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Problem workshop

One great problem, three kinds of immersion

From a story to your own conjecture

Problem stories open curiosity. The problem workshop is the next place, where curiosity becomes a testable idea.

  1. 1 · DiscoverMeet the questionLearn when it appeared and why it still holds people’s attention.
  2. 2 · ChallengeTest one caseMake your first observation in five minutes with a drawing, calculation, or colors.
  3. 3 · DevelopYou are hereBuild on ideasPublish an observation and grow it through comparisons, conjectures, and counterexamples.
Week 7 · MathOverflowPartial progress
Primary source

Lonely Runner Conjecture — A Moment of Solitude

Intermediate· Posed 1967

Problem

k+1k+1 runners on a unit circular track start at the same point with distinct constant speeds. For every runner, does there exist a time when they are at distance ≥1/(k+1)\ge 1/(k+1) from every other runner?

Why it matters

Posed independently by Wills (1967) and Cusick (1973), named by Goddyn (1998): "every runner becomes lonely at some moment." Looks geometric but is equivalent to deep questions in Diophantine approximation and lattice-point avoidance.

Progress so far

Proved for k≤6k \le 6 (up to 7 runners) by Barajas & Serra (2008). Open for k=7k = 7 (8 runners) and beyond. The boundary has advanced one runner at a time: k=4k=4 (Bienia–Goddyn 1998), k=5k=5 (Bohman–Holzman–Kleitman 2001), k=6k=6 (Barajas–Serra 2008).

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

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