Strategy / calculus riddle

Question

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A lady and a monster

Here is another rather famous riddle - I've seen it several times, but only once in its full form that I quote here:

A duck is located in the center of a circular pond, near which a hungry fox is waiting. The fox cannot swim - it can only run around the pond. The duck can fly only if it gets out of the pond (don't ask, it's probably a genetically altered duck). What's the minimal ratio of the fox to duck speed such that the fox will always catch the duck? What's the duck's strategy to escape the fox if the fox is slower than that?

The lower bound of $\pi$ is immediate. Upon some further reflection one gets $\pi+1$, but I heard one can do better than that. Any ideas?

Answer 1

Here is a very brief solution recently posted in the Usenet group alt.math.recreational by Ray Koopman to essentially the same problem; I’ve converted the straight ASCII to something more readable. (Note that in this version the duck is you, and the fox is a bear.)

If the lake has center $(0,0)$ and radius $1$, the bear is at $(-1,0)$, the speed ratio is $k$, you are at $\left(\frac1k,0\right)$, and the bear starts moving counterclockwise, then you should head toward $\left(\frac1k, \sqrt{1 - \frac1{k^2}}\right)$ instead of $(1,0)$. You will always be on the same side of the lake as the bear, so he won’t reverse direction. You will win the race as long as $$\pi + \arccos \frac1k > \sqrt{k^2 - 1}\;,$$ which works out to $k < 4.6$ (approximately).

For more detail, see the first two answers to this earlier question.

Answer 2

I found this page using google and it gives a number bigger than $\pi + 1$ but not much explanation.