The quick brown fox and the lazy duck

Calculus Level 2

A fox moving at a constant speed of ss m/s\text{ m/s} wants to eat a duck that is swimming at 1 m/s1 \text{ m/s} in a pond. The duck can only fly when he’s on land, and the fox can’t swim.

What is the maximal value of ss such that wherever the animals start, the duck can reach land without being caught by the fox?


  • The pond is a perfectly round circle.
  • The fox can start anywhere except inside the pond, but he can never enter the pond.
  • The duck must start somewhere on the pond.
  • If the duck reaches the edge of the pond at the exact same time as the fox, the fox catches the duck.
  • If the duck reaches land and the fox is not there, he instantly flies away and the fox loses his meal.
  • The duck and the fox see each other and take decisions in real time.

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