Run away from lions in a cage

Question

I came across an interesting problem:

There is a round cage and you are in it. Also two lions are in this cage too. The start position is that the distance between you and both lions is the diameter of the circle (you are on opposite sides of the cage). The speed of the lion is 1 m/s.

And the question is:

What is the minimal speed you need to have to always run away from lions and never be caught.

Probably it is a bit simpler to search the maximal speed when the lions will catch you. And the result of the original task will be the upper limit of that value.

I think the radius of the cage doesn't matter - it is only a scale problem. The only important thing is that the cage is round.

There is a similar problem here for one lion. But the answer links to buy some book and I couldn't find where to download it for free =).

And also I wonder if there is a solution for the generalized task with $N$ lions. That looks too complicated but I think the idea is the same - the lions should build a line when you can't run between any two of them and two lions on the ends of a chain will behave like the ones in the two-lions problem.

Answer 1

Let the centre of the ring be some origin. Let,at any instant, your position be U and those of the lions be $L_1$ and $L_2$. Suppose the minimum speed at which you must move be v. Let, at any instant, the x components of the velocities of the lions be a and $b$ and that of yours be c.

Then,

$v_{L_1} = <a,\sqrt{1- a^2}>$

$v_{L_2 }= <b,\sqrt{1- b^2}>$

$v_U=<c,\sqrt{v^2-c^2}>$

Now the relation between a, c and b, c is best represented by implicit functions. (You can't really tell whether you are watching the hungry lions run like mad to catch you and then decide how to move or the lions are smart enough to watch you run away from them in fear and than plan out a strategy). Also, if the lions are still smarter, a and b will be dependent on each other.(They can plan to get you together and then share!).

So you understand that the problem is not specific and leads to arbitrary possibilities. It would be better if you can restate the problem more formally.

As for example,(for the solution with N lions) you can think of a $N+1$-gon such that all $N_i$ vetrtices (i goes from 1 to N) move towards the N+1 th one with a constant rate. The N+1 th vertex must never coincide with a $N_i$ vertex while all the vertices are bounded in a circle.

Answer 2

First, i don't get why two lions are there. But supose they will move always directly to you. If you would move same speed as them around the cage, they would be coming closer and closer to you until they would finnaly cacht you. Or not? I thing they would be still-closing but never able to catch you, since it will end up that they will move 0.0000001mm just behind you. So the conclusion is that you need to move any larger speed than theirs.

Answer 3

It seems so that this is related to the optimization theory problem The Lion and Man and is in general a pursuit problem.

Peter

Answer 4

I would hope to avoid being cornered (metaphorically speaking) and try to get on a path that passes through the middle. The lions on the other hand, being team workers, would want to get to a position where they could home in for the kill. I see it going pretty much like this:

Initially we are all heading towards H, after a long chase in which the lions have just failed to catch me. At around the centre, the lions sense they are not going to catch me in a straight line, but notice the edge of the ring is not too far off, so they instinctively start to spread out along the paths CL.

The lions are just about to start homing in for the kill when I scupper their plans by doing an about turn towards J. Thinking that they have a chance to catch me they home in the other way towards the point E where they think they might get their meal.

Luckily I just avoid being eaten because I have a laser measuring device and a calculator with me.

So my conclusion is that the ratio of my speed to the lions needs to be slightly greater than HE:LE which would presumably be dependent on the diameter of the ring.