Six ducks in a pond

Question

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.

Answer 1

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 $i$.

Answer 2

Let's see if this answer makes sense. Let's say the ducks spread out as far as possible. It would make sense that to do this, they spread evenly around the circumference. Draw a circle of radius 5 and place 6 points along the perimeter to represent the ducks. Line segments connecting each point to its closest 2 neighbors creates a regular hexagon. Connecting these points to the center of the circle creates 6 equilateral triangles with a side length equal to the radius of 5.