quasi
Posts:
11,740
Registered:
7/15/05


Re: regular ngon runners problem
Posted:
Jul 23, 2013 7:32 AM


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 ngon 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 = i1 mod n (Thus, a_1,a_2, ..., a_n yield all possible residues mod n).
quasi

