Professor Stewart's Hoard of Mathematical Treasures (30 page)

You’ll notice that I’ve made two of the bodies black: this is for reference. At the first instant, the black bodies are one unit of space apart, with the top one on the left. At the next instant, they are one unit of space apart, with the top one on the right: they have swapped positions.

At which instant were they level with each other?

They weren’t. Because we are working with the smallest possible interval of time, what the pictures show is everything that happens. There is no ‘halfway time’ for the two black bodies to get level with each other. This problem is not insurmountable - we can just accept that a moving body makes this kind of ‘jump’, for instance. And maybe the whole neat and tidy classification of the paradoxes into four possibilities is misleading, and Zeno’s intentions were quite different.

‘Here’s a fine challenge for ye, me hearties!’ yelled Roger Redbeard, the pirate captain, who liked to keep his crew’s minds alert. If only to check they still had them.

He held up four coins, identical gold pieces-of-eight.

‘Now, me lads - what I wants ye to do is to place these four gold coins so that they be equidistant.’

Seeing the baffled looks on their faces, he explained. ‘What I means, lads, is that the shortest distance between any two coins’as to be the same as that between any other two coins.’

To his considerable surprise, the bosun immediately realised that it was no good ‘working in the plane’, and the solution required three dimensions of space. He quickly found an answer: place three of the coins touching each other in a triangle, and sit the fourth on top. All coins are touching, so all distances between them are zero, hence equal.

How to do it with four coins.

Redbeard, dismayed, thought for a moment. ‘So, ye think ye be smart? Try doin’ it with five coins, then. Make them all equidistant from each other!’

Eventually the bosun found an answer, but it wasn’t easy. What was it?

 

It is not widely known that you can work out the value of π by observing the stars. Moreover, the reasoning behind this feat is not based on astronomy, but on number theory - and it works, not because of a pattern in the stars, but because there isn’t one.

Suppose you pick two non-zero whole numbers at random, less than or equal to some upper limit. The probability should be uniform - that is, each number should have the same chance of being chosen. For instance, the upper limit might be a million, and the numbers you get might be 14,775 and 303,254, say, each with probability one in a million. Now ask: Do those two numbers have a common factor (greater than 1) - or not? In this case they don’t. In general, number-theorists have proved that the proportion of pairs with no common factor tends to 6/π
²
as the upper limit becomes arbitrarily large. This remarkable result is one of many properties of π that appear to have no connection to circles. It is exact, not an approximation, and it can (with some clever tricks) be deduced from the formula
In 1995, Robert Matthews wrote a short letter to the scientific journal Nature, pointing out that this theorem in number theory can be used to extract a reasonably accurate value of π from the stars in the night sky - on the assumption that the positions of the stars are random. His idea was to work out the angular distances between lots of stars (that is, the angle between the lines joining those stars to the observer’s eye) and then to transform those distances into large integers. (The actual formula he used was to take the cosine of the angle, add 1, and multiply by half a million.) If you ignore anything after the decimal point, and exclude zero, you get a list of positive integers between 1 and a million. Pick pairs at random, and let the proportion with no highest common factor be p. Then p is approximately 6/π
²
, so π is approximately
.

Matthews did this for the 100 brightest stars in the sky, producing a list of 4,095 integers between 1 and a million. From these he derived a million pairs of randomly chosen numbers, and found that p = 0.613333. Thus π should be approximately 3.12772. This isn’t as good as the school approximation 22/7, but it is within 0.4 per cent of the correct value. Using more stars should improve it. Matthews ended his letter by saying that ‘Latter-day Pythagoreans may take encouragement from learning that a 99.6 per cent accurate value for π can be found among the stars over their heads.’

In Sir Arthur Conan Doyle’s Sherlock Holmes story ‘Silver Blaze’, we find:

‘Is there any other point to which you would wish to draw my attention?’

‘To the curious incident of the dog in the night-time.’

‘The dog did nothing in the night-time.’

‘That was the curious incident,’ remarked Sherlock Holmes.

Here is a sequence:

Having taken Holmes’s point on board: what is the next number in the sequence?

 

There’s a snag with all these ‘find the next number in the sequence’ puzzles - the answer need not be unique. Carl Linderholm put his finger on this problem in the often-hilarious spoof Mathematics Made Difficult, published in 1971 when the ‘new math(s)’ was in vogue. In it, he remarks: ‘Mathematicians
always strive to confuse their audiences; where there is no confusion, there is no prestige.’ As an example, Linderholm defines the natural number system as a ‘universal pointed function’.

His take on ‘guess the next number’ puzzles is unusual but logical. For example, to find the next number after
he tells you to write down ‘the only answer any sensible person would put there’. Which is - what? Ah, that’s the clever part. As a clue, here are some more puzzles of the same type:

Here are the answers we would obtain using Linderholm’s method:

What is the justification for this bizarre set of answers? It is Lagrange’s interpolation formula, which provides a polynomial
p
(x) such that
p
(1),
p
(2), . . . ,
p
(
n
) is any specified sequence of length n, for any finite n. Some such p must fit the sequence
so the choice of 19 is justified by the polynomial. The same goes for all the other examples. As Linderholm explains, this answer is far superior to
because his procedure ‘is much the simpler, and is easier to use, and is obtained by a more general method’.

Why 19? Choose your favourite number and add 1. Why add 1? To ‘make it more difficult to determine your character defects by analysing your favourite number. No technique by which a person’s character may be found out from his secret number is known to the author, but of course someone may some day invent such a technique.’

In the spirit of Linderholm’s book, I really ought to show you Lagrange’s interpolation formula. So it’s on page 313.

Ron Graham has proved that any number greater than 77 can be expressed as a sum of distinct positive integers, whose reciprocals (1 divided by the appropriate integer) add up to 1. So this represents 1 as an Egyptian fraction (see page 76).

For example, let n = 425. Then
and 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 63 + 105 + 135 = 425.

On the other hand, Derrick Henry Lehmer showed that 77 cannot be written in this form. So here we have a special property of the number 77, in the context of Egyptian fractions.

If I arrange three equal circles so that each touches the other two, then it’s obvious that I need three colours if I wish to colour each circle so that circles that touch have different colours. The picture shows three circles, each touching the other two, so they all need different colours.

Three colours required.

Four equal circles in the plane can’t all touch each other, but that doesn’t mean three colours always work: there are more complicated ways to arrange lots of coins, and some of those might need four colours. What is the smallest number of equal circles that can be arranged so that four colours are needed? Again, the rule is that if two circles touch, they must have different colours.

 

In 2004, astronomers discovered asteroid 99942, and named it Apophis after the ancient Egyptian serpent who attacks the Sun-god Ra during his nightly passage through the eternal darkness of the Underworld.
³³
It was an appropriate name in some ways, because the astronomers also announced that there was a serious danger that the newly discovered asteroid might collide with the Earth on 13 April 2029 - or, if not, on 13 April 2036. The chance of a collision was initially estimated at 1 in 200, and peaked at 1 in 37, but is now thought to be highly unlikely.