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 to

prove 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 place

on a circular track of circumference 1, and proceed to run

counterclockwise along the track (forever). Assume the speeds

of 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 some

permutation 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, when

reduced to lowest terms, a_i/b say, we have b = 1 mod n and

a_i = i-1 mod n (Thus, a_1,a_2, ..., a_n yield all possible

residues mod n).

quasi