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

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 quasi Posts: 12,067 Registered: 7/15/05
Re: regular n-gon runners problem
Posted: Jul 23, 2013 3:42 AM

quasi wrote:
>
>Suppose n runners, n >= 3, start at the same time and place
>on a circular track, and proceed to run counterclockwise along
>the track (forever), each at a distinct positive constant speed.
>
>Conjecture:
>
>If there is an instant where the locations of the n runners are
>the vertices of a regular n-gon, then the speeds of the runners,
>arranged in ascending order, form an arithmetic sequence.
>
>Remark:
>
>It's easy to see that the converse holds.

The converse holds but not the conjecture.

It's not that simple.

Here's an easy counterexample:

Take 3 runners on a track of circumference 1, with respective
speeds, expressed in revolutions per unit time, of 1,5,15.
Then at time t=1/6, the runners form an equilateral triangle.

quasi

Date Subject Author
7/23/13 quasi
7/23/13 Virgil
7/23/13 quasi
7/23/13 quasi
7/23/13 quasi
7/23/13 quasi
7/25/13 quasi
7/23/13 William Elliot
7/23/13 quasi
7/23/13 William Elliot
7/23/13 quasi
7/23/13 James Waldby
7/23/13 quasi