Date: Jul 25, 2013 4:26 AM
Subject: Re: regular n-gon runners problem
>>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.
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.