Problem dossier · Graph Theory
Lonely runner conjecture
k runners with pairwise distinct constant speeds circle a unit track from a common start. The conjecture: each runner is, at some moment, at circular distance at least 1/k from every other runner.
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Status
Open. Proved for k ≤ 7; equivalent to a view-obstruction / Diophantine approximation problem for integer speeds.
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The Angle of Attack
Reduce to integer speed vectors and search the k = 8 case computationally; simulate trajectories for k = 3–8 and map the tight speed sets where loneliness is barely achieved.
Tags: diophantine approximation · view obstruction · simulation
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The Lab
No instruments built yet. When this problem gets tackled, its interactive instruments — explorers, searches, verifiers running in the browser — live here. See the Collatz dossier for what a fully tackled problem looks like.
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The Log
Empty. Work on this problem gets logged here as dated entries — constructions tried, code run, dead ends included. Dead ends are results.
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