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Rates and motion · Grades 7–10 · Unsolved

Lonely Runner Conjecture

Runners start together on a round track, all at different speeds. Will every runner, at some moment, be far from everyone else?

ABCB and C, both at the startStart/finish at topMoment shown: 30 secondsThree runners, one trackA runs 1 lap per minuteB runs 2 laps per minuteC runs 4 laps per minuteShaded arc: everywhere withina third of a lap of A. HereB and C are both at the start,outside it. A is “lonely.”
After 30 seconds, B and C are back at the start while A is halfway around. A is lonely.

Try it

2 runners: far means half a lap3 runners: far means a third of a lap4 runners: a quarter

What we know

It’s proved for up to seven runners.

For eight or more, nobody knows.

Ask your students

Where is each runner after half a minute? A third of a minute?

Act it out: three students, three speeds. Freeze! Who’s lonely?

Detailed one-pager (PDF) History, why it’s unsolved, and more classroom ideas

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