bigger goups?

Detailed history can be found here.

[horv]

The Lonely Runner Conjecture

The Lonely Runner Conjecture is one of the most well-known open problems in Mathematics... but a great example of how some problems can made accessible to anyone.

In 1954, Jorg Wills wrote a treatise on some problems in the field of Diophantine approximation, the branch of number theory which attempts (essentially) to approximate real numbers (or almost all numbers) with rational or whole numbers.

Of the few problems discussed in his paper, one has escaped solution. In the mathematical language provided by Wills, his conjecture asked the following question. Given a set of natural numbers k_i, let.

This is a math problem that someone would fail to understand just on looking at it. It uses terminology and symbols unfamiliar to the layperson, and some layering of functions which takes a little while to figure out for the mathematician. So why am I talking about this? T.W. Cusick noticed, some 15 years later, that this wasn't so bad after all, once you looked in the right place. Here's what he asked.

Suppose k runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for any given runner, is there a time at which that runner is distance at least  1/k (along the track) away from every other runner?

This is the exact same problem. Its even more general than Wills. Here, he transforms a a simple piece of abstract math into something accessible to anyone. Is it true that if a bunch of people run around a track and don't speed up or slow they will get lonely? 

Matt Stoffregen and I got interested in this problem by looking at Wikipedia. We've since made our contribution to its solution, but this would never have come without the intuition of a genius like Cusick. This is what I think good math is, or at least interesting math; transforming difficult and obscure problems into readily accessible ones that might be worked on by undergrads. Ultimately very few problems might be able to be treated this way, but it is always nice when they can be.