30
$\begingroup$

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.

  • 0
    If both lions are distant from you at a distance which is the diameter of the circle, both lions must be at the same point, i.e. you're glued on the wall of the cage and the lions are at the opposite direction of you in the circle, also on the wall. Maybe you meant you started in the center and both lions are on the walls to begin with?2012-01-05
  • 4
    @Patrick: Imaginary lions can share the same physical space and then split up.2012-01-05
  • 9
    Won't they follow the same path then? Unless they are intelligent lions and work out some badass optimal man-eating strategy.2012-01-05
  • 3
    There would be two versions. Co-op or competitive. There'd be some kind of Nash equilibrium or something. You could even have teams of lions if you have 3 or more. This problem is quite complicated AFAIK. An interesting variant involves trying to escape a pool while someone walks around the edge. Once you get out, you're faster than them. Obviously both of these problems can use a circle or a square or any shape. The main thing you can do is establish inequalities. Like find a strategy that always works for certain speeds. To get exact answers, you need some fancy calculus of paths...2012-01-05
  • 2
    Another interesting variant is if the man needs to eat food, and then he needs to catch the food dropped in the cage before the lions eat it. That looks like a pretty funny cellphone game. XD2012-01-05
  • 6
    To take @Asaf's point further, the lions in question are evidently point lions. Otherwise the distance between you and them would be less than the diameter of the cage by at least the width of a lion.2012-01-05
  • 0
    In the original problem 2 lions works as team (cooperative). In the case of concurency as they both are in the same situation, both of them will decide to have the same strategy and will behave as 1 lion in the very first problem. The cast with N lions is the same. So nothing interesting. But coperative strategy isn't obvious...2012-01-05
  • 6
    Why should both lions use the same strategy? By the was, the appropriate notion of strategy is [quite subtle](http://www.maths.qmul.ac.uk/~walters/papers/lion-and-man-journal.pdf).2012-01-05
  • 2
    "Why should both lions use the same strategy?" - because optimal strategy is single. If every lion works separately and on start they are in the same position, then the optimal strategy for both will be the same.2012-01-05
  • 1
    Just stay away from the point where the two lions intersect.2012-06-20
  • 1
    Here is a simulation with Sage: http://wiki.sagemath.org/animate?action=AttachFile&do=get&target=tamer.gif You can read the post here: http://wiki.sagemath.org/animate2012-07-01
  • 1
    Perhaps this person would know: http://www.huffingtonpost.com/2011/08/05/zoo-owner-alexander-pylys_n_918221.html2012-08-22
  • 0
    Given that start position, there's no solution where you can always run *away* from the lions because in the beginning you are already at the point of the arena which is the furthest away from the lions.2012-08-22
  • 1
    @OleksandrPshenychny: Given the symmetry of the problem (you can mirror the arena along the initial diameter connecting you and the lions without changing the problem), if the optimal strategy of the lions doesn't start by running exactly in your direction (with not starting to run immediately being a special case of that: running in your direction with speed 0), there must be *two* optimal strategies which are mirror images of each other. Therefore in that case each lion would have to choose one of them randomly, and there'd be a 50% chance for them to choose differently.2012-08-22

4 Answers 4