mathematics - Escaping a hungry lion you can't outrun

You are at the edge of an enormous circular arena. A hungry lion is eying you from the centre of this area. You are both capable of running at the same maximum speed, but constraint within the arena. The lion has worked out a strategy of always running at maximum speed in an outward direction such that he stays positioned on the line thru you and the center of the arena.

The starting signal sounds and the lion starts moving. You can't outrun the lion. How do you ensure you stay out of the lion's claws?

Clarifications:

• the arena is truly gigantic, and you can think of you and the lion as point objects constrained within a circle of unit radius


• there is no latency in the lion's reactions to your movements: at any moment in time you, the lion, and the center of the arena remain co-linear


• the lion catches you if, and only if, his position coincides with yours

Hint:

Running at maximum speed along any circular path centered on the center of the arena is not going to help you. Starting from the center, the lion will run at the same speed along a circular path of half the radius and catch you as soon as you have completed a quarter of a circle. (It is easy to check that these paths keep you, the lion and the center co-linear.)

Answer

First, run 1/3 of a unit toward the center of the arena. The radius of the arena is 1 unit and the lion runs at the same speed as you, so the lion will not catch you.

Now, we run along a series of straight line paths. The $n$-th path will be orthogonal to the the line segment connecting your position and the lion's (at the start of the path), and have a length of $\frac{1}{2n}$ units.

These paths will never take you outside the arena. To see this, if you are $d$ units from the center of the arena immediately before running along the $n$-th path, your distance from the center after the $n$-th path will be $$ \sqrt{d^2+\left(\frac{1}{2n}\right)^2}. $$ This means your distance from the center after the $n$-th path is $$ \sqrt{\left(\frac{2}{3}\right)^2+\left[\left(\frac{1}{2}\right)^2+\ldots+\left(\frac{1}{2n}\right)^2\right]}, $$ and this quantity is always less than 1.

Since each straight line path heads in a direction orthogonal to the lion, the lion cannot catch you.

Finally, the total distance you are running is $$ \frac{1}{3}+\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\ldots\right)=\infty $$ (this is essentially the fact that the harmonic series diverges), so you can continue to run along the chosen paths indefinitely.

In fact, this strategy enables you to evade the lion no matter how the lion decides to pursue you.