Date: Jul 23, 2013 7:32 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.
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.
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).