# 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.)

• Who cares how fast he runs, each time he reaches the previous point you were in, you've moved! Therefore he can never reach you! (I know, obviously incorrect, prove it wrong, I dare you! :D) – warspyking Jan 25 '15 at 4:29
• I think you may be right, any movement other than straight back or standing still. If you break up time into smaller and smaller segments and simulate, the points will never coincide and you can extend this to infinity. – Quark Jan 25 '15 at 4:43
• @Quark - you might want to consider the lion's path in response to a circular escape path (as mentioned in the hint I have added above). – Johannes Jan 25 '15 at 4:52
• Yeah well like I said in the solution, no matter what the lion can reach "close" to you within a set amount of time (not sure where a quarter circle or half the radius comes from; there are no givens), it's just the moment right before overlap because the human is always moving perpendicular to the lion's path the points will never quite overlap. If I'm wrong a counter argument would be why approaching a circular line (Julian's answer) is better than starting with one. – Quark Jan 25 '15 at 5:02
• @Quark - a circular path of the human causes the lion to aim for a point ahead of the human, causing the human no longer to move perpendicular to the lion's movement. – Johannes Jan 25 '15 at 5:10

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.

• I agree this is a viable way as well, is there a reason to start with the longer lines though? This solution converges to a circle so you can start running in a circle. – Quark Jan 25 '15 at 4:27
• I don't think running in a circle works. If you run along a unit circle with unit speed, the lion's path (as a function of time) looks something like $(\sin^2(t),\sin(t)\cos(t))$, and lion catches you after $\pi/2$ units of time. – Julian Rosen Jan 25 '15 at 4:30
• You can't really plot the lion's path with a function (exactly) though because it's still a feedback system. It's not really worth arguing the point though since yours is easier to understand. – Quark Jan 25 '15 at 4:32
• I don't think this works, you can run in an infinite discrete spiral but it tends to a circle at inifnity. And the lion catches you on a circular path? – Ewan Apr 28 '15 at 14:12
• Would it also work to start by running any arbitrarily small distance q the center of the arena and then travel as fast as possible along the path described by parametric equations in polar coordinates with respect to x: theta=x; rho=1-q/(1+x)? Basically figuring that the path gets closer and closer to being a circle, but as long the path is spiraling outward the lion won't quite catch the runner? – supercat May 7 '15 at 21:41

To stay out of the lions' claws:

Walk slightly away from the edge (can be any amount > 0), then run in a perfect circle concentric with the arena.

Explanation:

If the lion and human are truly points and the lion always runs on the line between the center and the human, then by running in a circle the human can stay away (although infinitesimally close) to the lion. The lion will reach (near) the human by $R/V$ seconds no matter what, where $R$ is the radius of the ring and $V$ is the velocity of the lion in units of $R$ per second.

Here's a picture to show what happens:

In the picture,

The lines are exaggerated, just extend that to a polygon with "infinite" sides with a circumscribed circle. There's probably a better way to prove this with calculus and $\partial \theta$ but I forget how, just remember seeing this puzzle before.

• "If [..] the lion always runs towards the humans' current position, then [..]". The lion doesn't aim for the human. Rather it ensures it maintains a position in between the center and the human. – Johannes Jan 25 '15 at 4:17
• Yeah I meant to say that but the solution is still the same/similar. Trying to edit it with that change but having issues with the image spoiler. – Quark Jan 25 '15 at 4:19