# When is a puzzle not a puzzle?;Mathematics

A man walked a mile due south, a mile due east, then a mile due north, and ended up where he started. Then he shot a bear. What colour was the bear ?

Most people would recognise this as a puzzle. (An old one in fact. In case you haven't heard it, the bear was white, as the only place the man could have ended up where he started is the North Pole.) But what is the difference between a puzzle and a maths question? The term puzzle covers a range of mental challenges - tricks, surprises, finding pretty patterns and so on - but puzzles are usually something you do for fun. On the other hand, a maths question from a 1968 O-level paper reads: "Calculate the area between the curve y = 12x2 - 2x, the x-axis and the line x = 4."

No doubt this question, unlike the one about the bear, would send shivers down the spine of any maths-phobic adult or child. Maths questions, such individuals would explain, are mind-numbing tests that usually involve algebra and rarely have anything to do with "real life".

So the difference between a maths question and a puzzle seems obvious. At least, until you start looking at the grey areas. For example, here is another question from that same O-level paper (which, incidentally, seems to have predicted the era of the motorway traffic jam): "A car is driven the 170 miles from Exeter to London at an average of 20mph and returns to Exeter at an average of 30mph. What is the average speed of the car over the whole journey?" Many candidates probably thought the journey there is at 20mph, the journey back is at 30mph, so the average is obviously in the middle - 25mph. But they would have been wrong. The correct answer is 24mph (340 miles in 14 hours and 10 minutes).

Many would find this answer surprising, and therefore rather interesting. And because it relates to real life and is slightly counter-intuitive, such a question is as likely to appear in a puzzle book as in a maths exam. Another important difference between traditional maths questions and puzzles is that while puzzles often deliberately mislead, maths questions usually point you in the direction of the technique you should use. ("Use the substitution rule to demonstrate that . . . hence or otherwise.") In maths questions, too, you can usually guarantee that all the information given will be useful. And if you find the answer to a question about a right-angled triangle is 16.1742 you might be suspicious because triangle answers in exams are usually nice integers.

Unfortunately much real-life problem solving isn't like that. In life we are constantly bombarded with useless information. There is rarely one right answer, and often what you are looking for (especially in business) is a short cut.

Entrepreneurs usually put their success down to creativity and lateral thinking, rather than working out an answer to three decimal places. In other words, real-life problem-solving is more like finding the answer to a puzzle than working out a traditional maths question.

There is the old puzzle about two trains on the same track. They start 20 miles apart and are travelling towards each other at 40mph. A fly that always travels at 60mph starts off on the first train and flies towards the second. When it gets to the second train it turns around and heads straight for the first. It continues this journey until it is squashed in the head-on collision between the two trains. How far does it fly in total?

The answer can be found using some elegant maths - calculate the distance on the first leg, then on the second, derive the equation for the series, then sum the series.

But a quicker way of solving the problem is to say the trains' speed relative to each other is 80mph, so it takes a quarter of an hour to cover the 20 miles to collision, and in a quarter of an hour the fly travels l5 miles. Richard Branson may not be a mathematician, but it is this kind of thinking that got him where he is today.

Lateral thinking (if that is what you want to call it) is a vital skill, and puzzles are one way of teaching it. The more experience you have of seemingly tough problems that have neat, short-cut solutions, the more open you become to looking for such solutions whenever you face a problem.

Posing interesting questions and discovering surprising outcomes lie at the heart of recreational mathematics. ("Recreational mathematics?" asked a friend of mine. "Surely that is a contradiction in terms.") It is at school that most children will be exposed to the recreational side of mathematics. But few maths teachers under 30 are likely to have even heard of, for example, the great popularisers of the subject, Martin Gardner, Sam Loyd and HE Dudeney.

So what is the difference between a puzzle and a maths question? Like beauty, it is all in the eye of the beholder. And in the end it does not really matter. The important thing is that children (and adults) are most likely to be interested in maths when it connects with real life or when it entertains, and this is what recreational maths does.

Experimental maths is now part of the formal curriculum, but for GCSE and A-level pupils, puzzles and maths recreations are not. It is vital that maths teachers are given a thorough grounding in the subject's well-documented recreational side, and then encouraged to use it. Maths is fun, but if the teachers don't believe it, their pupils certainly won't.

This article is a shortened version of a paper written for Teaching Maths and its Applications Robert Eastaway is co-author of The Guinness Book of Mind-benders