Math Puzzle: Lion in Circular Cage Puzzle

This is a problem you can be sure you've solved, only to find a twist in the road.<p>It's problem 1 in The Art of Mathematics by Béla Bollobás, a challenging puzzle book for serious math junkies. The solution in that book has a good explanation of (a) what you probably didn't think of, and sounds right; and (b) why it's wrong.<p>And just when you think Bollobás's book had the final word, you found out there are versions of the puzzle when both players can win. What? Don't take my word for it:<p><a href="http://arxiv.org/pdf/0909.2524.pdf" rel="nofollow">http://arxiv.org/pdf/0909.2524.pdf</a>

Assuming both players react instantaneously, and space is modelled as continuous (not discrete), I think the following strategy always wins for the lion.<p><pre><code> 1. Lion runs to centre of ring 2. Lion moves toward tamer, but always staying directly between tamer and centre. </code></pre> (2) is always possible, as the arc the lion has to move around to keep between tamer and centre is always shorter than any arc the tamer can move along since the lion is closer in. As a corollory, the lion can always get closer to the tamer unless the tamer moves direclty away from the centre.<p>Eventually the tamer reaches the edge of the ring and can no longer move directly away. Then the lion can continue to move outward, always between the tamer and centre, until they meet, and the lion eats.

For those curious about the mathematical objects involved: <a href="http://mathworld.wolfram.com/PursuitCurve.html" rel="nofollow">http://mathworld.wolfram.com/PursuitCurve.html</a> <a href="https://en.wikipedia.org/wiki/Pursuit_curve" rel="nofollow">https://en.wikipedia.org/wiki/Pursuit_curve</a><p>If you model this, you will probably get very different answers based on how well you discrete-ize (is there a better word for that idea?) the time and movement.<p>Also, it's not a turn based game, i.e. the man doesn't move and then the lion moves. They move together at the same time. I think if you could answer what happens if the man has his back to wall and the lion is as close as possible without catching him, then you would solve the problem. Because the lion has to predict where the person is going to move next, it feels like you won't ever know for sure what will happen. As time tends towards infinity, the lion will probably eat the guy due to sheer chance of both of them going the same direction at the right time.

You can conclude that the tamer will be eaten if the lion simply moves towards the tamer.<p>In traditional pursuit curves, the region in which the chase happens is unbounded. Here, this isn't the case.<p>The best that the tamer can do if the lion uses this strategy is to move away from the lion. If at any point he is unable to do so (i.e. when he hits a wall of the cage) then the distance shrinks.<p>Depending on the speeds at which they can travel and the size of the cage, the lion will either catch the tamer in a finite amount of time, or he will asymptotically approach the tamer.

I have seen a variant of this (but cannot remember where I saw it):<p>A duck is in the center of a circular lake. A fox is on the bank. The duck only needs to reach the bank (without simultaneously being caught by the fox) to win. The fox cannot swim.<p>If both move at the same speed, the fox wins easily. But what if the fox moves 4× as fast as the duck?

Mathematics aside, is this one of those puzzles where the answer is "of course the lion tamer is not caught - because he's a lion tamer and therefore the lion does not chase him."

I hate questions being phrased like this. I always end up thinking of things like "who entered the cage first", which is important in reality but has no bearing on the mathematical nature of the question.<p>If the lion tamer is in the cage before when the lion enters, then it's unlikely the that the Lion will catch the tamer. Unless there's some other factor, like the lion has been starved, or is nervous.

all the lion has to do is travel directly towards the tamer. any turns that the tamer does create an angle that the lion can cut through. of course this is the obvious answer, so i'm probably missing something.

If the lion just waits, I bet the tamer will die of dehydration first.