Professor Stewart's Hoard of Mathematical Treasures (28 page)

This type of proof doesn’t work in three dimensions, but the theorem is still true. It seems to have first been proved by Stefan Banach, Hugo Steinhaus, and others in 1938. A version about simultaneously bisecting n pieces in n dimensions was proved by Arthur Stone and John Tukey in 1942.

Here are two easier puzzles for you, which explain some of the limitations:

 

More on this theorem, and an outline of the proof, can be found at:

On planet Earth, and in those countries that play the game,
³⁰
cricket fans always get very upset when a batsman scores 49 and then gets out, because he has just missed a half-century. But this is a horribly decimalist way of viewing the situation.

The inhabitants of the distant alien world of Grumpius are a case in point. Oddly enough, when humans first made contact with their civilisation, it turned out that they were passionate about cricket. Astrobiologists speculate that the Grumpians must have picked up our satellite TV programmes during an exploratory trip through the Solar System.

Out for 49 — congratulations!

Anyway, whenever a Grumpian batsthing scores what we would write as 49, the crowd goes wild, and the batsthing raises its bat and bobs its tentacles in the Grumpian equivalent of a punched fist. Why?

 

The brilliant Hungarian mathematician Paul Erdős was distinctly eccentric. He never held a formal academic position, and he never owned a house. Instead, he travelled the world, living for short periods with his colleagues and friends. He published more collaborative papers than anyone else, before or since.

He knew the phone numbers of many mathematicians by heart, and would phone them anywhere in the world, ignoring local time. But he could never remember anyone’s first name—except for Tom Trotter, whom he always addressed as Bill.

One day, Erdős met a mathematician. ‘Where are you from?’, he enquired.

‘Vancouver.’

‘Really? Then you must know my friend Elliot Mendelson.’

There was a pause. ‘I am your friend Elliot Mendelson.’

Paul Erdős.

‘Ooooh! Jigsaws!’ yelled Innumeratus. ‘I love jigsaws!’

‘This one is special,’ said Mathophila. ‘There are 17 pieces, forming a square. I’ve laid them out on a square grid, and every corner of every piece lies exactly on the grid.’

Rearrange the pieces to form the same square . . . with one piece left over.

‘Now,’ she continued, ‘I’m going to take away one of the small squares, and your job is to fit the other 16 pieces back together again to make the same big square that we started with.’

Innumeratus saw no contradiction in that, and half an hour later he proudly showed his answer to Mathophila.

What was his answer, and how can he form the same square when one piece is missing? [Hint: it can’t really be exactly the same. And maybe that initial ‘square’ isn’t actually square . . . ]

 

A mathematician and an engineer are marooned on a desert island, which has two palm trees: one very tall, the other much shorter. Each has one coconut, at the very top.

The engineer decides to have a try for the more difficult
coconut, on top of the tall tree, while they still have the energy to reach it. He clambers up, scraping his legs raw, and eventually returns with the coconut. They smash it open with a rock and eat and drink the contents.

Three days later, both of them now weak with hunger and thirst, the mathematician volunteers to get the other coconut. He climbs the shorter tree, detaches the coconut, and brings it down. The engineer then watches bemusedly while the mathematician starts climbing the taller tree, groaning and sweating profusely, finally gets to the top, deposits the coconut there, and makes his way back down with even more difficulty. He is completely exhausted.

The engineer stares at him, then up at the distant coconut, and then back to the mathematician. ‘Whatever possessed you to do that?’

The mathematician glares back. ‘Isn’t it obvious? I’ve reduced it to a problem that we already know how to solve!’

Zeno of Elea was an ancient Greek philosopher who lived around 450 BC, and he is best known for Zeno’s Paradoxes - four thought experiments, each of which aims to prove that motion is impossible. Some of them may not have originated with Zeno, and others may not even have been stated by Zeno - the evidence is debatable - but I’ll list the traditional four, starting with the best known:

These two characters agree to have a race, but Achilles can run faster than the tortoise, so he gives the creature a head start. The tortoise argues that Achilles can never catch him, because by the time Achilles has reached the position where the tortoise was, it has moved ahead. And by the time Achilles has reached that position, the tortoise has moved ahead again ... So Achilles has
to pass through infinitely many locations before he can catch up, which is impossible.

Achilles in hot pursuit.

In order to reach some distant location, you must first reach the halfway mark, and before you do that, you must reach the quarterway mark, and before that ... So you can’t even get started.

At any instant of time, a moving arrow is stationary. But if it is always stationary, it can’t move.
³¹

This one is more obscure. Aristotle refers to it in his Physics, and says roughly this: ‘Two rows of bodies, each composed of an equal number of bodies of equal size, pass each other on a racecourse, proceeding with equal velocity in opposite directions. One row originally occupies the space between the goal and the middle point of the course; the other that between the middle point and the starting post. The conclusion is that half a given time is equal to double that time.’

What Zeno had in mind here is not at all clear.

Set-up for the stadium paradox.

As a practical matter, we know that motion is possible. While the tortoise is expounding his argument, Achilles shoots past him, oblivious to the impossibility of doing infinitely many things in a finite time. The deeper issue is: what is motion, and how does it happen? This question is about the physical world, whereas Zeno’s paradoxes are about mathematical models of the real world. If his logic were correct, it would dispose of several possible models. Is it correct, though?

Most mathematicians and school mathematics teachers resolve (that is, explain away) the first two paradoxes by doing a few calculations. For instance, suppose that the tortoise moves at 1 metre per second, while Achilles moves at 10 metres per second. Start with the tortoise 100 metres in front. Tabulate the events that Zeno considered:

 

 

The list is infinitely long - but why worry about that? Where is Achilles after, say, 12 seconds? He has reached the 120-metre mark. The tortoise is behind, at 112 metres. Indeed, Achilles gets level with the tortoise after exactly 11
seconds, because at that instant, both of them have reached the 111
-metre position.