A duck, pursued by a fox, escapes to the center of a perfectly circular pond. The fox cannot swim, and the duck cannot take flight from the water. The fox is four times faster than the duck. Assuming the fox and duck pursue optimum strategies, is it possible for the duck to reach the edge of the pond and fly away without being eaten? If so, how?
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asked Feb 1, 2014 at 12:33
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4$\begingroup$ The purpose of boldface and italics is to make certain words stand out, to make them special over the other words and catch the eye of the reader on a cursory glance. If everything is boldface and italicized then all the words are standing out, they are all special, in the same way. Therefore none stand out, and none is special. Therefore using boldface and/or italics through the entire post is... illogical. $\endgroup$– Asaf Karagila ♦Commented Feb 1, 2014 at 12:42
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$\begingroup$ Must be a duplicate, though I can't find it fast. Check dcg.ethz.ch/members/roger/puzzles $\endgroup$– MacavityCommented Feb 1, 2014 at 12:43
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$\begingroup$ Perhaps, this. $\endgroup$– David MitraCommented Feb 1, 2014 at 12:45
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1 Answer
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Definitions
Let $R$ be the radius of the pond. Let the velocities be $v$ for the duck, and $4v$ for the fox (see diagram).
Phase 1 - The Headstart
As long as the duck stays with a circle of radius $\frac{R}{4}$, he can ensure that he keeps the fox as far away as possible (on a diametral line to himself) by turning in a spiral, where his maximum outward velocity is given by:
$\dot{r} = v\sqrt{1 - \dfrac{16r^2}{R^2}}$
Phase 2 - The Escape
Assume now that the duck has reached the point $D$ (as shown) at a radius $r$ from the center (with the fox at point $F$), and wants to begin phase 2. His fastest route to shore takes him to point $S$ and covers a distance of $R-r$, while the fox must cover arc length $R\pi$ to reach $S$. Hence, for the two times:
$t_D = \dfrac{R-r}{v}$ for the duck, and $t_F = \dfrac{R\pi}{4v}$ for the fox.
If the duck is to make safety we need
$\dfrac{R-r}{v} < \dfrac{R\pi}{4v}$ or $r > (1 - \dfrac{\pi}{4}) R \approx 0.2146 R$. Since this is within the spiral zone $(r < \dfrac{R}{4})$,
the duck will be able to safely reach the shore.
