### Duck stuck in the pond

Yesterday, my friend and me decided to hit satyam to watch a late night movie after some hectic roaming in the bus from parrys to tambaram for data collection. The movie was Memoirs of a geisha, which was a decent watch. But this post is not about the movie. We reached satyam at around 9pm and had 1 hr to kill before the start of the movie. So the topic of discussion veered to people with attitude. We agreed that they were of two kinds: One kind who were pseudo pseud and the other who were gods in their own right. He gave me an example of the second kind. This guy, a compsci pass out last year is supposedly a stud in all walks of life, right from sports to acads. In an interview with amazon, he was asked to solve a supposedly tough and pseud math problem and he cracked it in 3 minutes.

The problem looks simple at first sight, but only when attempted will the difficulty be realised.

So here's the problem:

There is a pond of known radius, r. There is a duck at the centre of the pond and a man waiting to catch the duck when it comes out of the pond. The duck can move at a speed, v and the man can move around the circumference of the pond at a speed of 4 times that of the duck, 4v. Now the duck is waiting to come out of the pond.

The problem is to provide a solution for the path of the duck so that it comes out of the pond before the man can catch it.

I will rule out an infeasible solution at the outset.

Let the duck start out from the centre of the pond in the direction opposite to that in which the man is currently standing, that is the points represented by the man's position, the centre of the circle(pond) and the ducks position at delta t after the start of its motion are collinear.

Now, the man would require to cover half the circumference to reach the opposite end of the pond, where the duck is headed. The time taken by the man would be (pi*r)/4v

The time taken by duck to reach the same spot would be r/v.

Since pi/4 < 1, the man would reach the spot before the duck and hence the duck would be caught by the man.

I racked my head for quite some time after the movie before I could get the solution, but that may not be the case with everyone. So do give it a try...

## 6 comments:

Insti mind at work readers!

Its obvious that the duck can fly off...

Ignoring that possibility as I am not sure how tricky or straight forward this question is intended to be...

The man I presume follows the 'seeming' path of the duck and so goes to its predicted position based on its current path.

I have the solution but no enthu to prove it because the proof will be time consuming....

Now suppose the duck comes close to the edge of the pond and the man is there waiting to catch it but its just far enough for the man not to catch it. Let this be the starting point. Now one possible thing for the duck to do is go straight in the direction of the center till the man is at the corresponding diameter's end and then return. In which case they both would again arrive at the same time and so there is no advantage. However, if the duck goes towards the center till the man has reached the other end of the diameter and then takes a large turn to its left (say) such that its actually turned right..(think of an inverted 'AIDS awareness' ribbon) then the man will be running in the wrong direction and the duck can fool the man because he has to run a long distance. Alternatively.. the duck can trace out a parabola (x^2) that passes through the starting point and the point on the circle thats 60degrees way from the starting point taken at the center. In both cases the duck would reach first. Proof will be time consuming though because we have take the lengths etc....

Hey ihuman,

Nice attempt. Yes this ain't a twisted question. You are right in that the duck has to take a curve at some point.

Did you notice that the number '4' which is the number of times, the man is faster than the duck has a lot of significance as regards solving this problem or shall I say 'The feasibility of finding a feasible solution' to this problem.

Yeah and about the ribbon thing you mentioned: the duck doesn't have to go

out to the pond end before it starts off towards the centre and then takes a curve(x^2 or an arc of a circle) back. It can do that starting from the centre in the direction opposite to the man's position.

The proof isn't that difficult to understand , but it is difficult to arrive at as you need to recognise the two conditions or 'constraints' that ought to be satisfied for the feasibility of the solution, not to mention the optimality which can be second part of the problem...

Nice attempt again :-)

Hi Karthik...

Thanks for your compliment:). Nice to note the solution that you have mentioned...

I didnt realize your an IITian till I saw your latest posts...(didnt yet check the others out :P)

I am in IITM too.. names Jaganath and I am in 4th year b.tech Chem, Tapti... I am also called Karthik at home....

check therecordedthoughts.blogspot.com .. its my crappy blog :)

Ihuman

How about this?

Since, the man's path is not explicitly mentioned, then, one would have to obviously assume that he would take the shortest path to the point at which the line of duck's travel meets the circumference. Hence, at the starting point, if the man is stationed at clock position 12, and the duck starts moving towards clock position <6, he moves left, and for clock position >6, he moves to his right.

With this assumption, I would provide a very simple solution for the duck.

"Always move to that point which is exactly opposite to the line joining the duck and the man". This should result in some geometrical shape. If the duck takes this strategy, it shouldn't matter at what speed the man travels. The logic is as follows.

Let at any instant of time, the position of man and duck be labelled M and D respectively. At any instant, D moves in the exact opposite direction of M. And M always moves at some angle to the line DM. Which means, M is not moving towards D at any instant of time, and hence, will never be able to reach D.

Your solution is intuitive da...it may be true..even I arrived at the same solution..But it is not objective...

To solve it mathematically you need to recognise the two conditions required for the duck to get out of the pond and solve it from that point...

Post a Comment