Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Math Puzzle: Lion in Circular Cage Puzzle

pratikpoddarcse.blogspot.in

Post reply on HN
Print view View original

11–20 of 45 posts

Re: Math Puzzle: Lion in Circular Cage Puzzle

#11

Post by Cry_Wolf »

You can conclude that the tamer will be eaten if the lion simply moves towards the tamer.

In traditional pursuit curves, the region in which the chase happens is unbounded. Here, this isn't the case.

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.

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.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#12

Post by gcp »

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…

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.

This is all but obvious to me. I pick a random number from 0 to 2*Pi. If you guess my number correctly, you get to eat me. You can guess an infinite amount of times.

Who wins this game? Does the answer depend on the assumption of fundamental axioms?

Re: Math Puzzle: Lion in Circular Cage Puzzle

#13

Post by frooxie »

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?

Re: Math Puzzle: Lion in Circular Cage Puzzle

#14

Post by gcp »

Cry_Wolf wrote: post #11

You can conclude that the tamer will be eaten if the lion simply moves towards the tamer. In traditional pursuit curves, the region in which the chase happens is unbounded. Here, this isn't the case. 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. Depending on the speeds…

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.

Given that they're point masses, I can't see how speed and size of the cage could affect the solution. Unless the initial distance between lion and tamer also factors in.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#15

Post by onemoreact »

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?

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 and or Y on an XY grid at (1,0) and the lion is at (0,0) then the man must move to (2,Y) or risk being eaten which means his X must increase which mean he will eventually hit a wall.

PS: If the lion can always get within striking distance of the man then there is no difference between moving at the same time and alternating, because the lion can chose randomly and then catch back up on a miss which enables him to win eventually.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#16

Post by [deleted] »

gcp wrote: post #12
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…

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. This is all but obvious to me. I pick a random number from 0 to 2*Pi. If you guess my number correctly, you get to eat me. You can guess an infinite amount of times. Who wins this game? Does the answer depend on the assumption of fundamental axioms?

[deleted]

Re: Math Puzzle: Lion in Circular Cage Puzzle

#17

Post by [deleted] »

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?

[deleted]

Re: Math Puzzle: Lion in Circular Cage Puzzle

#18

Post by bradleyland »

gcp wrote: post #10
bradleyland wrote: post #6

Earlier quoted context omitted.

My problem with this representation is that it assumes the pursuer is unable to "lead" its target. The pursuit curve, as modeled in mathematical representations constrains the pursuer to traveling only on a path that points directly at the pursued. Lions are highly adapted hunters that frequently take down prey that are much faster than them. It would seem that they would have to develop strategies for optimizing the…

My problem with this representation is that it assumes the pursuer is unable to "lead" its target. How can the pursuer lead the target when the target moves randomly? At best, he can adjust his lead with an infinitesimal delay, but its not clear this allows him to catch the target in finite time.

And yet Lions catch faster prey all the time. Just because something occurs randomly doesn't mean it always results in a net loss for the pursuer. All the time spent in pursuit on an optimized (leading) path results in closed distance. Within the constraints of the cage the pursued can only choose to zig or to zag, so if the lion uses a leading path, and is correct in anticipating directional changes only half the time, he's still going to have lunch sooner rather than later. There are visual "tells" that allow the lion to anticipate directional shifts on the part of their prey, so I'd argue that his percentages will be significantly better.

I guess I'm struggling with understanding the value of the exercise. If we're talking inanimate bodies, I get it. This kind of physics is valuable. However, choosing arbitrary constraints in order to make interesting maths out of natural scenarios is low hanging fruit. I just don't see it as all that interesting.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#19

Post by screwt »

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 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.

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.

Re: Math Puzzle: Lion in Circular Cage Puzzle

#20

Post by ranit8 »

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…

Sorry, i'm too lazy to deeply check teh article. What happens if the lion starts from the center and always moves as to keep the center, himself and the tamer in a straight line? It should do some kind of spiral, but does it take infinite time to catch the tamer?

EDIT: someone came just before me. http://news.ycombinator.com/item?id=3617030

Post reply on HN