Lonely Runner Conjecture

Runners circle a track at different speeds. Does everyone eventually get some space to themselves?

Posed by Jörg Wills, 1967; named by Luis Goddyn, 1998.

In plain terms

Several runners start together on a circular track of length 1 and run forever at different constant speeds. Fix your attention on one of them. The conjecture says that at some moment, that runner is lonely: at distance at least 1/k from every other runner, where k is the total number of runners.

With two runners it is obvious — at some point the other one is on the far side of the track. With seven or eight, it stops being obvious, and the bound 1/k is exactly tight: with speeds 1, 2, …, k−1 the lonely moment happens, but with no room to spare.

What is known

Proved up to k = 7. A 2025 preprint by Rosenfeld claims k = 8, so that case is likely settled pending review, leaving k ≥ 9 as the frontier.

The problem reduces neatly: speeds can be assumed to be whole numbers, one runner can be parked at zero, and Tao showed it is enough to check speeds up to a computable bound. So each k is, in principle, a finite calculation — just an astronomically large one.

One belief about the problem turned out to be false. It was thought the only tight configurations were the obvious speeds 1, 2, …, k−1; Goddyn and Wong found others. This lab recovered those independently at k = 8, which is how it checked its own search was working.

Why it is hard

The finite bound is enormous, so brute force is hopeless beyond small k, and the tight cases sit right on the boundary — you cannot get away with an approximate argument, because the margin is exactly zero.

What a breakthrough would mean

This one has the richest connections of any problem here. It is equivalent to a question in view obstruction — whether a certain family of convex bodies covers space — and it has been linked to nowhere-zero flows in graphs and matroids, a genuinely applied corner of combinatorics. It also sits squarely in Diophantine approximation, since it is really a statement about how badly several numbers can be simultaneously approximated.

So a proof would not be an isolated curiosity. The techniques would be expected to say something about the covering and flow questions it is tied to. That breadth is unusual, and it is the honest reason to rate this problem above the others here on implications.

What this lab tried

Standing: Verified k=8 near-tight census to V=72: exactly 3 primitives below ML 13/100, all at ML=1/8. Goddyn-Wong instances recovered from scratch. k=9 is the frontier.

Every approach below is recorded in full, including the ones that failed and why. Attempt titles link to the complete record.

Go deeper

Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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