“Two, two…”
We were sitting in the Excavation, Simon eating Mackerel Norton again, but this time the authentic version, with Chinese-flavor packet rice. The stench creosoted your lungs. It’s as though Batchelors has thrown the Chinaman in with the rice. I was in the middle of the room, balanced on a dining chair that often drifts on the sea of rubbish, several feet north of the corner clothes cupboard. Simon was rocking on his bed, the stained and crumpled blankets pushed aside, his eyes crimped with delight.
“…two, two, two…”
Simon! What’s going on?
He forked up more fillet and boiled Chinaman from the plate on his lap. “…two, two, two…” exhaled in a cloud of fishy bits.
Somebody shoot him!
“…two, two, two, two…” cried Simon, louder.
Simon’s first ever mathematical memory is of sitting on his parents’ sofa, working out the value of two to the power of thirty, that is:
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 230
He doesn’t know why he started this snake of digits. He remembers not being able to stop. One moment he was fidgeting quietly on the sofa cushions; the next he was soaring into the stratosphere of the thousands, and lo! “My life as a mathematician had begun.”
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 210 = 1,024
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 211 = 2,048
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 212 = 4,096…
He still likes twos. He enjoys how they underpin half the number system, and the rest of mathematics teeters on top.
“…two, two…”
The Monster (as Simon deftly observes) is (take a deep breath) “a group of characteristic 2 type, with involution centralizer, structure 21+24Co1, where the group at the bottom of the normal series of the centralizer is of order 2; the group in the middle of order 224, and if one divides the order of the top group of the series by that of the Conway Group, one gets 225.”
That day, sunk in his mother’s sofa, he noticed that 225 (which is to say two, multiplied by itself twenty-five times over) equals 33,554,432, “and I liked the fact that it began with three pairs of digits.”
“…two, two, two…” he continues now.
Stop, Simon!
“…two to the twenty-six, two, two, two…” he sped up. “TWO!”
Deflated with satisfaction, he downed three congratulatory mouthfuls and bounced his lips to endure the heat.
“Two to the power of thirty is one, oh, seven, three, seven, four, one, eight, two, four.” Above the mid-hundred thousands, Simon rarely calls a number by its full name. He raises his eyes up and slightly to the right, and reads each digit off, as though they floated there, projected out of his mind onto a screen in the sky. “Excuse me, this rice is too hot. Can I use that napkin to blow my nose?”
When Simon first did this multiplication as a child, he liked the patterns the numbers made.
“What do you mean, ‘patterns’?”
“Two to the power of twenty-three was my favorite”:
2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 223 = 8,388,608
“Why?”
“It spelled ‘scissors.’”
Mathematics is the study of patterns. We use the word “two” to describe the pattern of “twoness”—the pattern we see if someone shows us first two of Simon’s socks with holes, then two empty mackerel tins, next two bus tickets to an orchid lecture in Pembrokeshire, after that two plastic bags of British Rail “Weekend Rover” holiday brochures bought in Torbay…The existence of this “twoness” pattern (and also the patterns for “oneness,” “threeness,” “forty-sevenness,” etc.) gradually led primordial man to develop the idea of a sequence of whole numbers, stretching from the small values, such as 1, 2 and 3, needed to tot up woolly mammoths and caves, through the midland quantities—10, 15, 20—better suited for sheep and birds’ eggs, and on to the infinite reaches of God.
What Simon means by saying that he likes the “pattern” that two to the power of twenty-three made “because it spells ‘scissors,’” is that the answer:
223 = 8,388,608
has its digits set out in the same arrangement as the letters in the word “scissors.”
Replace the S’s with eights:
And let 3, 6 and 0 swap with the remaining letters, as appropriate:
“So you see,” said Simon, who had meanwhile gone to the kitchen for more supper, “8,388,608 therefore has the same form as—”
“No it doesn’t,” I interrupted. “It can’t spell ‘scissors’ because it still leaves the ‘c’ unaccounted for. 8,388,608 has seven digits, and ‘scissors’ has eight letters.”
“It does when you spell ‘scissors’ the way I did: S-I-S-S-O-R-S.”
“But that’s cheating! I could make any number I like equal any word if I took that approach. It makes a nonsense of the whole game.”
Simon re-emerged from the kitchen in a fresh waft of headless fish and boiled Chinaman.
“It doesn’t when you’re four.”
Another puzzle-book game Simon played as a child—utterly pointless, vital to his mathematical development—was to turn phrases into sums. It’s important to understand how these games work. They’re the next step up from “sissors,” and they confirm (I think) the arrival of something critical—a sort of mischievous joy, a playfulness; I’m not quite sure what to call it—to Simon’s genius.
Simon’s love of mathematics as a child had nothing to do with logic. It was aesthetic and jocular. He enjoyed the subject in the same way a theatergoer applauds an energetic musical.
The most famous phrase-into-sum was invented in 1924 by Henry Dudeney, a civil servant:
The aim is to replace each letter with a unique number, and make the sum come out right. After a great deal of shredded paper and pencils hurled at the wall, you discover that the only possible substitution is:
These puzzles are like one of Simon’s anecdotes. Just when you think you’re getting somewhere, they’re abruptly over and you’re nowhere at all. The intriguing message you started with (why “SEND MORE MONEY”? Is Mr. Dudeney on the run from the police? Is he kidnapped, tied to a chair, with a pistol at his head?) turns abruptly to dust: it’s nothing more than 9,567 + 1,085 = 10,652. So what? You’re winded of excitement.
Computers are ideally suited to this type of problem. They chew through all the options for the letters at vast speed, and spit out the answer in milliseconds. The good schoolchild puzzler likes to discover shortcuts. He wants to exploit patterns to help him out. The most important thing is not the answer, but reaching the answer with slyness. He wants to creep round the back of the puzzle and give it a sharp pinch; then he’ll puff up his chest, pull at his shirt collar and consider himself a clever swell. A good mathematician has a lot in common with a seaside hoodlum.
Immediately, for example, he can tell that the “M” in “MONEY” must equal 1. This exploits a basic pattern in elementary arithmetic. If you take two single-digit numbers—say, 8 and 3,
that add together
to give a two-digit number
then that longer number will always begin with a 1 (3 + 8 = 11).
Like so many rules in arithmetic, this sounds fastidious and clunky when described at length, but in fact everybody knows it instinctively. It’s the rule that makes you furrow your brow when you’re being cheated at the coffee shop. “Three quid for a croissant and £8 for a cappuccino makes forty WHAT?”
The “M” of “MONEY” in the word puzzle must therefore be equal to 1.
If it didn’t equal 1, a pattern of mathematics would be broken, gout pills would walk by themselves, buses fly on wings, headless fish leap up from their tins and chase Simon around the kitchen. And if that “M” must be a “1,” then (by the rules of t
he game) so must the other “M”:
And so on. Pinch by pinch, the mathematician pesters the puzzle until it bursts into tears.
The first puzzle of this type that Simon invented, as a schoolboy, was:
Which is feeble. The words should make up an interesting phrase. Why is Simon Norton a hint? A hint at what? And what are those stars doing there? These puzzles are supposed to stick to ordinary letters. You can’t go adding stars to the alphabet (and allow them to stand for different digits—outrageous!) just because you can’t find something between A and Z to suit your purpose. But Simon was too amused by the chase after numbers to worry about that. SIMON + NORTON = NO*HINT would be closer to the mark.
Simon can solve these puzzles with startling speed.
“Don’t you have to think about it at all?”
“No.”
“You just know it? You can see instantly that the letter N must be a 9?”
“Yes.”
“It is somehow obvious to you at first glance that T will be an 8?”
“Yes.”
“‘Obvious’ in a logical sense,” I asked, “or ‘obvious’ in the sense of a sensation? Would you feel distressed as if you’d eaten a moldy tomato…”
“A raw one is disgusting enough.”
“…if someone suggested T might be 2, or 7?”
Simon looked perplexed. “This is a general phenomenon, isn’t it? When one does something often enough one learns how to do it without conscious thought.”
“But there’s a difference. Do something quickly because practice has made calculation easy, or do something quickly because you don’t need to calculate. The second is divination. That’s genius.”
“Huunnh,” Simon grunted, and stabbed at his fish in distress.
* * *
I’d read once about a journalist who’d decided to learn the trick of calculating the day of the week, for any date.
“July 15, 1843?” you would ask.
He’d reply, “Saturday.”
“December 30, 2076?”
“Wednesday.”
“March 19, 12693 BC?”
The formula was easy. He memorized it in an evening. Daily practice made his speed increase. In a month or two he could answer any question in ten seconds. What took time was the arrival of fluency, the ability to think without thinking: to be able to do the calculation so readily that even to himself no calculation appeared to be involved. That took three years.
After that, for this journalist, October 12, 1646, was no longer a Friday because the numbers said so but because it could not conceivably be anything else. “Friday” was the only possible word that evoked the sensation of “October 12, 1646.” It would be as foolish to mention “Tuesday” as it would be to put salt on your ice cream.
Is this what happened to Simon? Did his obsession with calculating two to the power of thirty when he was four years old rearrange the neurons in his brain (achieving in an afternoon what took the journalist three years—brains are so much more flexible at four years old) so that certain number problems became effortless after that—more a matter of sensation than calculation?
But this clumsy struggle to put my question into words had taken my focus off Simon. When I looked back at him now, he had put his plate to one side and was perched forward on his mattress, neck shrunk in, eyes staring straight ahead.
“Is that knocking?” he grunted. “Someone’s at the door.”
“Never mind that. We’re homing in on the origin of genius. ‘Sissors,’ calculating all those twos and finding patterns, phrase-into-sum puzzles—as I see it, the essential elements are a) playfulness, b) visual satisfaction, which would have made any attempt to force mathematical learning on you, like those ghastly aspirational parents, destructive by removing natural enjoyment, and c)—”
“The door!” Simon bounced up and down in frustration. The plate and cutlery clattered half a beat behind. “Someone is knocking!” he cried.
“—and c)” I pursued, “utter self-centeredness. Get the bloody door yourself.”
16 Simon Cuttlefish
For neither at Milan, nor at Pavia, nor in Bologna, nor in France or Germany, have I ever found a man who could successfully controvert or dispute me within the last twenty-three years. Yet I do not vaunt my powers on this account, for I think that, had I been made of stone, the same things would have come to pass. It is the result of the lack of clear thinking on the part of those who would challenge me, and no more a special dispensation to my own nature or to my own distinction, than it can be counted glorious for the cuttlefish to eject the shadows of its inky humour about the dolphin and force it to flee; that is merely the result of being born a cuttlefish.
Girolamo Cardano, The Book of My Life (1576)
When Simon was three and a half, his mother arranged an IQ test.
In terms of IQ, “genius” is supposed to cover those with a score above 140, about a quarter of 1 percent of the population (one in every 400 people), although some researchers put the mark at 158, the top 0.13 percent, which is one in every 800. Taking the stricter figure, that means there are 80,000 geniuses in Britain. That alone should indicate what a pile of idiocy these tests are. According to these statistics every village the size of Six Mile Bottom has Shakespeare and Darwin slouching down the chippy for haddock and peas. IQ tests are good for predicting one thing only: success at school, and even in that they are not the best guide. The most successful measure of how a child will do in exams and future academic life is parental income. Most people who are successful and rich and happy and winning prizes for creativity and inventive brilliance don’t have remarkable IQs.
But an IQ of 178 is eerie.
Such a person is not just very good; he is inconceivable. His aptitude is not beyond reach or comprehension; it is beyond description.
Simon’s test lasted just over an hour.
At three and a half,d he could count from 0 to 100 in twos:
(“Alex, a typo there! An extra ‘d’ on that last line.”)
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54…
In fives:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55…
And tens:
10, 20, 30, 40, 50, 60…
And threes:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99.
He could read books, write out “write, right, wright” and explain the meaning of each. The woman conducting the test took him out for a walk as part of the warm-up routine, and when they came back he spelled out “fire extinguisher” (in cursive, because he was “tired” of print writing).
There was only one area in which he lost marks: “bead-threading,” which involved stringing different-shaped objects onto a shoelace. She put square, circle, square, circle, then handed the string to Simon. “What’s next?”
Blissfully, he chose triangle.
Letter from Simon’s IQ examiner. Simon is arrowed in the photograph. On the other side of the letter, she reminds Helene (Simon’s mother, whom she calls “Elaine”) to keep the score to herself because it can “prove an unbearable strain even on the brightest child.” The emphasis on playing is interesting. Simon has always treated mathematics as a form of play.
“Simon Norton? The Simon Norton? That’s a blast from the past. Simon Norton—here?”
It’s a good thing I did find out who was knocking at the end of the last chapter. It was new tenants for Simon’s attic room. Shiny and Grubby, I named them. Standing in the living room (which, for reasons Simon cannot explain, is called the Ferret Room), they stared at me in a blur of brilliantine and bugs.
“I’d heard he’d gone mad,” said Shiny.
“No,” I corrected, “still quite sane.”
“Crazy as a Torquay pavement. Memorizes bus timetables.”
“Collects and reads them,” I admitted. “Some of the information
happens to stick in his brain, of course.”
“Catastrophic mental collapse.”
“Don’t be ridiculous! He still goes to conferences and publishes papers.”
“Of course, he’ll never get it back now,” said Shiny, with bright tragedy. “Mathematicians are over the hill by their mid-thirties.”
“Except Fermat,” pointed out Grubby. “He didn’t start until he was forty.”
“And Euler,” agreed Shiny. “Going into his seventies.”
“Like Littlewood.”
“And Gauss.”
“Archimedes.”
“Cayley.”
“Phew!” they exhaled. “But Simon Norton. The Simon Norton. He must be at least 100.”
Shiny or Grubby? One of them had to get the attic room. These were the only two who’d showed up and not run away. Simon said it was my decision. I couldn’t decide between them. They were both perched expectantly on the sofa, waiting for an answer.
Then I had an idea. I ran downstairs and brought Simon up from the Excavation.
“Set these two a math problem,” I said. “They don’t think you can hack it anymore. Give them a question to teach them a lesson.”
Without hesitation, Simon snatched a piece of paper from the duffel dangling from his arm, scribbled quickly, tore the sheet in half and, grinning his grinniest, handed one portion to each of the two candidates.
It was a new phrase-to-sum puzzle, but this time it involved multiplication:
Neither could do it—not on the spot.
But Shiny later phoned up to provide two solutions. Appalled, I gave the room to Grubby.
17
Symmetry is like a disease. Or, perhaps more accurately, it is a disease. At least in my case; I seem to have a bad case of it.