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


Re: regular ngon runners problem
Posted:
Jul 23, 2013 6:46 AM


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 ngon 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
(11)/(21) = 0/1 (21)/(21) = 1/1 (41)/(21) = 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
(11)/(21) = 0/1 (21)/(21) = 1/1 (61)/(21) = 5/1
the numerators of the reduced fractions include all the residues 0,1,2 mod 3.
quasi

