This number arises naturally, of course, as the hypotenuse of a right triangle with legs of length 1. Notoriously, √2 was found to be irrational by the Pythagoreans in around 500 BC. Their mathematics and philosophy was based upon natural numbers and small number ratios, and thus this discovery would have been very troubling: the Pythagoreans wouldn’t be the only ones to react badly to the imposition of the irrational.

The Frog Problem is a well-known exercise in problem solving that involves a lot of reasonably deep mathematics that is perhaps of more interest than learning a new integration technique. I want to try to expose this mathematics here, not that it is within the scope of all school students but so that it is recognised and possibly offered to at least the brightest pupils.

As an introduction to elementary programming for digital computers, it has been found advantageous to make a start with a desk machine. A very simple code was devised for this purpose. Interested sixth-formers tackled problems prominent in their ordinary mathematical work by this method and were soon able to make programmes and get results themselves. They were later able to visit a digital computer installation and to try their hands at using an established “autocode” (Sirius). They found that their previous experience with the desk machine was most valuable in initiating, very quickly, the kind of thinking required for the more sophisticated procedures. The total experience, although very brief, appears to have been well worth while, as a means of showing them something of the potentialities of modern computers in relation to problems within their own competence.