Professor Stewart's Hoard of Mathematical Treasures (47 page)

‘Let’s play a number game,’ said Mathophila.

Innumeratus, ever a sucker, took the bait. ‘What kind of game?’

Mathophila placed cards numbered 1-100 face up on the table.
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‘I’ll show you the rules.’ She wrote down:

‘OK,’ said Innumeratus. ‘You go first.’

‘Well, actually—’ Mathophila began, but then stopped. ‘Oh, very well.’ She picked up card 97 and discarded it.

Innumeratus, after some counting on fingers, said ‘That’s prime, isn’t it?’ When she nodded, he added ‘So I’ve got to choose card 1.’

‘Yes. The only other divisor is 97, and that’s gone. The smallest multiple is 194, and that’s too big.’

So Innumeratus picked up card 1, and discarded it.

Mathophila grinned, and picked up 89. ‘You lose.’

‘That’s prime, too?’ asked Innumeratus, who could sometimes be quite bright.

‘Yes.’

‘So I have to choose 1 again ... Oh. I can’t, it’s already gone.’ He paused. ‘That’s a silly game. The first player always wins.’

‘Yes, we call it the double-whammy tactic.’

Innumeratus thought for a moment. ‘OK, let me start. Now I’ll choose a prime.’ And he picked up card 47.

Mathophila, disdaining the 1, chose 94 instead.

‘Oops,’ said Innumeratus. ‘I hadn’t thought of that.’

‘The double whammy only works for big primes. Bigger than 50, which is half of 100.’

‘Right. So I have to choose 2 now. Because if I chose 1, you’d choose 97 again. Or 89. And I’d lose.’ So he chose 2. And eventually he lost. ‘It’s still a silly game,’ he protested. ‘I should have started with 97.’

‘True. But you were the one who insisted on playing before I told you the fourth rule, which is designed to prevent double whammies.’ And she wrote:

‘Now it’s a sensible game,’ said Mathophila. And they played quite a long game, without much regard for tactics, which illustrates the rules nicely.
I suggest that at this point you stop reading, make a set of cards, and play the game for a while. I’m going to ask you to figure out a winning strategy, and it helps to have played the game. Anyway, it’s a lot of fun.

Done that? Now we can get theoretical. Let’s look at a simplified version where the cards run from 1 to 40. That will get you started.

Some opening moves lose very quickly. For example:

An opening move of 34 suffers the same fate.

Some numbers are best avoided altogether - like 1 in the 100-card game. Suppose Mathophila is silly enough to play 5. Then Innumeratus strikes back:

Note that 25 must still be available when needed here, despite any previous moves, because it can be chosen only if the previous player plays 1 or 5.

There’s a hint of a winning strategy here. Mathophila knows she’s in trouble if she chooses 5, so she could try to force Innumeratus to choose 5 instead. Can she do this? Well, if Innumeratus chooses 7 then she can choose 35, and then Innumeratus has to choose 1 or 5, both of which lose.

Yes, but can she force Innumeratus to play 7? Well, if Innumeratus chooses 3 then Mathophila can choose 21, forcing Innumeratus to choose 7. Yes, but how does she make Innumeratus choose 3? Well, if he chooses 13, then Mathophila chooses 39 ...

Mathophila can keep building hypothetical sequences of moves, all of which force Innumeratus’s reply at every stage and lead to his inevitable defeat. The big question is: can she trap Innumeratus into such a sequence?

At some stage someone has to choose an even number, so we need to think about card 2. This is crucial because if Innumeratus chooses 2 then Mathophila can choose 26, forcing Innumeratus into the trap of playing 13. So now we come to the crunch: how can Mathophila force Innumeratus to choose 2?

She has to play an even number, and the more divisors this has, the more choices Innumeratus has, and Innumeratus might escape the trap. Anyway, the analysis gets complicated, too. Keep it simple. Suppose Mathophila opens with 22, twice a (smallish) prime. Then Innumeratus either chooses 2 and falls into Mathophila’s trap - the long sequence of forced moves just outlined - or he chooses 11. If Mathophila plays 1 she loses, so she chooses 33 instead. Now 11 has already been used, so Innumeratus is forced to choose 3 - and the trap is sprung. We already know how Mathophila can win when he does that. So Mathophila must win if she starts with 22.

That’s probably a bit confusing by now, so here’s a summary of Mathophila’s winning strategy. The two sets of columns deal with the two alternatives available to Innumeratus. For simplicity, I’ve assumed throughout that both players avoid 1, since it is an instant loss. With this choice eliminated, virtually every move is forced.

There is at least one other opening move for Mathophila that also lets her force a win: if she chooses 26, then the same kind of game develops, but with a few of the moves interchanged.

The crucial features of Mathophila’s strategy are the primes 11 and 13. Her opening move is twice such a prime: 22 or 26. It forces Innumeratus to reply either with 2 - at which point Mathophila is home and dry - or the prime. Then Mathophila replies with three times the prime, forcing Innumeratus to go to 3 - and she’s home and dry again.

So Mathophila escapes trouble because as well as twice the prime, there is exactly one other multiple of such a prime in the range being played, namely 33 or 39. This provides her with an escape route. Call these the medium primes - they lie between one-third and one-quarter of the number of cards. If Mathophila chooses twice a medium prime, then Innumeratus must choose that prime. Then she chooses three times that prime, forcing Innumeratus to play the number 3.

Here are two questions for you:

Certainly the answer depends on n. Mathophila wins when n is 3 or 8, whereas Innumeratus wins when n = 1, 2, 4, 5, 6, 7, 9. What about
n
= 100? What about all the values of n from 10 to 99? Can you solve the whole thing?

 

An engineer, a physicist, and a mathematician found themselves in a joke, very similar to many that you will have heard before, but did not immediately realise where they were.
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After a hasty back-of-the-envelope calculation, the engineer worked out what had happened and began to chuckle. Soon after, the physicist intuited where they were, based on a loose analogy with a particle confined in a box, and began laughing uproariously. The mathematician, however, seemed not to find their situation remotely funny. Eventually the others asked why.

‘I saw immediately that I was in an anecdote of some kind,’ he replied. ‘But it was only after I noticed characteristic structural features that I could be sure the anecdote was a joke. However, this joke is far too trivial a consequence of the general case to have any amusement value.’

Physicists are seeking a Theory of Everything (ToE) that will unify the two pillars of modern physics, relativity and quantum mechanics, while fixing certain inconsistencies between these two theories. The search has led them to speculate that our familiar 3-dimensional (3D) space is not actually 3D at all, but 10D or maybe 11D. The extra dimensions provide a place for fundamental particles to vibrate in (like a violin string), thereby giving rise to quantum numbers such as spin and charge (which are like the notes produced by the violin string). Now, you might think that it would be difficult for everyone to have been so wrong for so long about something so basic as the dimensionality of space. And in any case, surely space is space, and it can’t have 10 dimensions because there isn’t any room to fit 7 more in once we’ve sorted out the first 3.

However, it’s not that simple. Mathematicians have invented logically consistent geometries with 4, 5, 6, or even infinitely many, dimensions. Whichever number you like, including 10. So, on the face of it, there is nothing sacred about 3D space. It might be a historical accident, which could have been different in another run of the universe. It might be sacred after all, the only possibility for reasons we don’t yet appreciate. Or it might not actually be 3D, despite appearances. And even if it is, there’s no reason to expect it to be the neat, tidy 3D space of Euclid. In fact, thanks to Einstein’s general theory of relativity, we think space is curved, in ways that Euclid never dreamed of, and sort of mixed together with bits of time.