

Math Puzzle: Lion in Circular Cage Puzzle - pratikpoddar
http://pratikpoddarcse.blogspot.in/2012/02/lion-in-circular-cage-puzzle.html

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tylerneylon
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 of the puzzle when both players can win. What? Don't take
my word for it:

<http://arxiv.org/pdf/0909.2524.pdf>

~~~
frooxie
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?

~~~
onemoreact
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.

~~~
SonicSoul
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.

~~~
onemoreact
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.

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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.

~~~
yogsototh
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.

~~~
gcp
Nope, you're wrong.

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

~~~
gcp
Basically there is nothing preventing the lion from reaching the tamers'
coordinates exactly, i.e. it's not a case of the distance only closing "in the
limit".

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)

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zackzackzack
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 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.

~~~
aparadja
Related image from your first link
<http://mathworld.wolfram.com/images/gifs/purscir4.gif>

~~~
bradleyland
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 distance-to-intercept in order to be successful as a
species.

Thus, I believe the man is eaten rather quickly, even if the two are equal in
speed.

~~~
gcp
_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.

~~~
bradleyland
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.

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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.

~~~
gcp
_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.

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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?

~~~
spenceyboi
I believe you meant to say that the duck wins easily if they both move at the
same speed.

~~~
finnw
You're right, sorry (too late to edit)

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philbarr
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."

~~~
shpoonj
There are many holes in the question... Such as asking if it is POSSIBLE. We
know that all things are possible, however improbable, so the answer is yes.

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tomelders
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.

~~~
SonicSoul
the order does not matter because the Lion can move to any part of the cage
before starting his pursuit.

~~~
tomelders
my point is, I always mix the real with the hypothetical and get hooked up on
details that prevent me from answering questions like this. Lions are very
territorial, and respectful of territory. If a lion tamer enters a a cage with
a lion already in it then he is invading a lions territory. If a lion enters a
cage with a lion tamer in it, the the lion understands that he is invading
someone else's territory and is highly unlikely to attack.

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.

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SonicSoul
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.

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

