mathematics - The lake monster

You are on a rowboat in the middle of a large, perfectly circular lake. On the perimeter of the lake is a monster who wants to eat you, but fortunately, he can't swim. He can run (along the perimeter) exactly $4\times$ as fast as you can row, and he will always run towards the closest bit of shore to you. If you can touch shore even for a second without the monster already being upon you, you can escape. Suggest a strategy that will allow you to escape, and prove that it works.

Other notes:

• If two paths take the monster to this location equally quickly, he will arbitrarily choose one.


• The monster can reverse direction instantaneously, and you can turn your boat instantaneously.

Follow up: What is the minimum speed of the monster (relative to your boat) such that escape becomes impossible?

Answer

First of all, row out to a radius $R/4$ (where the lake has radius $R$) keeping you, the centre of the lake and the monster in a straight line - with you on the far side to the monster. This is always possible; radius $R/4$ is the first point where the angular speed you can achieve just matches that of the monster as he runs round to get you.

You are now a distance $3R/4$ away from the shore, directly opposite the monster so he needs to run a distance $\pi R$ to get you. You will take time $3R/4V$ at speed $V$ if you now row directly towards the nearest shore, and he will take $\pi R/4V$, which is fractionally greater.

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For the followup: If instead of $4\times$, the monster runs $N\times$ your speed... then you row out to radius $R/N$, you then take time $(N-1)R/NV$ to reach shore and he takes $\pi R/NV$ to reach the same point. You escape provided that $N < \pi + 1 \approx 4.1459$.