Issue No.8 Spring/Summer 1988

ZENO

Zeno, who lived during the earlier three-quarters of the fifth century BC, was a noted critic of the mathematicians, and showed that their knowledge was not so exact as they imagined. He was the originator of the well-known puzzle of Achilles and the tortoise.

This problem has been given in many forms, but the following will illustrate its main point.

CAN A DOG CATCH A RABBIT?

Suppose a dog at A is chasing a rabbit at B, that the distance between them is 100 metres, and the dog can travel twice as fast as the rabbit.

A______________100_______________B________50______C__25__D
Dog                                                             Rabbit

The rabbit will travel the 50 metres to C while the dog is travelling 100 metres to B. That is, the rabbit will be at C when the dog is at B. While the dog is travelling the 50 metres from B to C, the rabbit will travel 25 metres to D. That is, when the dog is at C the rabbit will be at D.

In the same way, when the dog reaches D, it will find the rabbit is 12.5 metres away; when the dog has travelled these 12.5 metres, it will find the rabbit is 6.25 metres in front, and so on.

Continuing in this manner, we find the distance between the dog and the rabbit lessens thus: 100, 50, 25, 12.5, 6.25, 3.125, 1.5625... etc., but the rabbit is always a little in front.

Will the dog ever catch the rabbit?