Fox vs. Duck

At a radius of slightly less than $\frac{r}{4}$, the duck can swim in circles, forcing the fox to run around.



Once the duck is at an angle of $\pi$ from the fox, it starts swimming towards the shore.

• The duck has to cover a distance of $\frac{3r}{4}$
• The fox has to cover a distance of $\frac{2\pi \cdot r}{2}$

Since the fox moves four times faster, the distance of the fox has to be larger than four times the distance of the duck.

As we can see, \begin{equation}\frac{3r}{4} * 4 < r* \pi \end{equation} \begin{equation} 3r  < 3.14r \end{equation} \begin{equation} 3  < 3.14 \end{equation} Therefore, the duck will survive!