Escape the Lake (Solution)

The first thing to realize is that you can’t just swim directly away from the monster. To make things easier on ourselves, let’s assume the lake has a radius of 1 and we swim at a rate of 1 per minute (so the monster runs at a rate of 4 per minute). If we swim directly away from the monster, it takes us 1 minute to reach the shore. The monster needs to run half the circumference of the circle, which is pi. Since the monster can run 4 in 1 minute and pi is less than 4, the monster will easily outrun us and be waiting at the shore to eat us when we arrive. Like so:

So what can we do? At first it seems impossible, but the solution is quite cute. The critical realization is that if we swim in a circle of radius 1/4, we can complete one full rotation in exactly the same time as the monster can run around the lake.

So if we swim in a circle of radius barely less than 1/4, then the monster will not be able to keep up with us. If we do this for long enough, eventually the monster will be on the opposite side of the circle as us. In other words, we will already be 1/4 of the way towards the opposite edge of the lake. Can we make a dash for it?

We will have to swim 3/4 to reach the other side and the monster needs to run halfway around the circle. The monster can run 3 in the time it takes us to run 3/4, but half the circle is more than that! So, indeed, this strategy works.

If it doesn’t make sense, watch this:

Can we do better? Is there a better strategy than dashing in a straight line directly away from the monster? It turns out yes. For a complete full derivation of the solution, I will refer you to this post, but I’ll tell you that the best strategy as follows:

Assume you swim at a speed of 1 and the monster runs at a speed of v.

1. Do the same circling procedure described above (at a radius of slightly less than 1/v) to get as far as possible away from the monster.
2. When you’re at your farthest point, instead of running straight away from the monster towards the shore, run at a right angle to that path. This causes the monster’s path to be the longest in order to reach you.

Take a look:

As you can see from the top right, in this simulation the monster’s speed is 4.55. At the limit, you can actually escape any speed lower than about 4.603338848. I set the speed somewhat lower to A) make the optimal strategy take less time and B) because there are only so many pixels on the screen and rounding means we can’t reach the theoretical limit.