Thursday, May 7, 2009

Rabbit and the Hunter

A rabbit is in a circular pond and on its edge stands a hunter who wants to catch it.

The hunter can move across the circumference of the pond at 4 times the speed at which the rabbit can swim. However the hunter cannot enter the pond and if the rabbit swims to the surface without the hunter being close by, then the rabbit escapes.

Can the rabbit outwit the hunter and escape?


The rabbit can escape.

If rabbit is one edge of the pond, and hunter is on the other side, then rabbit can easily come out of the pond and esacape.

Let's assume that, at the beginning, rabbit is at the center, and hunter is on the edge of the circular pond.

Let the radius of the pond be r.

Initially, rabbit moves a distance of r/4 from the center of the pond. If the rabbit rotates circularly, then the rotation speed would be same as the rotation speed of the hunter. Since, perimeter for rabbit path is 2*pi*r/4=pi*r/2. For Hunter, the perimeter is 2*pi*r. Since, hunter can travel 4 times faster than rabbit, both rotate at the same speed.

If rabbit is at a distance of (r/4-delta) (where delta is a very small number), then rabbit can rotate faster than hunter. So, eventually, at some point, rabbit would be at a distance of (r+r/4-delta) from the hunter. At that point, if rabbit moves to the nearest point to the edge of the pond, it has to travel 3*r/4 (ignoring delta, since it is a very small number). To reach that point, hunter has to travel pi*r. But, by the time rabbit travels 3*r/4, hunter can travel only 3*r, whereas the hunter should travel 3.14 * r to catch rabbit. So, rabbit can escape.

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