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 hundred 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?"
"Okay," said Root. "But I don't see how I'm going to do it. What other way is there besides just adding them up?"
"Who'd have guessed you're such a quitter," the Professor scolded. "Giving up before you've even tried."
"Fine. I'll try. But I can't promise I'll figure it out before the radio's done. I've got a lot of other stuff to do."
"We'll see," said the Professor, and he rubbed Root's head as he always did. "Oh!" he said suddenly. "I've got to make a note." He took a sheet of paper, wrote out their agreement, then clipped it to his lapel. There was something smooth and controlled in the way he held the pen and wrote the note, so different from his usual clumsy manner.
"But you have to promise to finish your homework before the game comes on; and to turn it off during dinner; and not to disturb the Professor while he's working." Root nodded grumpily as I listed each condition.
"I know," he said, "but it'll be worth it. The Tigers are good this year, not like last year and the year before when they were in last place. They even won their first game against the Giants."
"Is that right? Hanshin's having a good year?" the Professor said. "What's Enatsu's ERA?" The Professor looked back and forth between us. "How many strikeouts does he have?" Root waited for a moment before answering.
"They traded Enatsu," he said at last. "That was before I was born, and he's retired now." A jolt shot through the Professor and then he was still.
I had never seen him so distressed. He had always calmly accepted the way his memory failed him, but this time was different. This time he couldn't ignore the facts. Seeing him this way, I even forgot to worry about Root, who had received a shock of his own at causing the Professor such pain.
"But even after they traded him to the Carp, he was the best in the league." I hoped this would reassure him, but this new information distressed him even more.
"The Carp? What do you mean? How could Enatsu wear anything but the Hanshin pinstripes?"
He sat down and rested his elbows on the desk, running his hands through his freshly cut hair. Tiny clippings fell on his notebook. This time it was Root who rubbed the Professor's head. He smoothed the mussed hair as if trying to undo the trouble he'd caused.
Root and I were quiet on the way home that evening. When I asked him whether the Tigers had a game, his answer was barely audible.
"Who are they playing?"
"Taiyo."
"You think they'll win?"
"Who knows."
The lights were out in the barbershop and the park was empty. The formulas the Professor had scratched in the dirt were hidden in the shadows.
"I shouldn't have said those things," Root said. "But I didn't know he liked Enatsu so much."
"I didn't know, either," I said. And then, though it was probably wrong of me, I added, "Don't worry, it will all be back to normal by tomorrow morning. In the Professor's mind, Enatsu will be the Tigers' ace again and he won't remember anything about the Carp."
The problem that the Professor had posed to Root proved to be almost as difficult as the one that Enatsu had presented for all of us.
As the Professor had predicted, the man at the repair shop said that he had never seen such an old radio and that he wasn't sure he could fix it. But if he could, he said, he would try to have it done in a week's time. So every day, when I got home from work, I spent my evening looking for another way to find the sum of the natural numbers from 1 to 10. Root should have been working on the problem, too, but perhaps because he was upset over the incident with Enatsu, he gave up almost immediately and left me to find a solution. For my part, I was anxious to please the Professor, and I certainly didn't want to disappoint him any more than we already had. But the only way to please him, I suspected, was through numbers.
I began by reading the problem aloud, just as the Professor had insisted Root do with his homework: "1 + 2 + 3 +… 9 + 10 is 55. 1 + 2 + 3 +…" But this didn't seem to be much help-except to show that a simple equation could conceal a terribly difficult problem.
Next I tried writing out the numbers from 1 to 10 both horizontally and vertically and grouping them by odds and evens, primes and non-primes, and so on. I worked on the problem with matches and marbles, and when I was at the Professor's house, I jotted down numbers on the back of any piece of scrap paper, always looking for a clue.
To find an amicable number, all you had to do was perform the same sort of calculation again and again. If you had enough time, you'd eventually succeed. But this was different. I was constantly starting off in a new direction, looking for another way to approach the problem, only to wind up at a dead end, confused. To be honest, I wasn't always even sure of what I was trying to do. At times I seemed to be going around in circles and at others almost backward, away from a solution; and in the end, I was often simply staring at the scrap paper.
I'm not sure why I became so absorbed in a child's math problem with no practical value. At first, I was conscious of wanting to please the Professor, but gradually that feeling faded and I realized it had become a battle between the problem and me. When I woke in the morning, the equation was waiting-1 + 2 + 3 +… 9 + 10 = 55-and it followed me all through the day, as though it had burned itself into my retina and could not be ignored.
At first, it was just a small distraction, but it quickly became an obsession. Only a few people know the mystery concealed in this formula, and the rest of us go to our graves without even suspecting there is a secret to be revealed. But by some whim of fate, I had found it, and now knocked at the door, asking to be let in. Though I had never suspected it, from the moment I'd been dispatched by the Akebono Housekeeping Agency, I had been on a mission toward that door…
"Do I look like the Professor?" I asked Root, my hand pressed to my temple and a pencil clenched in my fingers. That day, I had covered the back of every flyer and handbill in the house, but I'd made no progress.
"No, not a bit," Root said. "When the Professor's solving a problem, he doesn't talk to himself the way you do, and he doesn't pull out his hair. His body's there but his mind goes somewhere else. And besides," he added, "his problems are a lot harder!"
"I know! But whose problem is this anyway? Maybe you could stop reading your baseball books for a minute and help me."
"But you're three times as old as I am! And besides, it's a crazy problem anyway."
"Showing the factors was progress. That was thanks to the Professor, wasn't it?"
"I guess so," said Root, looking at my work on the backs of the advertisements and nodding as though he found everything in proper order.
"I think you're on the right track," he said at last.
"Some help you are!" I laughed.
"Better than nothing," he said, turning back to his book.
Since he was very small, he'd often had to console me when I came home from work in tears-when I'd been accused of stealing, or called incompetent, or had the food I'd made thrown away right in front of me. "You're beautiful, Momma," he'd say, his voice full of conviction, "It'll be okay." This was what he always said when he comforted me. "I'm a beauty?" I would ask, and he'd say, feigning astonishment, "Sure you are. Didn't you know?" More than once I'd pretended to be crying just to hear these words; and he'd always play along willingly.
"But you know what I think?" he said suddenly. "When you're adding up the numbers, 10 is odd man out."
"Why do you say that?"
"Well, 10's the only one with two places."
He was right, of course. I had analyzed the numbers in many ways, but had not thought about how each number was special, different. When I looked at them again, it seemed terribly strange that I'd never noticed how odd 10 looked lined up against the others-the only one among them that could not be written without picking up the pencil.
"If you got rid of ten, you'd have a number in the center spot, which might be good."
"What do you mean, 'center spot'?"
"You'd know if you came to the last Parents' Day. We were doing gymnastics-that's my best sport-and in the middle of the exercise the teacher said, 'Double lines, face center.' The guy in the middle held up his arms and the rest of us lined up facing him. There were nine of us, so the guy in fifth place was the center, and the lines were even. For 10 it doesn't work. If you add just one guy, you don't have a center."
So now I tried leaving 10 aside and lining up the rest of the numbers. I circled five in the center, with four numbers before it and four after. The 5 stood, arms proudly extended, enjoying the attention of all the others.
And at that moment I experienced a kind of revelation for the first time in my life, a sort of miracle. In the midst of a vast field of numbers, a straight path opened before my eyes. A light was shining at the end, leading me on, and I knew then that it was the path to enlightenment.
The radio came back from the repair shop on Friday, the twenty-fourth of April, the day the Tigers were scheduled to play the Dragons. We put it on the center of the table and sat around it. Root twisted the knobs, and the broadcast of the game crackled out from the static. The signal was weak, but there was no doubt it was the baseball game-and the first sign of life from the outside world that had made its way into the house since my arrival. We let out a little cheer.
"I had no idea you could get baseball on this radio," said the Professor.
"Of course! You can get it on any radio."
"My brother bought it for me a long time ago, for practicing English conversation. I thought it would only pick up English."
"So you've never listened to the Tigers?" Root said.
"No, and I haven't got a TV, either…," murmured the Professor, as if confessing something awful. "I've never seen a baseball game."
"I don't believe it!" Root blurted out, nearly shouting.
"I know the rules, though," the Professor said, a bit defensively. But Root was not to be appeased.
"How can you call yourself a Tigers fan?"
"But I am-a big fan. When I was in college, I went to the library at lunch to read the sports pages. But I did more than just read about baseball. You see, no other sport is captured so perfectly by its statistics, its numbers. I analyzed the data for the Hanshin players, their batting averages and ERAs, and by tracking the changes, even miniscule shifts, I could picture the flow of the games in my head."
"And that was fun?"
"Of course it was. Even without the radio, I could keep every detail fixed in my mind: Enatsu's first victory as a pro in 1967-he beat the Carp with ten strikeouts; the game in 1973 when he pitched an extra-innings no-hitter and then hit a walk-off home run himself." Just at that moment, the announcer on the radio mentioned the name of the Tigers starting pitcher, Kasai. "So when is Enatsu scheduled to pitch?" the Professor asked.
"He's a little farther on in the rotation," Root answered without missing a beat. It surprised me to see him acting so grown-up. We'd promised that where Enatsu was concerned, we'd do anything to keep up the illusion. Still, it made us uncomfortable to lie to the Professor, and it was hard to know whether it was really in his best interest. But we could not bear to upset him again.
"We can tell him that Enatsu's back in the dugout, or that he's throwing in the bullpen," Root had said.
Since Enatsu had retired long before Root was born, he'd gone to the library to find out about him. He learned that he had a career record of 206 wins, 158 losses, and 193 saves, with 2,987 strikeouts. He'd hit a home run in his second at bat as a pro; he had short fingers for a pitcher. He'd struck out his great rival, Sadaharu Oh, more than any other pitcher, but he'd also surrendered the most home runs to him. In the course of their rivalry, however, he'd never hit Oh with a pitch. During the 1968 season, he set a world record with 401 strikeouts, and after the 1975 season (the year the Professor's memory came to an end), he'd been traded to the Nankai Hawks.
Root had wanted to know more about Enatsu, so he would seem more real to both of them as they listened to the cheers on the radio. While I had been struggling with the "homework" problem, he had been seeing to the Enatsu problem. Then one day, as I was flipping through a copy of Baseball Players Illustrated that he'd brought home from the library, I was stunned to find a picture of Enatsu, and see on his uniform the number 28. When he'd graduated from Osaka Gakuin and joined the Tigers, he'd been offered the three available numbers: 1, 13, and 28. He'd chosen 28. Enatsu had played his whole career with a perfect number on his back!
That evening, after dinner, we presented our solution. We stood before the Professor, pen and paper in hand, and bowed.
"This is the problem you gave us," said Root. "Find the sum of the numbers from 1 to 10 without adding them." He cleared his throat and then, just as we'd arranged the night before, I held the notebook while he wrote the numbers 1 to 9 in a line, adding 10 farther down on the page. "We already know the answer. It's 55. I added them up and that's what I got. But you didn't care about the answer."
The Professor folded his arms and listened intently, as if hanging on to Root's every word.
"So we decided to think about 1 to 9 first, and forget about 10 for right now. The number 5 is in the middle, so it's the… uh…"
"Average," I whispered in his ear.
"Right, the average. We haven't learned averages yet, so Momma helped me with that part. If you add up 1 through 9 and divide by 9 you get 5… so 5 × 9 = 45, that's the sum of the numbers 1 to 9. And now it's time to bring back the 10."
5 × 9 + 10 = 55
Root took the pen and wrote the equation on the pad.
The Professor sat studying what he had written, and I was sure then that my moment of inspiration must look laughably crude to him. I'd known from the start that I would never be able to extract something sublime and true from my poor brain cells, no chance of imagining something that would please a real mathematician.
But then the Professor stood up and began to applaud as warmly and enthusiastically as if we had just solved Fermat's theorem. He clapped for a long time, filling the little house with his approval.
"Wonderful! It's magnificent, Root." He folded Root in his arms, half crushing him.
"Okay, okay. I can't breathe," Root mumbled, his words nearly lost in the Professor's embrace.
He was determined to make this skinny boy with the flat head understand how beautiful his discovery was, but as I stood watching Root's triumph, I secretly felt proud of my own contribution. I looked at the line of figures Root had written. 5 × 9 + 10 = 55. And even though I'd never really studied mathematics, I knew that the formula became more impressive if you restated it in abstract form:
It was a splendid discovery, and the clarity and purity of the solution was even more extraordinary in light of the confusion it had emerged from, as if I'd unearthed a shard of crystal from the floor of a dark cave. I laughed quietly, realizing that I'd praised myself adequately, even if the Professor's compliments had been directed elsewhere.
Root was finally released, and we bowed again like two scholars who had just finished their presentation at an academic conference.
That day, the Tigers lost 2-3 to the Dragons. They had taken a two-run lead on a triple by Wada, but the Dragons responded with back-to-back home runs and won the game.