```Date: Jul 23, 2013 6:46 AM
Author: quasi
Subject: Re: regular n-gon runners problem

Here's a revised version ...Suppose n runners, n >= 3, start at the same time and placeon a circular track of circumference 1, and proceed to runcounterclockwise along the track (forever). Assume the speedsv_1,v_2, ..., v_n of the runners, expressed in revolutions per unit time, are positive real numbers such that v_1 < v_2 < ... < v_n.Conjecture:There is an instant of time where the locations of the n runnersare the vertices of a regular n-gon iff each of the n fractions    (v_i - v_1)/(v_2 - v_1)for i = 1,2,...,n is a rational number, and moreover, whenreduced to lowest terms the n numerators yield all possibledistinct residues mod n.Example (1):Speeds 1,2,4 cannot yield an equilateral triangle since, of thefractions   (1-1)/(2-1) = 0/1   (2-1)/(2-1) = 1/1   (4-1)/(2-1) = 3/1there is no reduced numerator congruent to 2 mod 3.Example (2):Speeds 1,2,6 do yield an equilateral triangle since, of thefractions   (1-1)/(2-1) = 0/1   (2-1)/(2-1) = 1/1   (6-1)/(2-1) = 5/1the numerators of the reduced fractions include all the residues 0,1,2 mod 3.quasi
```