Alan does maths

April 7, 2009

Fascinating TV doc last week, whereby comedian Alan Davies was inducted into the mysteries of maths by Oxford professor Marcus du Sautoy. After recalling the dusty dry way in which he was taught maths at school, Alan started by imagining a four-dimensional cube (hypercube – You Tube video) before being introduced to Euclid’s proof of the infinity of primes.

It’s a beautifully elegant proof, along the following lines: imagine there is a finite number of primes. Then there must be the biggest prime, called k (say), which means we can produce a list of the primes up to and including k (2, 3, 5, 7, … k). Let’s multiply all these primes together and add 1 to the result (2x3x5x7x … k+1). Now this new number, which we’ll call p, is not divisible by any of the primes we’ve multiplied together to create it. Why not? Because whichever prime we divide p by, there will always be a remainder of 1. There has to be, since that is how we created the number p. That means p is either prime itself (and hence is a prime not on our original list), or is divisible by a prime which is not on our original list. Either way, the list we drew up was incomplete – as must be ANY such finite list of the primes. Hence there must be an infinite number of primes, because every time we try to list them all, we can always do the above trick to show we’ve missed one out.

The programme then looked briefly at the Riemann hypothesis and the “Let’s Make a Deal” scenario (a highly counter-intuitive example of probability also known as the Monty Hall problem).

In the meantime, both Alan and the Prof. underwent brain scans while pondering maths problems, to see which areas of the brain are involved in mathematical thinking. I was relieved to see Marcus get the sum “8+7” wrong, although he did much better on the more advanced questions.

It was a couple of these questions which stuck in my mind, as they were numerically straightforward but required a bit of thinking to work out. Here’s one: “A woman gives her age as 30, not counting Saturdays or Sundays. How old is she really?”. And a second one: “What’s the probability of rolling two dice and getting a pair of numbers that differs by exactly 1?”.

I’ll confess now that I got neither correct in the time allowed, but the light came on after a few moment’s thought.

To take the second one first – when rolling two dice, there are 36 possible combinations in total. Let’s list the occasions when the pairs of numbers differ by exactly 1: 1,2; 2,3; 3,4; 4,5; 5,6. There are five of them. But of course, we must take into account that the pair of numbers “1,2” can arise in two ways – dice A can be “1” and dice B can be “2”, or vice versa. It is the same with all five pairs – they can arise in two ways. That means there are 10 ways in which we can roll two dice and end up with pairs of numbers differing by 1. Since there are 36 possible combinations, the probability of rolling such as pair is 10 in 36 (or 5 in 18).

Now let’s look at the first problem. The woman counted weekdays only to arrive at her “age” of 30. Weekdays constitute five-sevenths of a week. So the age she has told us is only five-sevenths of the true total. Now we’re cooking! We know that 30 is five-sevenths of the correct answer. That means the correct answer is seven-fifths of 30 (why?) and even I can do the sum to arrive at the answer of 42.

What I enjoyed about the programme was the evident enthusiasm of Prof du Sautoy and the way this infected Alan (perhaps best known these days as the self-confessed dunce on Stephen Fry’s upmarket quiz show QI).

Any of the above problems (apart from the Riemann hypothesis!) are suitable for introduction into the journalism classroom to introduce concepts in number theory, probability or geometry in a painless way. Best of all, at no point do the words “number theory”, “probability” or “geometry” have to be used. It is often the vocabularly of maths which gets in the way of understanding.

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