The Fox, The Duck and a Circular Pond

A Duck that is being chased by a Fox saves itself by sitting at the centre of a circular pond of radius r. The Duck can only fly from land and not able to fly from the water. Furthermore, Fox cannot swim. The speed of the Fox is four times the speed of the Duck. Assuming that the Duck and Fox are perfectly smart, is it possible for the Duck to ever reach the edge of the pond and fly away to its escape from the ground?

Solution:

  • The Duck can’t swim directly away from the Fox because for that the Duck would have to swim a distance r and till that time Fox would have cover half the circumference of the pond i.e., (pi * r). Since the speed of the Fox is four times the speed of the Duck, therefore, he will be able to catch Duck as we know that

    (pi * r) < (4 * r)
  • The solution for this puzzle is that Duck could swim in concentric circles closed to the centre of the pond. Due to this Duck would cover the small circumference of the pond and the Fox will have to cover the larger circumference of the pond.
  • Using the above strategy the Duck will make tiny concentric circles around the center and due to this, the Fox will not be able to cope up with the Duck as he has to cover the entire pond.
  • As the speed of the Fox is four times the Duck, therefore the Duck must move in a concentric circle of radius r/4 such that the distance cover by Duck and the Fox is same. As long as Duck stays within that concentric circle the Duck will gain some distance over Fox after some time and the Fox will be unable to keep up with the Duck.
  • As soon as the Fox is 180 degrees behind the Duck, the Duck would cover the remaining distance (3*r / 4) as distance cover the Fox is less than the distance cover by Duck because
      (pi * r) > 4 * (3r/4)
    
  • At last, The Duck would able to reach the edge of the pond and fly away.
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