# Puzzle #11 – Terra Fermat

I recently watched The Hunger Games: Catching Fire. By means of a convenient plot device, Katniss ends up back in the arena. She is immediately presented with a test of her physical intuition.

In the centre of the arena is a circular lake. The lake is divided into twelve circular sectors of equal size by rocky pathways. The pathways converge on an island bearing the cornucopia, housing a bounty of weaponry.

The twenty-four tributes start in one of the circular sectors on a small pedestal, some distance from the cornucopia.

Suppose Katniss can swim at speed $u$ and run at speed $v$. For simplicity we’ll assume both $u$ and $v$ are constant. Assume also that $v>u$.

Question: At what angle should Katniss approach the nearest rocky pathway such that she reaches the cornucopia in the least possible time?

Hint: It is easiest to define the angle as being that which Katniss’ path subtends with the normal to the pathway.