PopCo (17 page)

I am nodding. ‘Yeah,’ I say.

‘What is it?’ asks Dan.

‘In maths, it is a function that only goes one way,’ I explain. ‘It is sometimes called a “trap door function” as well. The idea is that, like falling into a trap door, it is something that is easy to do but hard to get out of. You can feed a number into a one-way function and get a result, but, for whatever reason, you can’t easily take the result, feed that into anything and get the number you started with. So the function works one way.’

He looks blank. I am not explaining it very well.

‘It’s like, if I take the function
x
+5, and say the result equals
y
, I can always find
x
again by taking 5 away from y. But many one-way functions are so complicated, or lead to numbers so big, that you
simply can’t unravel them. It’s the mathematical equivalent of mixing paint. If I have a tin of blue paint and a tin of yellow paint and I mix them, I will get two tins of green paint. Once I have that green paint, there’s no way of getting the tin of blue paint back again. You can’t un-mix paint.’

‘That’s it exactly,’ Grace says. ‘With GM technology, you could mess around mixing up genetic equivalents of the blue paint and yellow paint not even realising that you’d never be able to get those paints back again. We already see super weeds, resistant to any kind of herbicide or predator – they already exist. You can’t undo the spread of mutation once it’s there. It’s terrifying. Don’t even get me started on nanotechnology …’

‘I heard that some of the bio-tech companies have made plants with no seeds,’ Esther says. ‘So that the growers have to go back to the company each time for more. I imagine that if that spread it would be the end of the world. If all plants stopped producing seeds …’

‘Well, at least that can’t happen,’ Grace says. ‘Think about it. If there are no seeds, the attribute can’t spread. Barrenness seems to be the one thing nature forbids. It can’t spread because there are no seeds.’

‘Oh yeah,’ says Esther. She looks embarrassed. ‘I still don’t like the idea of eating all that stuff, though. I don’t like the idea of having vegetables with locust genes in them or whatever, especially since I’m a vegan.’

‘Me neither,’ says Grace.

For a few minutes we lie on the grass in silence, all our eyes looking up at the sky. I think about how I am looking at the same sky I looked at when I was a kid, but everything underneath it has changed. And when you are a child you know things will change, because everyone says that things do, and they do, too, but slowly enough for you not to notice. Political regimes change, things blow up and people die and suddenly the world is completely different. But the sky stays the same, and the moon waxes and wanes the same each month. But if people could change those things they would. Imagine if you could advertise using the moon. It could be – what? – a giant hamburger, or some company’s logo. Usually when I think things like this I get a tingle and then think about something else. For some reason today I vow that if this happens in my
lifetime I will seriously think about killing myself. What sort of person would sell the moon if they could? For a million pounds, would I sell the moon if I could?

‘No one ever cracked that Go problem, did they?’ Dan says lazily.

‘No,’ Grace says. ‘That would be such a breakthrough. It’s not just PopCo offering prizes for the person who works out how to get a machine to play Go properly. I think Microsoft has a huge prize, too. Half of us in Robotics and AI are working on something in our spare time but it’s pretty impossible. Are any of you any good at Go?’

‘Alice is,’ Dan says.

I shake my head. ‘I’m not
that
good.’

Esther’s rolling a joint. ‘You know that guy I was talking to before? Hiro? Well it turns out that he’s the reigning PopCo champion. Cool, huh?’

‘You should play him,’ Dan says to me.

‘I’m really not that good,’ I say again. ‘I bet Grace is good.’

‘Are you?’ Esther says.

‘Not really. I’d never played before I came to work here,’ Grace says. ‘But I play almost every day now. It really is the ultimate AI geek game, it turns out. Don’t know how I never played it before.’ She grins.

‘Why can’t computers play Go?’ Dan asks. ‘I’ve never really understood that.’

‘Pattern recognition,’ Esther says, frowning. ‘Or something like that.’

‘Yeah,’ says Grace. ‘Machines can’t recognise subtle patterns the way humans can. It’s one of the main things that separates people from machines, actually: machines process data a lot faster than normal humans, but humans can recognise faces and voices in a way that no one can get computers to match – or even come close to. You could pick out your best friend in a crowd but a computer would only see light and shade. With Go, a lot of it is about seeing patterns. The Go masters think a lot about the shapes they create on a board, and strive for beauty as well as victory. Computers can’t do that. Another problem is that computers can’t understand that sometimes you may have to sacrifice some territory to make gains later on. You know all that vaguely Zen stuff about not being able to lose without winning and vice versa? That’s what you can’t teach computers.’

Dan frowns. ‘Can’t they be taught to do something like a risk assessment, to work out the consequences of every possible move? Then, if it seemed that the computer would be successful from making what seems to be a “losing” move, it would simply see it as a winning move and play it anyway?’

‘Well, that’s how chess programs work,’ Grace explains. ‘They call it the “brute force” method. A chess programme simply runs a set of
what ifs
to see whether particular moves would be successful. But there are too many possible moves in Go. A chess-board only has 64 places you can move but a Go board has 361. The computing power required to work out all the possible combinations of moves would be pretty staggering. But it’s like the whole thing with recognising faces. A good human player can look at a Go board and instinctively know whether territory can be captured or not. It seems to be almost impossible to teach a computer to do the same thing. People are just better at seeing patterns.’

361
. The square of 19. Still looking at the sky, I find myself thinking about prime numbers. It used to be a habit – almost an obsession – of mine, to immediately wonder if a number is prime or not. I even just did it with 361, even though I know it’s a square. Perhaps it’s because I have a prime number birthday: the 19th of July 1973.19 is prime, as is 7, as is 1973. These numbers have no whole number divisors apart from one and themselves. There aren’t many prime years in which to have been born, actually. In the twentieth century you’ve basically got 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997 and 1999. When I realised I had a prime number birthday, I then wanted everything about my life to be prime. The primes are, after all, the most mysterious and beautiful numbers in the universe. You can’t ever break them apart, but every other number breaks down into its prime factors eventually. They are the building blocks of everything.

So I am sitting here on sun-warmed earth, leaning against grey rock and I close my eyes. Behind them, all I can see, suddenly, are building blocks abstracted in the dark. The rock behind me, the rocks under the ground, the blocks in all the structures on the PopCo Estate below me. You can build something and you can smash it up and bury it, but the blocks always stay the same. Prime numbers, genes, atoms. The blocks have to stay the same, don’t they?

My grandfather is making marmalade while my grandmother works in her study.

‘What does she do up there all day?’ I ask, keeping away from the big pan like I have been told to do.

‘She does maths,’ he says simply.

‘What sort of maths?’

‘Complicated maths.’

‘What sort of complicated maths exactly?’

‘She’s trying to prove the Riemann Hypothesis.’

‘The
what
?’

He laughs. ‘Quite. And she says
I
set myself impossible tasks.’

I don’t know what he means by this.

At the end of the afternoon I am allowed to help slip pieces of muslin over the jars and secure them with elastic bands. Then we write on the labels,
Orange, 1983
and put them in the larder. Soon after that, my grandmother comes down from her study and yawns, which is my grandfather’s cue to pour her a whisky over ice.

‘What’s the Riemann Hypothesis?’ I ask her immediately.

She laughs. ‘It’s the work of the Devil.’

‘Is it important?’ I ask next.

‘Yes, to some people,’ she says, with an amused expression.

It’s always hard to know what to talk about with my grandmother. It’s not that she is frightening but she really is incredibly busy, all the time. My grandfather will chat with me about anything: how weather works, Ian Botham, electrical circuits, the right way to sand wood, how to mix paint and so on; but my grandmother has always been frustratingly enigmatic. Occasionally I have shyly asked her questions like, ‘What are we having for supper?’ or ‘Do you think it’s going to rain?’ and she has just absent-mindedly said something back like, ‘Oh, um, ask your grandfather,’ and then disappeared upstairs to her study. Once, to avoid this response, I asked her what her favourite colour was. She just looked at me with a really puzzled expression and then simply said she didn’t know. I think she likes me, but definitely not as much as my grandfather does. Anyway, I have asked her about the Riemann Hypothesis
because this is obviously the thing she is most interested in and perhaps she will like me more if I understand the thing she is most interested in. But answers are not forthcoming, so I change tack.

‘What’s the most important maths anyone has ever done?’ I ask.

My grandfather comes and sits opposite me on his favourite chair. ‘Now there’s a question,’ he says. ‘There’s a question indeed.’ He glances over at my grandmother, and then back at me. ‘The most important maths. Hmm.’

‘Euclid?’ says my grandmother, more to him than me.

‘Hmm. It has to be Bletchley Park, really, doesn’t it?’

She looks sad for a second. ‘Well …’

He looks at me. ‘Have you ever heard of Bletchley Park?’

I shake my head, imagining ducks in a pond.

‘This was classified information until very recently …’

‘Is it to do with the war?’ I ask, instantly thrilled.

‘Oh, yes.’

My grandmother sips her drink while my grandfather starts telling me all about how, during the Second World War, the most intelligent mathematicians, linguists, crossword addicts, music theorists and chess players were rounded up and sent to this secret mansion between Oxford and Cambridge to crack German codes. He tells me in such detail about this mansion, with its outside units called ‘huts’ and its ballroom and its gardens that it almost seems as if he was there himself. My grandmother is quiet as he speaks but occasionally she nods and raises her eyebrows, as if confirming what he is saying. He tells me all about something called the Enigma machine, which turned messages into (supposedly) unbreakable code, and how the German operators often made mistakes in its use so that it was easier for the British cryptanalysts to break their messages.

‘The German keys changed at midnight,’ he says. ‘Intercepted messages would start pouring in and then the race would be on to find that day’s key …’

‘What do you mean, the key?’ I ask.

‘The setting for the Enigma machine,’ he says. ‘Once you know what the setting is, you can unscramble the message. There was a certain amount of information that the cryptanalysts would use to their advantage, like how the same setting couldn’t be used more than twice, that the new setting couldn’t use consecutive wheels and
so on … Enigma would never encipher a letter as itself, so that also helped to narrow things down … But people really did think Enigma was unbreakable. Sometimes, British Intelligence forces even had to stage events – the movement of a particular fleet, perhaps – so that they would know what the German messages for that day would say. Of course, once you know roughly what a message says, it is easier to unravel the cipher version of it. The cryptanalysts would also look for encrypted messages that looked like they might be weather reports. After all, everyone knows what the weather has been. But with Enigma, a key is only valid for twenty-four hours and then it changes. The Allied Forces needed to find a way to crack the code, not just get individual keys.’

‘So how did they do it?’ I ask.

He chuckles. ‘Bombes.’

‘Bombs?’

He spells it out. ‘Bombes. They were early computers, originally conceived by a Polish scientist but refined by Alan Turing, who was the most well-known cryptanalyst at Bletchley. In fact, Turing’s machines formed the basis for all computing today.’

I have never seen a computer, although I have heard that James has something called a ZX Spectrum on which he plays games to do with space. Before I became shipwrecked and moved, there was a craze at my school for hand-held games, which are like little computers. I had a go on my friend’s Frogger game once, but my frog died immediately and then she took the game back.

‘Turing,’ says my grandmother wistfully. ‘He really was a genius. How could they have let that happen to him?’

‘What happened to him?’ I ask.

‘He killed himself. With a poisoned apple.’

‘Like Snow White?’

‘Just like Snow White.’

I shudder. I’ve only recently become aware of people killing themselves and the very concept gives me nightmares. I cannot imagine any reason that someone would actually want to die and I don’t think I’ll understand it as a grown-up, either.

‘Turing was a great man,’ my grandfather says. ‘As well as his bombes, he invented “mind machines”, imaginary computers that showed what would happen with certain maths problems.’

‘But they did actually have computers in the war?’ I say. I thought
they had only just invented computers a few years ago. Everyone talks about them as if they are new.

‘Yes, sort of,’ says my grandfather. ‘Anyway, it was because Turing was such a mathematical genius that he was able to construct all these ways of cracking the Enigma ciphers. Of course, there were lots of other people there too, working in the same sorts of ways, but they say that without his input the Allied war effort would have been put back several years. Interestingly, Turing also wanted to prove the Riemann Hypothesis. Anyway, there you go. The most important maths. It won the war for us, or so people say.’

‘Why did they have musicians and crossword people at Bletchley Park?’ I ask, not yet finished with this conversation.

‘Because people like that are good at seeing patterns. A lot of cryptanalysis is about having half of something – or less – and being able to guess the rest. Obviously, crossword people would be good at that, and linguists. Meanwhile, the mathematics experts were able to finish patterns using their knowledge of the patterns of numbers, and probability and so on. Those who know a lot about music theory understand music mathematically, and music has patterns based on maths. Musicians understand intuitively which notes fit where in a melody. A lot of code-breaking demanded those sorts of skills as well. And music is, after all, completely based around numbers …’

‘Is it?’ My eyes must be wide. I thought music was the furthest thing from maths in the world. I thought music was all about letters: A, B, C and so on, all the way up to G, which is where the notes run out and you have to start again.

‘Oh yes. Have you ever heard of Pythagoras and his urn?’

‘No.’

‘Have you heard of Pythagoras at all?’

‘No.’

‘Well, Pythagoras was a famous mathematician in Ancient Greece. He invented something called Pythagoras’s Theorem, which will be very important when you do geometry and trigonometry in a couple of years …’

‘Can you explain it now?’

‘No, we’ll get so far from our starting-point we’ll never find our way back.’ My grandfather smiles. ‘I can show you tomorrow if you’re still interested. Anyway, Pythagoras filled his urn with water
and banged it with a stick – or something like that. The urn made a pleasing sound. A
musical note
, is how we would describe what he got. Pythagoras experimented with water in the urn until he came upon the following observation. If, after playing his first note, he tipped exactly half of the water out of the urn and hit it with his stick, the note he got was very pleasing next to the first one. If he tipped half the water out again, so he was left with a quarter of the amount he started with, and banged it with his stick, this note was also pleasing when played next to the first two. If he tipped out two-thirds of the original amount, leaving him with a third, this was a pleasing note too. If, however, he left some other quantity of water in the urn, an amount that wasn’t a precise fraction of the original amount, he got
dissonance
, which means that the note sounded wrong. And that’s the basis of music theory.’

‘Did he use lots of urns?’ I ask.

‘What do you mean?’

‘Well, rather than keep filling and emptying the same urn …’

‘Yes. I’ve never really thought about that but yes, he must have done.’ My grandfather looks pleased, like I have just solved one of the riddles he always gives me, but it’s so obvious. In my mind, when I imagine this man Pythagoras, I don’t see him all flustered, trying to mark off points on one urn, with his robes all twisted and his face all red. I imagine him calm and serene, with a row of urns lined up like keys in a glockenspiel, all measured out perfectly, playing beautiful music. But something else about this doesn’t make sense to me.

‘Why does a third work?’ I ask.

My grandfather makes his concentrating face. ‘Sorry?’

‘You said he kept halving the amount of water. One, a half, a quarter and so on … Why a third?’

‘Oh, Alice,’ he says. ‘You do ask good questions.’ He laughs and glances over at my grandmother, who is also smiling. ‘The pattern is 1, ½, ⅓, ¼, ⅕, ⅙ and so on. It doesn’t reduce by half every time, but by whole number denominators of one …’

‘Whole number …
What
?’

‘In a fraction, the bit on the underneath is called a denominator,’ my grandmother explains. ‘That’s the bit that goes down by one every time.’ And this is amazing, because it’s the first time she has ever explained anything to me – and she has made an effort to
explain it in child-language as well. At this moment, I feel closer to her than I ever have before and I resolve to really try to understand what she does, so we can talk about it all the time.

Also: I bet I could solve the Riemann Hypothesis, whatever that is. I bet I could solve anything if I tried hard enough. I am definitely going to be famous for solving a mystery or a puzzle that has stumped grown-ups for ever. This is my plan. Sometimes, when my grandfather is out and my grandmother is working, I make up recipes in the kitchen. I am absolutely certain that one day I will stumble on the special combination of ingredients that will make me rich and famous. Magic biscuits that make you fly; invisible blancmange; mould that has healing properties. I imagine having the Riemann Hypothesis explained to me and the answer arriving in my brain a split-second later, like it was running for a train.

I look at my grandmother, wisps of her grey hair escaping from her long plait, and my grandfather, with his blue shirt-sleeves still rolled up, and they suddenly exchange a comfortable, happy smile. The house still smells heavy and sweet with freshly made marmalade and the sun is now setting outside. I know that in about five minutes my grandfather will get up and switch on the electric light and my grandmother will play one of her records, probably something by Bach. But just now, as they exchange their smile and sip their drinks, I imagine us suspended here in this moment of happiness for ever and I have no shipwrecked feelings at all.

The next day my grandfather receives a package.

‘Aha,’ he says. ‘This is it.’

‘What is it?’ I ask him, intrigued. His face is lit up like Christmas. ‘Is it a present?’

‘What? No. Not a present. But very exciting. Look.’

He shows me pages of what seems to be a copy of an old-looking manuscript. There are handwritten words that I can’t read, and strange pictures of plants and people and animals. Something about it makes me feel strange. I don’t quite know why. Perhaps it’s because I don’t recognise anything on the pages. It looks like you should be able to recognise things: the words, the illustrations and so on; but it’s like a book you’d find in a dream; almost real but not quite.

‘What is it?’ I ask.

‘This is the Voynich Manuscript,’ my grandfather says proudly.

‘And what do you do with it?’

‘You try to read it.’

‘Is it in code, then?’

‘Oh yes. Or at least, that’s what people think. This …’ He fans the pages carefully. ‘This is one of the oldest unbroken codes in history. And I am going to break it.’

‘Can I help?’ I immediately ask, unable to stop myself.

‘Yes,’ he says. ‘You certainly can help.’

What
? I am actually being allowed to help with an important, secret, grown-up project? Bloody hell. Slightly dazed, I go upstairs and immediately sharpen all my pencils.

Once school starts, I fall into the following routine. I get up at seven-thirty, which leaves me just enough time to get dressed and have breakfast before I have to run for the village bus. Tracey waits at the same bus stop as me for her special school bus (which I will also take next year when I move up to the comprehensive) but we never speak. Sometimes on the bus I reply to letters from Rachel, who is back at her boarding-school. More often, though, I take a fragment of the Voynich Manuscript that I have traced the night before, and study that. After school, I walk to the bus stop in town, breathing the bonfirey, marshmallowy smells of autumn. I love this time of year, when people start to rehearse for Christmas plays and pantomimes, and the air feels like it’s full of magic spells. This is the time of year when arriving home after school feels cosy, like going back to bed.