TITLE: Six ducks in a pond
QUESTION [9 upvotes]: This puzzle has got me stumped.
$6$ ducks are swimming on a pond radius $5$. Show that at any moment there are two ducks a distance at most $5$ apart.

REPLY [6 votes]: If one duck sits at the center of the pond we are done. Otherwise each duck has a well defined argument (polar angle), and we can number the docks cyclically according to increasing arguments. In this way we have
$$\arg(D_i)-\arg(D_{i-1})\geq0,\quad \sum_{i=1}^6 \bigl(\arg(D_i)-\arg(D_{i-1})\bigr)=2\pi\ .$$
It follows that $\arg(D_i)-\arg(D_{i-1})\leq{\pi\over3}$ for at least one $i\in[6]$, and since a circular sector of radius $5$ and central angle ${\pi\over3}$ has diameter $5$ it follows that $|D_i-D_{i-1}|\leq5$ for such $i$.