The housekeeper and the professor (3 page)

"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.

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?"

"Yes, and why human beings seem so foolish and frail compared
to whoever or whatever created these numbers." The Professor
sat back in his chair and opened one of his journals.

"Well, hunger makes you even more foolish and frail, so we
need to feed that brain of yours. Dinner will be ready in a minute."
Having finished grating some carrots to mix into his hamburger, I
carefully slipped the peelings into the garbage pail. "By the way," I
added, "I've been trying to find another pair of amicable numbers
besides 220 and 284, but I haven't had any luck."

"The next smallest pair is 1,184 and 1,210."

"Four digits? No wonder I didn't find them. I even had my son
help me. I found the factors, and then he added them up."

"You have a son?" The Professor sat up in his chair; his magazine
slipped to the floor.

"Yes."

"How old is he?"

"Ten."

"Ten? He's just a little boy!" The Professor's expression had
quickly darkened, he was becoming agitated. I stopped mixing
the hamburger and waited for what I was sure was coming: a
lesson on the significance of the number 10.

"And where is your son now?" he said.

"Well, let's see. He's home from school by now, but he's probably
given up on his homework and gone to the park to play baseball
with his friends."

"
'Well, let's see'!
How can you be so nonchalant? It'll be dark
soon!"

I was wrong, there would be no revelations about the number
10, it seemed. In this case, 10 was the age of a small boy, and nothing
more.

"It's all right," I said. "He does this every day."

"Every day! You abandon your son every day so you can come
here to make hamburgers?"

"I don't abandon him, and it's my job to come here." I wasn't
sure why the Professor was so concerned about my son, but I went
back to my recipe, adding some pepper and nutmeg.

"Who takes care of him when you're not home? Does your
husband come home early from work? Does his grandmother
watch him?"

"No, unfortunately there's no husband or grandmother. It's just
the two of us."

"So he's at home all alone? He sits and waits for his mother in a
dark house while you're here making dinner for a stranger? Making
my
dinner!"

No longer able to control himself, the Professor jumped up from
his chair and began circling the table. The notes on his body trembled
as he ran his hand nervously through his hair. Dandruff sprinkled
on his shoulder. I turned off the soup just as it began to boil.

"You really don't need to worry," I said, trying to sound calm.
"We've been doing this since he was much younger. Now that he's
ten, he can manage for himself. He has the phone number here,
and if he needs help, he knows to ask the landlord downstairs—"

"No, no, no!" The Professor cut me off as he paced around the
table. "You should
never
leave a child alone. What if the heater
fell over and started a fire? What if he choked on a candy? Who'd
be there to help? Oh! I don't want to think about it. Go home
right now! You should make dinner for your child. Go home!" He
grabbed my arm and tried to pull me toward the door.

"I'll go," I said, "but I just have to make these hamburgers for
you."

"Are you going to stand there frying hamburgers while your
child could be dying in a fire? Now listen to me: beginning tomorrow
you'll bring your son along with you. He can come straight here
from school. He can do his homework, and be near his mother. And
don't think you can fool me just because I'll forget by tomorrow."

He pulled off the tag that read "the new housekeeper" and
fished a pencil from his pocket. Under the portrait, he added the
words "and her son, ten years old."

I left that evening—or rather, I was chased out—without having
time to wash my hands, let alone clean the kitchen properly.
The Professor appeared even angrier than when I had interrupted
his thinking. But his anger seemed to hide a deep fear, and I hurried
home wondering what I would do if I found the apartment in
flames.

Any reticence or wariness I felt for the Professor vanished the moment
I saw him with my son, and from that point on I trusted him
completely. As I'd promised the evening before, I gave my son a
map to the house and told him to come directly from school. It
was against agency rules to bring children to the workplace, but
there was no denying the Professor.

When my son appeared at the door the next day with his
schoolbag on his back, the Professor broke into a wide grin and
opened his arms to embrace him. I didn't even have time to point
at the line he'd added to his note—"and her son, ten years old."
As a mother, it was a joy to see someone so completely embrace
my child, and I felt a slight twinge of jealousy that my welcome
from the Professor was always much more reserved.

"I'm so glad you've come!" he said, without any of the questions
he asked me every morning. Bewildered by the unexpected greeting,
my son stiffened, but managed a polite answer. The Professor
took off my son's Hanshin Tigers baseball cap and rubbed his
head. Then he gave him the nickname before he'd even learned his
real one.

"I'm going to call you Root," he said. "The square root sign is a
generous symbol, it gives shelter to all the numbers." And he
quickly took off the note on his sleeve and made the addition:
"The new housekeeper ... and her son, ten years old,
."

At first I made us name tags, thinking that if the Professor
weren't the only one with notes clipped to him he might feel less
anxious. I told my son to change his school name tag for one I
made that read "
." The experiment proved less successful
than I'd hoped. No matter how much time passed, I was always
the young woman who made painfully slow progress with numbers,
and my son would be the boy who simply appeared, and was
embraced.