The Lonely Runner Conjecture

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There are runners going around a circular track of unit length. They all start together from the same spot and each one has their own distinct speed. The distance between two runners is the distance around the track between them. A runner is defined as lonely if his distance to any other runner is at least .


The Lonely Runner conjecture states that each runner becomes lonely at some time [1]. The conjecture has been proven for up to seven runners [2].

This Demonstration lets you verify this conjecture. Select a number of runners and start the race. When a runner becomes lonely, his position and his marker on the speed gauge turn red and two red arcs of length are displayed fore and aft of his position.

Each new race or renewed selection of the number of runners generates a different set of random speeds.


Contributed by: Erik Mahieu (May 2014)
Open content licensed under CC BY-NC-SA




[1] T. S. Roberts. "Lonely Runner Conjecture." Unsolved Problems in Number Theory, Logic, and Cryptography. (May 21, 2014)

[2] S. Czerwiński, "Random Runners Are Very Lonely."

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