```Date: Jul 25, 2013 4:26 AM
Author: quasi
Subject: Re: regular n-gon runners problem

quasi wrote:>quasi wrote:>>>>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 ona circular track of circumference 1, and proceed to run alongthe track (forever). The runners run at pairwise distinctconstant 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.Conjecture:There is an instant of time when the locations of the runnersare the vertices of a regular n-gon iff for some permutationv_0,v_1, ..., v_(n-1) of the n speeds, (1) Each of the n fractions (v_k - v_0)/(v_1 - v_0) fork = 0,1, ..., (n-1) is a rational number.(2) For k = 0,1, ..., (n-1), letting a_k/b_k denote a reducedfraction form of the fraction (v_k - v_0)/(v_1 - v_0), eachb_k is relatively prime to n, and a_k = k*(b_k) mod n.quasi
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