by Yoko Ogawa
I happened to glance at some of the notes pinned to his suit: "… the failure of the analytic method…," "… Hilbert's thirteenth…," "… the function of the elliptical curve…" Shuffled in among the fragments of obscure numbers and symbols and words was one scrap that even I could understand. From the stains and bent corners of the paper and the rusted edges of the binder clip, I could tell that this one had been attached to the Professor for a long time: "My memory lasts only eighty minutes," it read.
"I have nothing to say," he said, turning suddenly and speaking in a loud voice. "I'm thinking at the moment. Thinking. And to have my thoughts interrupted is like being strangled. Don't you know that barging in here when I'm with my numbers is as rude as interrupting someone in the bathroom?"
I bowed and apologized repeatedly, but I doubt he heard a word of what I said. He had already returned to his fixed point somewhere off in space.
To be shouted at like that on the first day could be a serious problem, and I worried that I might become the tenth star on his file card before I'd even started. I promised myself that I would never disturb him again while he was "thinking."
But the Professor was always thinking. When he came out of the study and sat at the table, when he was gargling in the bathroom, or even when he did his strange stretching exercises, he continued thinking. He ate whatever was set in front of him, mechanically shoveling the food in his mouth and swallowing almost without chewing. He had a distracted, unsteady way of walking. I managed to find the right moment to ask him about things I needed to know-where he kept the wash bucket or how to use the water heater. And I avoided making any unnecessary noise, even breathing too loudly, as I moved about that unfamiliar house, waiting for him to take even a short break from his thinking.
I made a cream stew for dinner, something with vegetables and protein that he could eat with just a spoon-and that he could eat without removing bones or shells. Perhaps it was because he'd lost his parents at such a young age, but he had less than perfect table manners. He never said a word of thanks before he started eating, and he spilled food with almost every bite. I even caught him cleaning his ears with his dirty napkin at the table. He did not complain about my cooking, and he remained silent as he ate. Each time he plunged the spoon into the stew, he looked as if he might lose it in the bowl.
"Would you like some more? I've made plenty." It was careless of me to speak up suddenly like that, to take such a familiar tone, and all I got by way of an answer was a burp. Without so much as a glance in my direction, he got up and disappeared into his study. There was a small pile of carrots at the bottom of his bowl.
At the end of my first day, I noticed a new note on the cuff of his jacket. "The new housekeeper," it said. The words were written in tiny, delicate characters, and above them was a sketch of a woman's face. It looked like the work of a small child-short hair, round cheeks, and a mole next to the mouth-but I knew instantly that it was a portrait of me. I imagined the Professor hurrying to draw this likeness before the memory had vanished. The note was proof of something, that he had interrupted his thinking for my sake.
Over the next few days, I introduced myself by pointing to the note on his cuff. The Professor would be silent for a moment, comparing my face with the picture on his sleeve, trying to recall what the note had meant. At last he would make a little huffing sound and ask me my shoe size and telephone number. But I realized that something dramatic had changed when, at the end of my first week, he came to me with a bundle of papers covered with formulas and numbers, and asked me to send it off to the Journal of Mathematics.
"I'm terribly sorry to bother you, but…"
His tone was polite, and completely unexpected after the way he had scolded me in his study on my first day. It was the first request he had made of me, and he was no longer "thinking," for the moment.
"It's no trouble at all," I told him. I carefully copied the mysterious foreign address onto the envelope and ran off happily to the post office.
When I returned, the Professor wasn't thinking anymore. He was stretched out in the easy chair by the kitchen window, and as he rested I was finally able to clean the study. I opened the windows and took his quilt and pillow out into the garden to air. And then I ran the vacuum cleaner at full throttle. The room was cluttered and chaotic, but comfortable.
I was not surprised to find balls of hair and moldy Popsicle sticks behind the desk, or a chicken bone resting on top of one of his bookshelves. And yet, the room was filled by a kind of stillness. Not simply an absence of noise, but an accumulation of layers of silence, untouched by fallen hair or mold, silence that the Professor left behind as he wandered through the numbers, silence like a clear lake hidden in the depths of the forest.
But despite its relative comfort, if you had asked me whether it was an interesting room, I would have had to say no. There was not a single object to spark the imagination, no trinkets from the Professor's past, no mysterious photographs or decorations that might have amused a housekeeper.
I attacked the bookshelves with the duster. Group Theory. Algebraic Number Theory. Studies in Number Theory… Chevalley, Hamilton, Turing, Hardy, Baker… So many books and not one I wanted to read. Half of them were in foreign languages, and I couldn't even make out the titles on the spines. A few notebooks were stacked on the desk, along with a scattering of pencil stubs and binder clips. How could he think at such a characterless desk? The residue from an eraser was the only evidence of the work that he had done here.
As I wiped away the dust, arranged the notebooks, and gathered up the clips, it occurred to me that a mathematician ought to have some sort of expensive compass you couldn't find in an ordinary stationery shop, or an elaborate slide rule. The seat of the chair was worn down where the Professor sat.
"When is your birthday?"
That evening after dinner, he did not disappear immediately into his study. Though I was busy cleaning up, he seemed to be looking for a topic of conversation.
"February twentieth."
"Is that so?"
The Professor had picked the carrots out of his potato salad and had left them on the plate. I cleared and wiped the table, noticing that he still seemed to spill a great deal, even when he wasn't thinking. It was spring, but still chilly once the sun set, so the oil heater was burning in the corner.
"Do you send a lot of articles to magazines?" I asked.
"I wouldn't call them 'articles.' They're just puzzles for amateur mathematicians. Sometimes there's even a prize. Wealthy men who love mathematics put up the money." He looked down, checking his suit in various places, and his gaze fell on a note clipped to his left pocket. "Oh, I see. I sent a proof to the Journal of Mathematics today."
It had been much more than eighty minutes since I'd made my trip to the post office.
"Oh, dear!" I said. "If it's a contest, I should have sent it express mail. If it doesn't get there first, I suppose you don't get the prize."
"No, there was no need to send it express. Of course, it's important to arrive at the correct answer before anyone else, but it's just as important that the proof is elegant."
"I had no idea a proof could be beautiful… or ugly."
"Of course it can," he said. Getting up from the table, he came over to the sink where I was washing the dishes and peered at me as he continued. "The truly correct proof is one that strikes a harmonious balance between strength and flexibility. There are plenty of proofs that are technically correct but are messy and inelegant or counterintuitive. But it's not something you can put into words-explaining why a formula is beautiful is like trying to explain why the stars are beautiful."
I stopped washing and nodded, not wanting to interrupt the Professor's first real attempt at conversation.
"Your birthday is February twentieth. Two twenty. Can I show you something? This was a prize I won for my thesis on transcendent number theory when I was at college." He took off his wristwatch and held it up for me to see. It was a stylish foreign b
rand, quite out of keeping with the Professor's rumpled appearance.
"It's a wonderful prize," I said.
"But can you see the number engraved here?" The inscription on the back of the case read President's Prize No. 284.
"Does that mean that it was the two hundred and eighty-fourth prize awarded?"
"I suppose so, but the interesting part is the number 284 itself. Take a break from the dishes for a moment and think about these two numbers: 220 and 284. Do they mean anything to you?"
Pulling me by my apron strings, he sat me down at the table and produced a pencil stub from his pocket. On the back of an advertising insert, he wrote the two numbers, separated strangely on the card.
220
284
"Well, what do you make of them?"
I wiped my hands on my apron, feeling awkward, as the Professor looked at me expectantly. I wanted to respond, but had no idea what sort of answer would please a mathematician. To me, they were just numbers.
"Well…," I stammered. "I suppose you could say they're both three-digit numbers. And that they're fairly similar in size-for example, if I were in the meat section at the supermarket, there'd be very little difference between a package of sausage that weighed 220 grams and one that weighed 284 grams. They're so close that I would just buy the one that was fresher. They seem pretty much the same-they're both in the two hundreds, and they're both even-"
"Good!" he almost shouted, shaking the leather strap of his watch. I didn't know what to say. "It's important to use your intuition. You swoop down on the numbers, like a kingfisher catching the glint of sunlight on the fish's fin." He pulled up a chair, as if wanting to be closer to the numbers. The musty paper smell from the study clung to the Professor.
"You know what a factor is, don't you?"
"I think so. I'm sure I learned about them at some point…"
"For 220 is divisible by 1 and by 220 itself, with nothing leftover. So 1 and 220 are factors of 220. Natural numbers always have 1 and the number itself as factors. But what else can you divide it by?"
"By 2, and 10…"
"Exactly! So let's try writing out the factors of 220 and 284, excluding the numbers themselves. Like this."
220: 1 2 4 5 10 11 20 22 44 55 110 142 71 4 2 1: 284
The Professor's figures, rounded and slanting slightly to one side, were surrounded by black smears where the pencil had smudged.
"Did you figure out all the factors in your head?" I asked.
"I don't have to calculate them-they just come to me from the same kind of intuition you used. So then, let's move on to the next step," he said, adding symbols to the lists of factors.
220: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = =142 + 71 + 4 + 2 + 1: 284
"Add them up," he said. "Take your time. There's no hurry."
He handed me the pencil, and I did the calculation in the space that was left on the advertisement. His tone was kind and full of expectation, and it didn't seem as though he were testing me. On the contrary, he made me feel as though I were on an important mission, that I was the only one who could lead us out of this puzzle and find the correct answer.
I checked my calculations three times to be sure I hadn't made a mistake. At some point, while we'd been talking, the sun had set and night was falling. From time to time I heard water dripping from the dishes I had left in the sink. The Professor stood close by, watching me.
"There," I said. "I'm done."
220: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
220 = 142 + 71 + 4 + 2 + 1: 284
"That's right! The sum of the factors of 220 is 284, and the sum of the factors of 284 is 220. They're called 'amicable numbers,' and they're extremely rare. Fermat and Descartes were only able to find one pair each. They're linked to each other by some divine scheme, and how incredible that your birthday and this number on my watch should be just such a pair."
We sat staring at the advertisement for a long time. With my finger I traced the trail of numbers from the ones the Professor had written to the ones I'd added, and they all seemed to flow together, as if we'd been connecting up the constellations in the night sky.
2
That evening, after I'd got home and put my son to bed, I decided to look for "amicable numbers" on my own. I wanted to see whether they were really as rare as the Professor had said, and since it was just a matter of writing out factors and adding them up, I was sure I could do it, even though I'd never graduated from high school.
But I soon realized what I was up against. Following the Professor's suggestion, I tried using my intuition to pick likely pairs, but I had no luck. I stuck to even numbers at first, thinking the factors would be easier to find, and I tried every pair between ten and one hundred. Then I expanded my search to odd numbers, and then to three-digit numbers as well, still to no effect. Far from being amicable, the numbers seemed to turn their backs on each other, and I couldn't find a pair with even the most tenuous connection-let alone this wonderfully intimate one. The Professor was right: my birthday and his watch had overcome great trials and tribulations to meet each other in the vast sea of numbers.
Soon, every inch of the paper was filled with figures. My method was logical, if a little primitive-yet I ended up with nothing to show for all my work.
I did make one small discovery: the sum of the factors of 28 equals 28.
28: 1 + 2 + 4 + 7 + 14 = 28
Though I wasn't sure this amounted to anything. None of the other numbers I'd tried were the sum of their own factors, but that didn't mean there weren't more out there. I knew it was an exaggeration to call it a "discovery," but for me it was just that. This one line of numbers stretched across the page as if pulled taut by some mysterious intention.
As I got into bed, I finally glanced at the clock. It had been much more than eighty minutes since we'd had our talk about amicable numbers. By now he'd have forgotten all about our secret, and he'd have no idea where the number 220 had come from. I found it difficult to fall asleep.
From a housekeeper's perspective, working for the Professor was relatively easy: a small house, no visitors or phone calls, and only light meals for one man who had little interest in food. At other jobs, I always had to do as much as possible in a short amount of time; but now I was delighted to have so much time to do a truly thorough job of cleaning, washing, and cooking. I learned to recognize when the Professor was beginning a new contest, and how to avoid disturbing him. I polished the kitchen table to my heart's content with a special varnish and patched the mattress on his bed. I even invented various ways to camouflage the carrots in his dinner.
The one thing about the job that was always a little tricky was understanding how the Professor's memory worked. According to the old woman, he remembered nothing after 1975; but I had no idea what yesterday meant to him or whether he could think ahead to tomorrow, or how much he suffered.
It was clear that he didn't remember me from one day to the next. The note clipped to his sleeve simply informed him that it was not our first meeting, but it could not bring back the memory of the time we had spent together.
When I went out shopping, I tried to return home within an hour and twenty minutes. As befit a mathematician, the device in his brain that measured those eighty minutes was more precise than any clock. If an hour and eighteen minutes had passed from the time I walked out the door to the time I got back, I would receive a friendly welcome; but after an hour and twenty-two minutes, we were back to "What's your shoe size?"
I was always afraid of making some careless remark that might upset him. I nearly bit my tongue once when I started to mention something the newspaper had said about Prime Minister Miyazawa. (For the Professor, the prime minister was still Takeo Miki.) And I felt awful about suggesting that we get a television to watch the summer Olympics in Barcelona. (His last Olympics were in Munich.) Still, the Professor gave no sign that this bothered him. When the conversation veered off in a direction he couldn't follow, he
simply waited patiently until it returned to a topic he could handle. But, for his part, he never asked me anything about myself, how long I'd been working as a housekeeper, where I came from, or whether I had a family. Perhaps he was afraid of bothering me by repeating the same question again and again.
The one topic we could discuss without any worry was mathematics. Not that I was enthusiastic about it at first. In school, I had hated math so much that the mere sight of the textbook made me feel ill. But the things the Professor taught me seemed to find their way effortlessly into my brain-not because I was an employee anxious to please her employer but because he was a such a gifted teacher. There was something profound in his love for math. And it helped that he forgot what he'd taught me before, so I was free to repeat the same question until I understood. Things that most people would get the first time around might take me five, or even ten times, but I could go on asking the Professor to explain until I finally got it.
"The person who discovered amicable numbers must have been a genius."
"You might say that: it was Pythagoras, in the sixth century B.C."
"Did they have numbers that long ago?"
"Of course! Did you think they were invented in the nineteenth century? There were numbers before human beings- before the world itself was formed."
We talked about numbers while I worked in the kitchen. The Professor would sit at the kitchen table or relax in the easy chair by the window, while I stirred something on the stove or washed the dishes at the sink.
"Is that so? I'd always thought that human beings invented numbers."
"No, not at all. If that were the case, they wouldn't be so difficult to understand and there'd be no need for mathematicians. No one actually witnessed the first numbers come into being-when we first became aware of them, they'd already been around for a long time."
"And that's why so many smart people try so hard to figure out how they work?"