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 place

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

counterclockwise along the track (forever). Assume the speeds

v_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 runners

are 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, when

reduced to lowest terms the n numerators yield all possible

distinct residues mod n.

Example (1):

Speeds 1,2,4 cannot yield an equilateral triangle since, of the

fractions

(1-1)/(2-1) = 0/1

(2-1)/(2-1) = 1/1

(4-1)/(2-1) = 3/1

there is no reduced numerator congruent to 2 mod 3.

Example (2):

Speeds 1,2,6 do yield an equilateral triangle since, of the

fractions

(1-1)/(2-1) = 0/1

(2-1)/(2-1) = 1/1

(6-1)/(2-1) = 5/1

the numerators of the reduced fractions include all the

residues 0,1,2 mod 3.

quasi