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Topic: regular n-gon runners problem
Replies: 12   Last Post: Jul 25, 2013 4:26 AM

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Posts: 12,067
Registered: 7/15/05
Re: regular n-gon runners problem
Posted: Jul 23, 2013 6:46 AM
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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.


There is an instant of time where the locations of the n runners
are the vertices of a regular n-gon 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

(1-1)/(2-1) = 0/1
(2-1)/(2-1) = 1/1
(4-1)/(2-1) = 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

(1-1)/(2-1) = 0/1
(2-1)/(2-1) = 1/1
(6-1)/(2-1) = 5/1

the numerators of the reduced fractions include all the
residues 0,1,2 mod 3.


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