Except that this isn’t what happened, because my mother was already dead. ‘Long, hot summer, 1982,’ it says on the back of the photos. 1982. The year before I went to live with them. The year before my father disappeared. I didn’t see my grandparents much during that year, especially after that big argument that my father had with my grandfather.
The kettle is whistling, so I put down the photos, pick it up and pour water into my mug. Then I add a pinch of green tea from the caddy on the shelf. Especially after that big argument that my father had with my grandfather. Of course … That argument was about the solution to the Stevenson/Heath puzzle. My father was asking my grandfather to tell him the answer so he could sail off and get the treasure. Shit. When was that? Autumn 1982? Winter? I don’t think it was in the summer. What did we do that summer? I certainly didn’t see my grandparents. What did I do in the long hot summer of 1982? Come on, Alice, your last summer with your father. It’s too difficult to remember. I remember one summer when we went to Margate with Nana Bailey, but I don’t think it was that one. Except … Oh, God. Of course. 1982 was the summer that we went to Wales and stayed in a caravan with a horrible woman called Sandy, and her son Jake. My father wanted us to all become ‘a family’: him, me, Sandy and Jake. But on the last day of the holiday, a cold, rainy bank holiday Monday, a big man with tattoos came and asked Sandy to ‘come home’. She did. And now I remember something else: it was the night we got home from that holiday that my grandfather came over with my necklace. It was still raining. So much for the long hot summer. And of course my father and grandfather argued then as well. ‘Why do you keep spying on me?’ my father shouted at my grandfather, before leaving the flat wearing his long, black raincoat. It was while he was out – at the phone box telephoning Sandy – that my grandfather gave me the necklace and told me never to take it off.
Back upstairs, I switch on the lamp and settle down in my grandfather’s chair with the photographs. The one that most intrigues me is the one where he is sitting at his desk, before my grandmother comes in with her bathing suit. You can see everything on his desk, including the piece of paper he was reaching to take down from his notice board. I recognise the piece of paper anyway: it’s in the trunk – I saw it as I was rifling through before – but it’s something that my grandfather has always had pinned up around his various desks. It’s a copy of Fletcher Pratt’s ‘Frequency of Occurrence of Letters in English’. There are other interesting things in the photograph as well, items scattered on my grandfather’s desk: a World Atlas, a red pamphlet, a collection of Edgar Allan Poe’s tales and stories, and a book I remember well, the one about Gödel’s Incompleteness Theorem that I read when I was doing all those prime factorisations the first year I lived with my grandparents (and the one that inspired me to try to write ‘I love you’ in Gödel’s code). There are other bits and pieces, mainly sheets and sheets of paper. His fountain pen is there, lid off, resting on the frequency table document.
I sit and look at the photograph for about twenty minutes. It is extraordinary really, and I am sure that no one else would see what I can see in it, but it is clear now that this photograph is a perfect snapshot of my grandfather’s desk just after he cracked the Stevenson/Heath puzzle. The red pamphlet is, of course, the one containing the Stevenson/Heath manuscript. My brain feels creaky, but it is still registering the significance of all this. My grandfather gave me my necklace at the end of August 1982, which means he must have arrived at the answer earlier that summer. The World Atlas on his desk implies that, at the moment this photograph was taken, he had reached the stage where he had coordinates to plot. So what do the other things mean? Why would he be taking down the frequency table? What would he have been doing with it?
I go over to the trunk and find the frequency table, the same faded photocopy my grandfather used for all those years. Of course, it is different from the cleaner, newer version of itself captured in the photograph. It now has about five holes in the top, from the drawing pins with which it has been secured to various notice boards over the years. There is a doodle in the top right-hand corner – a few cubes and a triangle – and somewhere in the middle of the right-hand margin is that kind of faded-but-becoming-clearer blue squiggle you get when you try to get ink to flow out of a new biro. In the bottom left-hand corner there is a phone number beginning 01, the old London code. For a moment I consider ringing this number (would it now be 020 7 or 020 8, I wonder) until I notice another way in which the piece of paper differs from the photograph from 1982. This one has tiny blue dots by several of the letters on the table: E, T, A, R, I, S, L, C.
For a few moments I look from the frequency table in my hand to the photograph. Then I get up and take my grandfather’s magnifying glass out of the drawer. I study the photograph again and, now it’s magnified, I can see something peculiar in it. In the frequency table in the photograph, just by the lid of my grandfather’s pen, it is now possible to pick out the letter E, at the top of the table. It has a blue dot by it. But none of the other letters have blue dots. Could it be that this was what he was doing when he was disturbed by the photographer: marking off this set of letters? But why? It doesn’t make any sense to me. It’s 1982, and he’s just finished cracking the Stevenson/Heath code, which uses numbers, anyway, not letters. He’s got his Atlas on his desk. So why would he be needing to use a frequency table at that point? And why mark off letters on it? I know my grandfather well. He hadn’t started on something else, not with his desk still cluttered with Stevenson/Heath-related stuff. Whatever he had on his desk – his most precious working space – would have been there for a reason.
And of course there’s the other thing: although he had that frequency table pinned up by every desk he ever had, I never saw him use it. He knew those frequencies by heart.
My head hurts. Perhaps there’s nothing here for me to see.
I finish my tea, which is now cold. The fire is almost dead. I was right about all the memories in here. It’s not just my grandfather I remember, though, but myself as a child – even though I was never a child here. I remember the crazy recipes I used to concoct, thinking I would chance upon something that would make me invisible, or give me amazing superpowers. I would have been about nine or ten, and I would spend whole afternoons in the kitchen, mumbling to myself as I tipped things into a mixing bowl. ‘Hmmm,’ I’d say to myself. ‘The merest pinch of plain flour combined with a small teaspoon of jam. And then all you need is a pint of water, a few grains of baking powder and an eggcupful of hundreds and thousands …’ At that age I believed that all inventions were accidents. Of course, I’d been brought up on stories of serendipitous scientific discoveries: Alexander Fleming, I knew, had accidentally grown life-saving mould on some unwashed plates; but so many other things had been discovered by accident. Ice lollies were invented by a kid who left a beaker of soft drink outside with a stirring stick in it all night, and Velcro was conceived when George de Mestral got covered in cockleburs while out on a walk. I knew that if I kept fiddling around in the kitchen, doing random things, eventually I would have my own moment of serendipity. But it never came. And, since then, I think I can honestly say that I have never had a ‘happy accident’. Every achievement I have made has been through logic and deduction. And although I always take shortcuts, these are shortcuts through a definite system.
This is why I can’t believe that what I have found now is significant. I have not used any system here. But that doesn’t stop me writing those letters – E, T, A, R, I, S, L, C – down on a piece of paper and seeing what anagrams can be made from them. Of course, I do this methodically, trying to begin words with E, then T, than A. And although you can make other things from this jumble of letters, including ‘recitals’, it is the word ARTICLES that I find first. And then it all starts to fall into place.
And is this a happy accident? No. Because my grandfather taught me the following two important lessons: if you see a big number, prime factorise it, and if you see a jumble of let
ters, start making anagrams. Perhaps seeing the photo was a happy accident. Perhaps that was it.
Of course, it takes me the rest of the weekend to really work it out. ‘Articles’, of course, is the answer, up to a point. It refers to the Articles of the pirate ship, a document that John Christian would certainly have had in his possession, and the text that Francis Stevenson used to create his number-code. And now that I know this was the text used I feel like it was so obvious all along. All those clues my grandfather gave me … He never said it was a book, ever, and he said that the ‘text’ didn’t exist any more and that he’d had to put it together himself, backwards. If only I had sat down and made a list of all books and texts mentioned in the story. Perhaps I would have got it then.
So how does this relate to my necklace? Why did my grandfather make me wear it for all those years? Could I ever have worked back from that ridiculous number and got ‘Articles’ as a solution? No. But, gradually, over the weekend, I work out what it does mean, and why it was so clever.
I remember when I tried to write ‘I love you’ in Gödel’s code and how frustrated I got with the big numbers I was generating. Of course, ‘I love you’ does contain various letters that are way down in the alphabet, and that will generate very big numbers, like 1325 and 1921. But even a simple word can lead to huge numbers if you try to encode it this way, and I gave up on Gödel’s code pretty quickly. The alphabet was just too complicated for it, with 26 letters all needing to be assigned a numerical value. And it didn’t help that T, at number 2 in any English frequency table (which meant you were going to see it a lot), was always going to be number 20 in the alphabet.
But my grandfather had always been very fascinated with Gödel’s code and its potential use in cryptography, so it’s not surprising that he did actually use it this one time, when he simply had an eight-letter word to encipher. But even the word ‘Articles’ is problematic when you try to put it into Gödel’s code. It contains the letters T and S – and S is the eighth letter, which would imply having to calculate 1919 right at the end of the process. And this is why my grandfather used Fletcher Pratt’s ‘Frequency of Occurrence of Letters in English’ to assign numbers to letters, rather than posi tion in the alphabet.
A 3
R 6
T 2
I 7
C 13
L 11
E 1
S 8
I wonder what my grandfather thought as he finished formulating his idea for a coded proof that would show that he knew what the missing document was. Did he think that he’d be able to create a sensible number that he could have engraved on a necklace for me? Something I would know to prime factorise, then recognise as a Gödel code and then compare to the frequency table? He can’t have done. The number that you get when you run that lot through Gödel’s code is 49 digits long. It doesn’t even fit on a calculator properly …
Sometime on Saturday afternoon I actually do it, though: I get my calculator and I type in the calculation. 23 × 36 × 52 × 77 × 1113 × 1311 × 171 × 198. And when I see the result, I almost pass out. It is exactly what is on my necklace: 2.14488156Ex48, a number that, all my life, I have known better than my phone number. The number is not correct, of course, but it is what you get when you type that sum into a calculator. And at that moment I understand. My grandfather, for whatever reason, never intended me to break the necklace code and get the answer. The necklace code was the way you checked the answer, not the way you obtained it. My grandfather must have hoped that I would, at some point, come up with the word ‘Articles’, and that I would know how to run it through my necklace, via Fletcher Pratt, via Gödel, and understand that it was the right answer.
Or maybe not. Maybe he always meant to simply tell me how to do it, but never got around to it. After all, towards the end of his life he was so obsessed with Voynich that nothing else really mattered. No. I think I know what the necklace was actually for: it’s obvious. If anyone disputed the fact that my grandfather had come up with the answer – or, say, if someone else said they’d come up with it first – all he’d have to do would be to get the necklace, prove I’d been wearing it since 1982 (and, yes, of course there are pictures of me wearing it then) and then show his challenger that if you put the word ‘Articles’ through Gödel’s code in the way he did, you get exactly what is on my necklace. He knew that the people who would challenge him would not necessarily be mathematicians, and might be from the press, or even the police. And the last thing you would want to use as proof in a case like this would be a number too big to show on a normal pocket calculator. This way was perfect. My grandfather would have imagined it, I know he would. He would have imagined standing there with his calculator, putting the numbers in, and coming up with exactly what was on my necklace.
Late on Sunday night I am still sitting by the trunk, looking through my grandfather’s papers. And it’s about 1 am, after a takeaway pizza and lots of green tea, that I finally see the actual number. It’s at the top of a ten-page document: a list of other, smaller numbers. It says: 2144881560001920185896805344125304809323777694600. And that’s it, the 49-digit number. And yes, if you prime factorise it, you get 23 × 33 × 52 × 77 × 1113 × 1311 × 171 × 198. So maybe my grandfather did imagine that one day I would find it, this piece of paper, and that I would know what to do with it.
I smile, deliberately casting my eyes upwards (to Heaven? The ether?), and I wish he was here so I could tell him that I have already done it: I’ve worked out his code. But I know what he’d say to me next. He’d say, ‘Go on then, clever clogs, do the next bit. Work out what the Articles say, and then work out where the treasure is.’ I sigh, knowing that this bit is going to take the most time, even if I don’t have to construct the Articles from scratch as he had to. No. The Articles are here in this document, with the Gödel number for ‘Articles’ at the top. What I have to do now is work out which text my grandfather would have used to encode it. And so I look at the photograph again, I consider everything he ever taught me about code-breaking, and then I go to the shelf and take down the book.
Acknowledgements
Thank you to Tom Boncza-Tomaszewski, as usual; Simon Trewin, for facing the world on my behalf; Leo Hollis, for rescuing me back in 2000; Jenna Johnson, for doing such a great and thoughtful job with the American edit; Suzi Feay, for giving me the best job in the world and all the amazing books that go with it; Hari Ashurst-Venn, for the guitar sessions; Emilie Clarke, for letting me treat her bookshop like a library; Lucy Wright, for talking to me about seeds; Allen Clarke, for teaching me about sailing; Jason Kennedy, for the sanctuary; Sam Ashurst, for one particular conversation; Mel McMahon, for all the homeopathic books and advice; Tony Mann, for checking the solution for me, and providing about twenty missing zeros; Francesca Ashurst for reading; and Couze Venn for looking after me.
Thanks also to the American Cryptogram Association for letting me use their name, and that of their publication, The Cryptogram. Thanks to Stewart Dean for letting me use his Life game on my website. Thanks also to the Torbay Local History Library and all the Torbay librarians. I couldn’t have done this without all the books and the many hours of research.
I couldn’t possibly list all the books I have used in this project. Many of them, although useful, don’t deserve a mention. They are the marketing books, the trend studies and the guides on selling products to children. These will be recycled. The following list contains the books which I most loved during the PopCo project, and the ones I think readers may also enjoy.
The Code Book by Simon Singh (4th Estate, 2000)
The Codebreakers by David Khan (Weidenfeld & Nicolson, 1967)
The Magical Maze by Ian Stewart (Weidenfeld & Nicolson, 1997)
The Music of the Primes by Marcus du Sautoy (4th Estate, 2003)
Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter (Penguin Books, 1981)
The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Math
ematical Truth by Paul Hoffman (4th Estate, 1999)
The Colossal Book of Mathematics: Classic Puzzles, Paradoxes And Problems by Martin Gardner (W. W. Norton, 2001)
Six Degrees of Separation: The Science of a Connected Age by Duncan J. Watts (William Heinemann, 2003)
Obedience to Authority by Stanley Milgram (Tavistock, 1974)
Toy Wars: The Epic Struggle Between G. I. Joe, Barbie and the Companies Who Make Them by G. Wayne Miller (Times Books, 1998)
No Logo by Naomi Klein (Flamingo, 2000)
Fences and Windows by Naomi Klein (Flamingo, 2002)
The Algebra of Infinite Justice by Arundhati Roy (Flamingo, 2002)
Animal Liberation by Peter Singer (Pimlico, 1995)
Fast Food Nation by Eric Schlosser (Penguin, 2002)
So Shall We Reap by Colin Tudge (Allen Lane, 2003)
Woman on the Edge of Time by Marge Piercy (Women’s Press, 1979)
Between Silk and Cyanide by Leo Marks (HarperCollins, 1998)
Nancy Wake: The Inspiring Story of One of the War’s Greatest Heroines by Peter Fitzsimons (HarperCollins, 2002)