quasi
Posts:
11,467
Registered:
7/15/05


Re: regular ngon runners problem
Posted:
Jul 25, 2013 4:26 AM


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 on a circular track of circumference 1, and proceed to run along the track (forever). The runners run at pairwise distinct constant 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 runners are the vertices of a regular ngon iff for some permutation v_0,v_1, ..., v_(n1) of the n speeds,
(1) Each of the n fractions (v_k  v_0)/(v_1  v_0) for k = 0,1, ..., (n1) is a rational number.
(2) For k = 0,1, ..., (n1), letting a_k/b_k denote a reduced fraction form of the fraction (v_k  v_0)/(v_1  v_0), each b_k is relatively prime to n, and a_k = k*(b_k) mod n.
quasi

