49 replies to this topic

[bonanova]

An ogre with a top land  speed of 4 mph, but incapable of swimming, stalks the shore of a circular lake. The fair maiden out in the boat can outrun the ogre on land, but her rowing speed is nothing to write home to mother about. How fast must she be able to row to effect an escape?

 

Make the assumption that the ogre is hungry [or otherwise desirous of the girl] and uses a best-strategy pursuit.

[ThunderCloud]

Spoiler for

If I were the girl in the boat, I think I'd start by centering the boat in the lake. Then, I would row directly away from where the ogre is standing. The girl will have to row a distance equal to the radius of the lake (r), in less time than than the ogre can run a distance equal to half the circumference of the lake (Pi*r). Since it will take the ogre (Pi*r / 4) hours to complete the distance (assuming the radius is expressed in miles, of course), the girl must row with minimum speed S so that r / S = Pi * r / 4. With a little algebra, S = 4 / Pi mph.

Edited by ThunderCloud, 20 February 2013 - 10:01 PM.

[CaptainEd]

Spoiler for smart maiden outwits ogre!

Assuming that the maiden's goal is merely to get out of the lake without encountering the ogre, that she has no particular direction she is obliged to go.

Wouldn't it behoove the maiden to perform continuous reevaluations of her course and steer to the spot currently opposite the ogre at every moment? I'll bet she'd have to be terribly slow oarswoman to lose this race. (However, I don't have the numbers...   

Edited by CaptainEd, 20 February 2013 - 10:21 PM.

[ThunderCloud]

Spoiler for smart maiden outwits ogre!

Assuming that the maiden's goal is merely to get out of the lake without encountering the ogre, that she has no particular direction she is obliged to go.

Wouldn't it behoove the maiden to perform continuous reevaluations of her course and steer to the spot currently opposite the ogre at every moment? I'll bet she'd have to be terribly slow oarswoman to lose this race. (However, I don't have the numbers...   

Drat! You're right...

Spoiler for

She could "spiral" her way out of the lake at slower speed. The question is how much slower... this problem is trickier than I thought.

[Prime]

Spoiler for a good exercise

I would recommend the maiden should first find the radius r where she could circle in the same time it takes the Ogre to run around the entire lake of the radius R. She must do so while keeping the center of the lake between herself and the Ogre in a straight line. Once she has achieved that, she should make a dash for the shore in a straight line of the length R - r. While the Ogre will have to run around half the lake (Pi*R) distance to catch her.

If the speed of the maiden is M and the Ogre - O, then

 

1). Pi*R/O = Pi*r/M; ==> r = M*R/O; (equation for the circle)

2). (R-r)/M = Pi*R/O (equation for the dash.)

 

Solving for the M, we get: M = O/(Pi + 1);

If she cannot row faster, she must use her cell phone to call Ogre Protection Agency.

Nice problem.

Edited by Prime, 20 February 2013 - 10:53 PM.

[CaptainEd]

Spoiler for smart maiden outwits ogre!

Assuming that the maiden's goal is merely to get out of the lake without encountering the ogre, that she has no particular direction she is obliged to go.

Wouldn't it behoove the maiden to perform continuous reevaluations of her course and steer to the spot currently opposite the ogre at every moment? I'll bet she'd have to be terribly slow oarswoman to lose this race. (However, I don't have the numbers...   

Drat! You're right...

Spoiler for

She could "spiral" her way out of the lake at slower speed. The question is how much slower... this problem is trickier than I thought.

Drat! You're right...I thought she could certainly spiral her way out. But a sufficiently slow-paddling maiden could actually be driven back into the center. Trickier than I thought, too.

Edited by CaptainEd, 20 February 2013 - 11:06 PM.

[bonanova]

Spoiler for a good exercise

I would recommend the maiden should first find the radius r where she could circle in the same time it takes the Ogre to run around the entire lake of the radius R. She must do so while keeping the center of the lake between herself and the Ogre in a straight line. Once she has achieved that, she should make a dash for the shore in a straight line of the length R - r. While the Ogre will have to run around half the lake (Pi*R) distance to catch her.

If the speed of the maiden is M and the Ogre - O, then

 

1). Pi*R/O = Pi*r/M; ==> r = M*R/O; (equation for the circle)

2). (R-r)/M = Pi*R/O (equation for the dash.)

 

Solving for the M, we get: M = O/(Pi + 1);

If she cannot row faster, she must use her cell phone to call Ogre Protection Agency.

Nice problem.

 

Are you sure she has to row that fast?

[Prime]

 

Spoiler for a good exercise

I would recommend the maiden should first find the radius r where she could circle in the same time it takes the Ogre to run around the entire lake of the radius R. She must do so while keeping the center of the lake between herself and the Ogre in a straight line. Once she has achieved that, she should make a dash for the shore in a straight line of the length R - r. While the Ogre will have to run around half the lake (Pi*R) distance to catch her.

If the speed of the maiden is M and the Ogre - O, then

 

1). Pi*R/O = Pi*r/M; ==> r = M*R/O; (equation for the circle)

2). (R-r)/M = Pi*R/O (equation for the dash.)

 

Solving for the M, we get: M = O/(Pi + 1);

If she cannot row faster, she must use her cell phone to call Ogre Protection Agency.

Nice problem.

 

Are you sure she has to row that fast?

The question in the OP is: "How fast she must she be able to row..." (meaning top speed.)

Spoiler for not to say

Her average speed could be less. She could take time getting onto the perimeter of the small circle. However, once there, R - r distance for maiden vs. Pi*R for Ogre seems to be the optimal.Her speed has to be ~ 4/4.14 mph. I'd say I mph to maintain some distance. (Ogres have long hands.)

Alternatively, maiden coulld make smaller circles while the Ogre runs all the way around the lake, until he drops from exhaustion.

[Prime]

Spoiler for On second thought,

Once she is off her small cirle making her dash for the shore, and the Ogre chose his direction; she could continue spiraling away. Would it buy her an additional advantage? How well versed are Ogres in Calculus?

[bonanova]

We have a sufficient condition.

Escape is possible if the ogre's speed ratio advantage is not greater than [ 1 + pi  ].

 

Is it also an necessary condition?

I.e. can she escape if the speed ratio is higher?  How much higher?

 

That's the "twist" mentioned in the title.