The Professor was unstinting with his praise for Root. He never seemed to lose patience when time passed and they were making little progress; and like a miner sifting a speck of gold from the muddy river bottom, he always found some small virtue to compliment, even when Root was stuck.
"Well then, suppose we draw a picture of this little shopping trip. First, there are two handkerchiefs; then two pairs of socks—"
"Those aren't socks!" Root interrupted. "They look more like overweight caterpillars. Let me draw them."
"I see what you mean. That does look more like a caterpillar." "He bought the same number of handkerchiefs the second time but more socks. Five pairs is a lot to draw.... Mine are starting to look like caterpillars, too."
"No, they're fine. And you're right, only the number of socks increases, along with the price. Why don't we check to see how much the price went up?"
"So, you'd subtract ¥380 from ¥710...."
"Always show your work, and do it neatly."
"I usually just scribble on the back of scrap paper."
"But every formula and every number has meaning, and you should treat them accordingly, don't you think?"
I was sitting on the bed, doing some mending. Whenever they started Root's homework, I tried to find something to do in the study in order to be near them. I would iron the Professor's shirts, or work on a stain in the rug, or snip string beans for supper. If I was working in the kitchen and heard their laughter drift in from the other room, I felt terribly excluded—and I suppose I wanted to be there when anyone was showing kindness to my son.
The sound of the rain seemed louder in the study, as if the sky were actually lower there. The room was completely private, thanks to the lush greenery that grew up around the house, and there was no need to close the curtains even after dark. Their reflections appeared dimly in windowpanes, and on rainy days the musty smell in the study was stronger than usual.
"That's right! Then it's just a matter of simple division and you've got it."
"So, you get the price of the socks first: ¥110."
"Okay, but you've got to be careful now. The handkerchiefs seem innocent, but they may turn out to be tricky."
"Right. But it's easier to do the sums when the numbers are small."
The desk was a bit too high, and Root was forced to sit up very straight as he leaned over his problem, a well-chewed pencil clutched tightly in his hand. The Professor sat back, legs crossed and looking relaxed, and his hand drifted to his unshaven chin from time to time as he watched Root work. He was no longer a frail old man, nor a scholar lost in his thoughts, but the rightful protector of a child. Their profiles seemed to come together, superimposed on one another, forming a single line. The gentle patter of the rain was punctuated by the scratching of pencil on paper.
"Can I write out the equations separately like this? Our teacher gets mad if we don't combine them all into one big formula."
"If you're doing them carefully and correctly, he has no reason to get mad."
"Okay, let's see.... 110 times 2 is 220. Subtract that from 380.... That's 160 ... 160 divided by 2 ... is 80. That's it. One handkerchief costs ¥80."
"That's right! Well done!"
As the Professor rubbed Root's head, Root glanced up into his face, not wanting to miss the look of approval and pleasure.
"I'd like to give you a problem myself," said the Professor. "Would you mind?"
"What?"
"No long faces now. Since we're studying together, I feel like playing the teacher and giving you homework."
"That's not fair," said Root.
"It's just one little problem. All right? Here it is: What is the sum of all the numbers from 1 to 10?"
"Okay, I'll let you give me homework if you'll do something for me. I want you to get the radio fixed."
"The radio?"
"That's right. I want to listen to the ball games. You don't have a TV and the radio's broken. And we're coming down to the pennant race."
"Oh, I see ... baseball." The Professor let out a long, slow breath, his hand still resting on Root's head. "What team do you like?" he asked at last.
"Can't you tell from my hat?" Root said, picking up the cap he'd left with his backpack and pulling it over his head. "The Tigers!"
"The Tigers? Is that right? The Tigers," the Professor murmured. "Enatsu! Yutaka Enatsu, best pitcher of all time."
"Yes! Good thing you don't like the Giants. Okay, we've got to get the radio fixed," Root insisted. The Professor seemed to be muttering something to himself, but I closed the lid of the sewing box and stood up to announce it was time for dinner.
3
I finally managed to get the Professor out of the house. Since I'd come to work, he had not so much as set foot in the garden, let alone gone for a real outing, and I thought some fresh air would be good for him.
"It's beautiful outside today," I said, coaxing him. "It makes you want to go out, get some sun." The Professor was ensconced in his easy chair with a book. "Why don't we take a walk in the park and then stop in at the barbershop?"
"And why would we do that?" he said, glancing up at me over his reading glasses.
"No particular reason. The cherry blossoms are just over in the park and the dogwood is about to bloom. And a haircut might feel good."
"I feel fine like this."
"A walk would get your circulation going, and that might help you come up with some good ideas for your formulas."
"There's no connection between the arteries in the legs and the ones in the head."
"Well, you'd be much handsomer if you took care of your hair."
"Waste of time," he said, but eventually my persistence got the better of him and he closed his book. The only shoes in the cupboard by the door were old leather ones covered in a thin layer of mold. "You'll stay with me?" he asked several times as I was cleaning them off. "You can't just leave me while I'm having my hair cut and come home."
"Don't worry. I'll stay with you the whole time." No matter how much I polished, the shoes were still dull.
I wasn't sure what to do with the notes the Professor had clipped all over his body. If we left them on, people were bound to stare, but since he didn't seem to care, I decided to leave them alone.
The Professor marched along, staring down at his feet, without a glance at the blue sky overhead or the sights we passed along the way. The walk did not seem to relax him, he was more tense than usual.
"Look," I'd say, "the cherry blossoms are in full bloom." But he only muttered to himself. Out in the open air, he seemed somehow older.
We decided to go to the barbershop first. The barber recoiled at the sight of the Professor's strange suit, but he turned out to be a kind man. He realized quickly that there must be a reason for the notes, and after that he treated the Professor like any other customer. "You're lucky to have your daughter with you," he said, assuming we were related. Neither of us corrected him. I sat on the sofa with the men waiting in line for their haircuts.
Perhaps the Professor had an unpleasant memory of going to the barber. Whatever the reason, he was clearly nervous from the moment the cape was fastened around his neck. His face went stiff, his fingers dug into the arms of the chair, and deep creases lined his forehead. The barber brought up several harmless topics in an attempt to put him at ease, but it was no use.
"What's your shoe size?" the Professor blurted out. "What's your telephone number?" The room fell silent.
Though he could see me in the mirror, he craned around from time to time, checking to see that I'd kept my promise to stay with him. When the Professor moved his head, the barber was forced to stop cutting, but he would wait patiently and then go back to work. I smiled and gave a little wave to reassure the Professor that I was still there.
The white clippings of hair fell in clumps on the cape and then scattered to the floor. As he cut and combed away, did the barber suspect that the brain inside this snowy head could list all the prime numbers up to a h
undred million? And did the customers on the sofa, waiting impatiently for the strange old man to depart, have any notion of the special bond between my birthday and the Professor's wristwatch? For some reason, I felt a secret pride in knowing these things, and I smiled at the Professor just a bit more brightly in the mirror.
After the barbershop, we sat on a bench in the park and drank a can of coffee. There was a sandbox nearby, and a fountain and some tennis courts. When the wind blew, the petals from the cherry trees floated around us and the dappled sunlight danced on the Professor's face. The notes on his jacket fluttered restlessly, and he stared down into the can as if he'd been given some mysterious potion.
"I was right—you look handsome, and more manly."
"That's quite enough of that," said the Professor. For once he smelled of shaving cream rather than of paper.
"What kind of mathematics did you study at the university?" I asked. I had little confidence that I would understand his answer; maybe I brought up the subject of numbers as a way of thanking him for coming out with me.
"It's sometimes called the 'Queen of Mathematics,' " he said, after taking a sip of his coffee. "Noble and beautiful, like a queen, but cruel as a demon. In other words, I studied the whole numbers we all know, 1, 2, 3, 4, 5, 6, 7 ... and the relationships between them."
His choice of the word queen surprised me—as if he were telling a fairy tale. We could hear the sound of a tennis ball bouncing in the distance. The joggers and bikers and mothers pushing strollers glanced at the Professor as they passed but then quickly looked away.
"You look for the relationships between them?"
"Yes, that's right. I uncovered propositions that existed out there long before we were born. It's like copying truths from God's notebook, though we aren't always sure where to find this notebook or when it will be open." As he said the words "out there," he gestured toward the distant point at which he stared when he was doing his "thinking."
"For example, when I was studying at Cambridge I worked on Artin's conjecture about cubic forms with whole-number coefficients. I used the 'circle method' and employed algebraic geometry, whole number theory, and the Diophantine equation. I was looking for a cubic form that didn't conform to the Artin conjecture. ... In the end, I found a proof that worked for a certain type of form under a specific set of conditions."
The Professor picked up a branch and began to scratch something in the dirt. There were numbers, and letters, and some mysterious symbols, all arranged in neat lines. I couldn't understand a word he had said, but there seemed to be great clarity in his reasoning, as if he were pushing through to a profound truth. The nervous old man I'd watched at the barbershop had disappeared, and his manner now was dignified. The withered stick gracefully carved the Professor's thoughts into the dry earth, and before long the lacy pattern of the formula was spread out at our feet.
"May I tell you about something I discovered?" I could hardly believe the words had come out of my mouth, but the Professor's hand fell still. Overcome by the beauty of his delicate patterns, perhaps I'd wanted to take part; and I was absolutely sure he would show great respect, even for the humblest discovery.
"The sum of the divisors of 28 is 28."
"Indeed ... ," he said. And there, next to his outline of the Artin conjecture, he wrote: 28 = 1 + 2 + 4 + 7 + 14. "A perfect number."
"Perfect number?" I murmured, savoring the sound of the words.
"The smallest perfect number is 6: 6 = 1 + 2 + 3."
"Oh! Then they're not so special after all."
"On the contrary, a number with this kind of perfection is rare indeed. After 28, the next one is 496: 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248. After that, you have 8,128; and the next one after that is 33,550,336. Then 8,589,869,056. The farther you go, the more difficult they are to find"—though he had easily followed the trail into the billions!
"Naturally, the sums of the divisors of numbers other than perfect numbers are either greater or less than the numbers themselves. When the sum is greater, it's called an 'abundant number,' and when it's less, it's a 'deficient number.' Marvelous names, don't you think? The divisors of 18— + 2 + 3 + 6 + 9—equal 21, so it's an abundant number. But 14 is deficient: 1 + 2 + 7 + 10."
I tried picturing 18 and 14, but now that I'd heard the Professor's explanation, they were no longer simply numbers. Eighteen secretly carried a heavy burden, while 14 fell mute in the face of its terrible lack.
"There are lots of deficient numbers that are just one larger than the sum of their divisors, but there are no abundant numbers that are just one smaller than the sum of theirs. Or rather, no one has ever found one."
"Why is that?"
"The answer is written in God's notebook," said the Professor.
Everything around us was glowing in the sunlight; even the dried shells of the insects floating in the fountain seemed to glitter. The most important of the Professor's notes—the one that read "My memory lasts only eighty minutes"—had come loose, and I reached over to adjust the clip.
"I'll show you one more thing about perfect numbers," he said, swinging the branch and drawing his legs under the bench to make more room on the ground. "You can express them as the sum of consecutive natural numbers."
6 = 1 + 2 + 3
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7
496 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31
The Professor reached out to complete the long equation. The numbers unfolded in a simple, straight line, polished and clean. The subtle formula for the Artin conjecture and the plain line of factors for the number 28 blended seamlessly, surrounding us where we sat on the bench. The figures became stitches in the elaborate pattern woven in the dirt. I sat utterly still, afraid I might accidentally erase part of the design. It seemed as though the secret of the universe had miraculously appeared right here at our feet, as though God's notebook had opened under our bench.
"Well then," the Professor said at last. "We should probably be getting home."
"Yes, we should," I said, nodding. "Root will be there soon."
"Root?"
"My son. He's ten years old. The top of his head is flat, so we call him Root."
"Is that so? You have a son? We can't dawdle then. You should be there when he gets home from school." With that, he stood to go.
Just then, there was a cry from the sandbox. A little girl stood sobbing, a toy shovel clutched in her hand. Instantly, the Professor was at her side, bending over to comfort her. He tenderly brushed the sand from her dress.
Suddenly, the child's mother appeared and pushed the Professor away, picking the girl up and practically running off with her. The Professor was left standing in the sandbox. I watched him from behind, unsure how to help. The cherry blossoms fluttered down, mingling with the numbers in the dirt.
"I did the problem and I got it right. So now you have to keep your promise and fix the radio." These were the first words out of Root's mouth as he came through the door. "Here, look," he said, holding out his math notebook.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 55
The Professor studied Root's work as though it were a sophisticated proof. Unable to recall why he had assigned this problem or what connection it had to repairing the radio, he was perhaps looking for an answer in the sum itself.
The Professor carefully avoided asking us questions about things that had happened more than eighty minutes ago. We would have happily explained the meaning of the homework and the radio if he had asked, but he preferred to examine the facts before him and draw his own conclusions. Because he had been—and in many ways still was—such a brilliant man, he no doubt understood the nature of his memory problem. It wasn't pride that prevented him from asking for help but a deep aversion to causing more trouble than necessary for those of us who lived in the normal world. When I realized why he was so reluctant to bring up the subject of his memory, I
decided I would say as little as possible about it, too.
"You've added up the numbers from 1 to 10," he said at last.
"I got it right, didn't I? I checked it over and over, I'm sure it's right."
"Indeed it is!"
"Good! Then let's go get the radio fixed."
"Now just a minute," said the Professor, coughing quietly as if to give himself time to think. "I wonder if you could explain to me how you got the answer?"
"That's easy! You just add them up."
"That's a straightforward way to do it; perfectly reliable, and no one can argue with that." Root nodded proudly. "But think for a minute: what would you do if a teacher, say, a mean teacher, asked you to add the numbers from 1 to 100?"
"I'd add them up, of course."
"Naturally you would. You're a good boy, and a hard worker. So I'm sure you'd come up with the right answer for 1 to 100, too. But what if that teacher was really cruel and made you find the sum for 1 to 1,000? Or 1 to 10,000? You'd be adding, adding, and adding forever while that teacher laughed at you. What would you do then?" Root shook his head. "But you can't let that evil teacher get to you," the Professor continued. "You've got to show him you're the better man."
"But how do you do that?"
"You need to find a simpler way to get the answer that works no matter how big the numbers get. If you can find it, then I'll get the radio fixed."
"That's not fair!" Root objected, kicking his chair leg. "That wasn't part of the deal."
"Root!" I interrupted. "Is that any way to act?" But the Professor didn't seem to notice his outburst.
"A problem isn't finished just because you've found the right answer. There's another way to get to 55; wouldn't you like to find it?"
"Not really ... ," said Root, sulking.
"All right, here's what we'll do. The radio is old, and it may take them a while to get it working again. So how about a contest to see whether you can find another way to get the sum before the radio is fixed?"