Date: Jul 25, 2013 4:26 AM
Author: quasi
Subject: Re: regular n-gon runners problem
quasi wrote:

>quasi wrote:

>>

>>Here's a revised version ...

>

>Close but still not quite right.

>

>I'll make one final revision.

No, the statement is still flawed.

I was careless, but I understand it better now.

Here's a fix ...

Suppose n runners, n >= 3, start at the same time and place on

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

the track (forever). The runners run at pairwise distinct

constant real speeds, measured in revolutions per unit time,

with the sign of each speed -- positive, negative, or zero,

determining the direction of motion -- counterclockwise,

clockwise, or stationary, respectively.

Conjecture:

There is an instant of time when the locations of the runners

are the vertices of a regular n-gon iff for some permutation

v_0,v_1, ..., v_(n-1) of the n speeds,

(1) Each of the n fractions (v_k - v_0)/(v_1 - v_0) for

k = 0,1, ..., (n-1) is a rational number.

(2) For k = 0,1, ..., (n-1), letting a_k/b_k denote a reduced

fraction form of the fraction (v_k - v_0)/(v_1 - v_0), each

b_k is relatively prime to n, and a_k = k*(b_k) mod n.

quasi