Dec 10, 2009

Catch me if you can?

Source: Asked to me by Ram Kumar, a frequent visitor to this blog.

Problem:
There's a duck swimming in a circular lake with a wolf at the edge. wolf won't enter water. Duck in water is slower than Wolf on land. Duck on land is faster than the Wolf on land. Given the lake diameter 'd' and wolf speed 'w', what is the minimum speed of the duck in water so that he can escape? what should be his escape strategy?

Hint:
1) Its less than w/4.
2) There is pi involved. :)

Solution:
Answer given by Vivek Jha (EE, IITB) in the comments. Proof of the solution given by me in the comments of one of the earlier posts: http://pratikpoddarcse.blogspot.com/2009/11/polyas-urn-problem.html?showComment=1258400372028#c8784580092886919079

1. i did not get the meaning?
as the wolf wont enter the water,it can never catch the duck unless it comes out of water.

2. its written "duck needs to escape". It has to get out of water and run :)

3. the duck comes to the center of the lake.Then it oves in a direction diametrically opposite to that of wolf.
the min required speed of duck is w/2*pi.

4. @teja.. you need to do better.. I said that the answer is less than w/4.

5. Its w/(pi+1).....I think.

6. Same proof here.....btw nice solution for that urn problem....nice discussion on the thread also.

7. This comment has been removed by the author.

8. I believe, that the answer is 'slightly greater than zero'.the strategy for the duck is to move perpendicular to the velocity of the wolf at all times,such that his distance from land decreases (assuming initial distance of the duck from wolf is >=d/2,which can be easily achieved).As radial relative velocity is never changing,the distance between duck and wolf will remain same.duck will follow a spiral path till it reaches the land.
Comment if you find anything wrong with this strategy.

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