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I have seen a variant of this (but cannot remember where I saw it): 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. If both move at the same speed, the fox wins easily. But what if the fox moves 4× as fast as the duck?
I have seen a variant of this (but cannot remember where I saw it): 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. If both move at the same speed, the fox wins easily. But what if the fox moves 4× as fast as the duck?
I believe you meant to say that the duck wins easily if they both move at the same speed.
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. 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.
the order does not matter because the Lion can move to any part of the cage before starting his pursuit.
It's for this reason that you'll never see a Lion tamer enter a cage full of lions. He or She will always enter the cage first, and the lions will follow.
It has no bearing on a hypothetical question, but in reality it would make a huge difference.
I remember as a kid, a math teacher carefully explaining to me why my answer to a question that involved cooking was wrong. The maths assumed you could just increase the quantity of ingredients proportionally to the number of people you were cooking for, but that's not always the case, and in this example I was right in reality, but wrong mathematically.
Earlier quoted context omitted.
Nope, you're wrong.
Can you elaborate? I was wondering the same thing as yogsototh.
The paper linked elsewhere here gives an illustrative example where this happens in what is clearly finite time.
(This is if the tamer is on the border as parent claimed - the paper also shows that the lion can't win if he doesn't do this)
Earlier quoted context omitted.
The pursuit curve assumes that the lion always moves directly towards the tamer. The better strategy for the lion is your back-to-the-wall position. The lion should be able to get infinitesimally close to the tamer. Pragmatically and finally, the delta will be smaller the reach of the lion's pawns. Lion wins.
"...back-to-the-wall position." Can you expand on this? My thought was that if the lion 'zig-zagged' somewhat, it would effectively force the tamer to constantly change direction[1] and ultimately the lion would catch up. Is that what you meant by back-to-the-wall? [1] Assuming that the tamer's only strategy is to maximise distance from the lion.
The lion has to move further to the side than the tamer, so the tamer switches directions after each "step". This works, because within a circle the tamer with his back to the wall has to move a little towards the lion while moving sideways. The lion moves a little bit more to the side a little bit less towards the tamer. Over time the lion comes closer and the steps become smaller. When it gets really close however, the round wall looks increasingly like a flat wall. Lion and tamer end up just moving sideways without ever touching? We need some more math for this.