The Chronicles of Circles: The Wolf, the Girl and the Lake

damionhonegan
Posers and Puzzles 28 May '12 01:42
1. 28 May '12 01:42
1) A girl is swimming in the middle of a perfectly circular lake. A wolf is running at the edge of the lake waiting for the girl. The wolf is within a fence surrounding the lake, but it cannot get out of the fence. The girl can climb over the fence. However if the wolf is at the edge of the lake where the girl touches it,
then it will eat her. The wolf runs 2 times faster than the girl can swim. Assume the wolf always runs toward the closest point on the edge of the lake to where the girl is inside. Can the girl escape? If so, what path should she swim in?

2) A girl is in the centre of a circular lake. There is a wolf on the edge that runs 4 times faster than she swims. He will always take the shortest path to the closest point on the shore to your current location. How does she escape?
2. talzamir
Art, not a Toil
28 May '12 12:01
If the girl swims a hair's breadth from the center of the lake and waits, the wolf will move to the nearest point. From there, the distance for the girl to the point of the shore farthest from the wolf is R, and for the wolf along the shoreline pi x R, so the girl gets there and escapes even if the wolf is twice as fast, or even three times as fast as she.

The angular speed of the girl around the center point of the lake depends on her distance from the center. If the wolf is four times as fast as the girl, the angular speed of the girl is larger than the wolf's as long as her distance from the center of the lake is less than R/4. Therefore she can swim in a spiral, keeping the center of the lake between herself and the wolf, as long as she is no farther than that from the middle point. From there dashing to the shore is a distance 3/4 R, so when the girl makes a run for it at speed v it will take her (3/4 R) / v = .75 (R/v) to reach the shore; and for the wolf, pi R / (4 v) = pi/4 (R/v) which is about 0.77 (R/v), so the wolf is too slow.

A natural follow-up question is whether there could be a wolf that is fast enough, and if so, how great is the minimum speed it needs to have? Seems that 4v is almost, but not quite, enough.
3. AThousandYoung
28 May '12 18:52
Originally posted by talzamir
If the girl swims a hair's breadth from the center of the lake and waits, the wolf will move to the nearest point. From there, the distance for the girl to the point of the shore farthest from the wolf is R, and for the wolf along the shoreline pi x R, so the girl gets there and escapes even if the wolf is twice as fast, or even three times as fast as she. ...[text shortened]... w great is the minimum speed it needs to have? Seems that 4v is almost, but not quite, enough.
A natural reponse to that question is 2 pi.
4. forkedknight
Defend the Universe
29 May '12 18:36
Originally posted by talzamir
The angular speed of the girl around the center point of the lake depends on her distance from the center. If the wolf is four times as fast as the girl, the angular speed of the girl is larger than the wolf's as long as her distance from the center of the lake is less than R/4. Therefore she can swim in a spiral, keeping the center of the lake between hers ...[text shortened]... and for the wolf, pi R / (4 v) = pi/4 (R/v) which is about 0.77 (R/v), so the wolf is too slow.
How far does the girl have to swim, and how far does the wolf have to run before the girls escapes in this second (4v) case?
5. SwissGambit
Caninus Interruptus
31 May '12 05:04
Originally posted by damionhonegan
1) A girl is swimming in the middle of a perfectly circular lake. A wolf is running at the edge of the lake waiting for the girl. The wolf is within a fence surrounding the lake, but it cannot get out of the fence. The girl can climb over the fence. However if the wolf is at the edge of the lake where the girl touches it,
then it will eat her. Th ...[text shortened]... ard the closest point on the edge of the lake to where the girl is inside. Can the girl escape?
No, because she doesn't have a calculator. ðŸ™‚
6. talzamir
Art, not a Toil
31 May '12 13:51
Originally posted by AThousandYoung
A natural reponse to that question is 2 pi.
2 pi = 6.28 or so, meaning that a wolf six times as fast as the girl is too slow?
Sounds a bit much?

v1 = girlspeed
v2 = wolfspeed
R = radius of the lake
r = max radius to which the girl can get so that the center point of the lake is between her and the wolf.

If the girl circles the center point at some radius x, one circle takes her 2 pi x / v1. The wolf stalking her needs 2 pi / v2 per lap. If the girl has a better laptime she can use some of her speed to get closer to the shore while keeping the wolf farthest from the point on the shore nearest to the girl. This strategy works until

2 pi x / v1 = 2 pi R / v2
x = R v1 / v2 = r

When the girl can't expand x anymore so that the wolf stays at the point farthest from her, her best chance would seem to be to dash rapidly to the nearest point on the shore. The distance is R - r and the time (R - r) / v1.

(R - r) / v1 < pi R / v2
R - R v1 / v2 < pi R v1 / v2
v2 - v1 < pi v1
v2 < (pi + 1) v1

So a wolf 4.1416 times as fast as the girl catches her, at least if she uses she uses the strategy of spiraling as far as she can and then swimming straight for the shore. Perhaps there is a better strategy for the girl?

(diving of course would be smart and would, if the lake is deep enough all the way to the shore, elude very fast wolves indeed.)
7. AThousandYoung
02 Jun '12 03:38
pi + 1, interesting.
8. forkedknight
Defend the Universe
04 Jun '12 20:20
Originally posted by damionhonegan
2) A girl is in the centre of a circular lake. There is a wolf on the edge that runs 4 times faster than she swims. He will always take the shortest path to the closest point on the shore to your current location. How does she escape?
If the bolded part is true, then the girl can just dither back and forth, keeping wolf running alternately clockwise and counterclockwise, instead of swimming in a spiral. It seems like the wolf would have to be much faster than 4.16x in order to prevent this from working.
9. Pianoman1
Nil desperandum
05 Jun '12 06:58
Originally posted by damionhonegan
1) A girl is swimming in the middle of a perfectly circular lake. A wolf is running at the edge of the lake waiting for the girl. The wolf is within a fence surrounding the lake, but it cannot get out of the fence. The girl can climb over the fence. However if the wolf is at the edge of the lake where the girl touches it,
then it will eat her. Th ...[text shortened]... e shortest path to the closest point on the shore to your current location. How does she escape?
If the girl swims in a tiny circle around the centre of the lake, them the wolf will have to run all the way around the lake to be always at the closest point to her. The wolf will soon collapse with exhaustion allowing the girl to escape!
10. talzamir
Art, not a Toil
05 Jun '12 08:00
Then again, wolves are well known for their endurance while swimming girls aren't..

How does oscillating about the middle of the lake help? The wolf will at all times be running away from the point on the shore farthest from the girl. Swimming in a spiral will keep the wolf at the point that is the least advantageous for it, that is, the point on the shore farthest from the girl. Is that not ideal for the girl? That way the girl can get to (r, 0) while the wolf is at (-R, 0). What is not obvious to me is what is the girl's best strategy from that point on. Heading straight for (R,0) suffices for any wolf with speed less than pi + 1 times the speed of the girl. But the wolf is approaching either clockwise or counter-clockwise but not both, so it could be that the girl could improve her odds a bit more by choosing to do something else. Or not.

This reminds me of another puzzle that is far simpler. A hunter shoots at a lion that is due west from where the hunter is standing. The lion is not killed, just gets angry and attacks; the gun jams. The hunter knows that there is a river running east-west somewhere to the north and notes two things about the lion - that it's twice times as fast as he is, even when wounded, and smart enough to run straight for the interception point along the hunter's current course, rather than along a pursuit curve. In what direction should the hunter run so as to make it to the river before the lion catches him - the answer is not "straight north", which has me think that the girl could also have better strategies than a mad dash for the shore.
11. sonhouse
Fast and Curious
07 Jun '12 15:13
Originally posted by Pianoman1
If the girl swims in a tiny circle around the centre of the lake, them the wolf will have to run all the way around the lake to be always at the closest point to her. The wolf will soon collapse with exhaustion allowing the girl to escape!
What she doesn't know is the wolf is a robot powered by energizer batteries.
12. talzamir
Art, not a Toil
09 Jun '12 11:02
And the girl clearly isn't as her batteries would short-circuit in the water..
13. talzamir
Art, not a Toil
11 Jul '12 05:18
I've been looking into this with a friend of mine. There is a fairly simple strategy for the girl that defeats even faster wolves, but the exact value of it is hard to say as though the path of the girl is simple, the equation for it is nasty. Numerically the value is easy to find though, so it seems that the girl can escape a wolf that is approximately 4.59 times as fast as she; a lot better than pi + 1. The solution can probably be improved still, making the challenge, can the girl escape from a wolf five times as fast as she?