```Date: Jul 23, 2013 7:32 AM
Author: quasi
Subject: Re: regular n-gon runners problem

quasi wrote:>>Here's a revised version ...Close but still not quite right.I'll make one final revision.This one's right -- I'm sure of it. In fact, I can see how toprove it, but for now, I'll just state it as a conjecture.The revision ...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 speedsof the runners, expressed in revolutions per unit time, are pairwise distinct positive real numbers.Conjecture:There is an instant of time where the locations of the runners are the vertices of a regular n-gon iff for somepermutation v_1,v_2, ..., v_n of the n speeds, 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, a_i/b say, we have b = 1 mod n anda_i = i-1 mod n (Thus, a_1,a_2, ..., a_n yield all possibleresidues mod n).quasi
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