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Math Puzzle: Lion in Circular Cage Puzzle

pratikpoddarcse.blogspot.in

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Re: Math Puzzle: Lion in Circular Cage Puzzle

#31

Post by zackzackzack »

hythloday wrote: post #30
zackzackzack wrote: post #2

For those curious about the mathematical objects involved: http://mathworld.wolfram.com/PursuitCurve.html https://en.wikipedia.org/wiki/Pursuit_curve 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. Also, it's not a turn based game, i.e. the man doesn't move and then the lion moves. They move together at t…

is there a better word for that idea? Quantize. :)

Thank you! I knew it had to be a word that I knew and had shamefully forgotten.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#33

Post by SonicSoul »

onemoreact wrote: post #15
frooxie wrote: post #13

Earlier quoted context omitted.

Am I missing something? My reasoning is that the lion will easily catch the lion tamer by moving straight towards him/her. If the tamer moves in the same direction, he/she will eventually hit a wall. If the tamer moves in any other direction, the lion will be able to move at least slightly closer to it. Repeat until caught. Or?

It really just depends on how you model time and space. Assume that the lion and man are on a hex grid and the lion is 1 unit from the man. The man can now move to 3 spaces without being caught and change his orientation relative to the lion, which assuming the grid is reasonably large allows for him to move along a closed loop though space which means he can stay alive. If on the other hand the man is moving +/- 1 X…

what part of this question makes it a possibility that they're on a grid and can only move in 4 directions? in regular world (which is used to host this riddle) humans and animals can move diagonally as well.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#34

Post by SonicSoul »

tomelders wrote: post #27

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.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#35

Post by andylei »

frooxie wrote: post #13
tylerneylon wrote: post #9

This is a problem you can be sure you've solved, only to find a twist in the road. 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. And just when you think Bollobás's book had the final word, you found out there are versions…

Am I missing something? My reasoning is that the lion will easily catch the lion tamer by moving straight towards him/her. If the tamer moves in the same direction, he/she will eventually hit a wall. If the tamer moves in any other direction, the lion will be able to move at least slightly closer to it. Repeat until caught. Or?

from the article:

> We split time into a sequence of intervals, of lengths t1, t2, t3, . . .. At the ith step the man runs for time ti in a straight line that is perpendicular to his radius vector at the start of the step. He chooses to run into the half plane that does not contain the lion (if the lion is on the radius then either direction will do). So certainly the lion does not catch the man in this time step. The man then repeats this procedure for the next time step, and so on.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#36

Post by onemoreact »

SonicSoul wrote: post #33
onemoreact wrote: post #15

Earlier quoted context omitted.

It really just depends on how you model time and space. Assume that the lion and man are on a hex grid and the lion is 1 unit from the man. The man can now move to 3 spaces without being caught and change his orientation relative to the lion, which assuming the grid is reasonably large allows for him to move along a closed loop though space which means he can stay alive. If on the other hand the man is moving +/- 1 X…

what part of this question makes it a possibility that they're on a grid and can only move in 4 directions? in regular world (which is used to host this riddle) humans and animals can move diagonally as well.

Let's change the rules slightly. Take a ring one mile in diameter and assume the lion needs to get within 1 foot of the tamer. They both move at 1 inch per tick of game clock in any direction. To make things simple the the lion goes to the center of the ring, then aims for 2 inch closer to the ring than the tamer. Well the tamer will either go closer to the wall or the lion get's closer to him every second. But while he runs along the wall the lion takes a shorter path closer to the center of the ring, catches him and eats him. Also, if the tamer ever goes closer to the center of the ring in any turn the lion get's closer to him that turn because the lion stays on a line between the center of the ring and the tamer.

The reason my example does not work for this puzzle is simply there is no distance closer the the center of the ring the lion can aim for and still get to eat him. At which point you need to think about what space looks like for points that are really close to each other.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#37

Post by finnw »

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?

Re: Math Puzzle: Lion in Circular Cage Puzzle

#38

Post by gcp »

yogsototh wrote: post #21
screwt wrote: post #19

Assuming both players react instantaneously, and space is modelled as continuous (not discrete), I think the following strategy always wins for the lion. 1. Lion runs to centre of ring 2. Lion moves toward tamer, but always staying directly between tamer and centre. (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 s…

I believe it works only if the Lion and the tamer have a non null size (not represented as point). I am not sure of this, but if you represent the Lion and the tamer as points, the Lion can go as close as it want to the tamer, but never really reach him. Of course considering the tamer is on the border.

Nope, you're wrong.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#39

Post by spenceyboi »

finnw wrote: post #37

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.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#40

Post by skyo »

gcp wrote: post #38
yogsototh wrote: post #21

Earlier quoted context omitted.

I believe it works only if the Lion and the tamer have a non null size (not represented as point). I am not sure of this, but if you represent the Lion and the tamer as points, the Lion can go as close as it want to the tamer, but never really reach him. Of course considering the tamer is on the border.

Nope, you're wrong.

Can you elaborate? I was wondering the same thing as yogsototh.
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