Free The housekeeper and the professor by Yōko Ogawa Page A

The
Professor's hand was trembling slightly. The black marks seemed
to float up in the half-light. "So then, if we total up the number of
dots in each triangle, we get 1, 3, 6, 10, 15, and 21. And if we write
these as equations:
    1
    1 + 2 = 3
    1 + 2 + 3 = 6
    1 + 2 + 3 + 4 = 10
    1 + 2 + 3 + 4 + 5 = 15
    1 + 2 + 3 + 4 + 5 + 6 = 21
    "In other words, a triangular number is the sum of all the natural
numbers between 1 and a certain number. Then, if you put two
of these triangles together, things get even more interesting. Why
don't we look at the fourth one, 10, so we don't have to draw too
many dots?"

    It wasn't particularly cold in the hall, but the trembling in his
hand had grown worse and the dots had slightly smudged. His
whole being seemed concentrated in the tip of his pencil. A few of
the notes on his suit were smeared with blood and now illegible.
    "Look at this. When you put two of the four-row triangles together,
you get a rectangle that is 4 dots high and 5 dots wide; and
the total number in the rectangle is 4 × 5 or 20 dots. Do you see
that? And if you divide that in half, you get 20 ÷ 2 = 10, or the sum
of the natural numbers from 1 to 4. Or, if you look at each line of
the rectangle, you get:

    "And once you know that, you can use this relationship to figure
out the tenth triangle—the sum of the numbers from 1 to
10—or the hundredth or any other. For 1 to 10 it would be:

    "And for 1 to 100,

    "And 1 to 1000,

    "And 1 to 10,000...."
    The pencil rolled out of his hand and fell at his feet. The Professor
was crying. I believe it was the first time I saw him in tears,
but I had the feeling that I'd seen these emotions many times before.
I placed my hand on his.
    "Do you understand?" he said. "You can find the sums of all
the natural numbers."
    "I understand."
    "Just by lining up the dots in a triangle. That's all there is to it."
    "Yes, I see that now."
    "But do you really understand?"
    "Don't worry," I told him. "Everything's going to be all right.
How can you cry, look at these beautiful triangular numbers."
    Just then the door to the examination room opened and Root
emerged.
    "See!" he said, giving his bandaged hand a wave. "I'm fine."
     
    Leaving the clinic, we suddenly realized that we were starving,
so we decided to eat out. Since the Professor hated crowds, we
went to the emptiest restaurant in the arcade near the station
and had a bowl of curry and rice. There were almost no other
customers, so we might have guessed that the curry wouldn't be
particularly good; but Root, who almost never ate out, was delighted.
He also seemed pleased to have such a dramatic bandage
(for a relatively minor injury), as if he were the hero of
some great battle.
    "I won't be able to help with the dishes or even take a bath for
a while," he said, a bit full of himself.
    The Professor carried him home on his back, and Root was less
worried about being seen now that it was dark. Perhaps he was
just being considerate of the Professor. Whatever the reason, he
climbed on without objection and rode happily. A thin sliver of
moon hung above the row of sycamores glowing under the street
lamps. A pleasant breeze was blowing, our stomachs were full,
and Root's hand would heal. I felt a great sense of contentment.
My footsteps fell in with the Professor's, and Root's tennis shoes
swung back and forth in time.
     
    After seeing the Professor home, we headed back to our apartment.
For some reason Root was suddenly in a bad mood. He went
straight to his room and turned on the radio, and refused to answer
when I called to tell him to take off his bloodstained clothes.
    "Are the Tigers losing?" I asked. He was standing at his desk,
glaring at the radio. They were playing the Giants. "They lost yesterday,
didn't they?" Still no answer. The announcer informed us
that the score was tied 2-2 in the bottom of the ninth, with Nakata
and Kuwata locked in a pitchers' duel. "Does it hurt?" I asked.
He bit his lip and kept his eyes on the radio. "If it