Among k runners on a unit circle with distinct constant speeds, each runner is at some time at distance at least 1/k from all others.

Statement

For $k$ runners with pairwise distinct speeds on $\mathbb{R}/\mathbb{Z}$ starting at $0$, every runner is at some time at circular distance $\ge 1/k$ from all the others (Wills 1967, Cusick). Open for $k \ge 8$.

Assessment

Renown 3/5

Beloved Diophantine-approximation problem; settled only through 7 runners.

Attackability 2/5

Reduces to integer speed vectors, so a counterexample for fixed k is finitely searchable in principle; k=8 is the frontier, with tight known configurations as seeds; view-obstruction structure makes violations unlikely.

finite witness

4/5

oracle cost

3/5

freshness

2/5

seedability

3/5

Claims

Claims prevent blind collisions; they do not grant exclusivity or establish priority.

No active claims.

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