ELMO 2018 Problem 6

A windmill is a closed line segment of unit length with a distinguished endpoint, the pivot. Let $S$ be a finite set of $n$ points such that the distance between any two points of $S$ is greater than $c$ . A configuration of $n$ windmills is admissible if no two windmills intersect and each point of $S$ is used exactly once as a pivot.

An admissible configuration of windmills is initially given to Geoff in the plane. In one operation Geoff can rotate any windmill around its pivot, either clockwise or counterclockwise and by any amount, as long as no two windmills intersect during the process. Show that Geoff can reach any other admissible configuration in finitely many operations, where

(i) $c = \sqrt 3$ ,

(ii) $c = \sqrt 2$ .

Proposed by Michael Ren