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loaf and fish had
somehow been thawed and eaten, and the dirty dishes had been
washed and put away in the cupboard. I was sure that the old
woman took care of the Professor when I wasn't there, but as long
as I was around, she never made an appearance. I had no idea why
she had placed such a firm restriction on communication between
her house and the Professor's, but I decided that my next challenge
was to figure out how to get to know her.
    The Professor's problems, on the other hand, were all mathematical.
He never seemed particularly proud of his accomplishments,
even when he had spent a long time solving an equation
that had won both the prize money and my praise.
    "It was just a little puzzle," he would say, "a game"; and his
tone sounded more sad than modest. "The person who made the
problem already knew the answer. Solving a problem for which
you know there's an answer is like climbing a mountain with a
guide, along a trail someone else has laid. In mathematics, the
truth is somewhere out there in a place no one knows, beyond all
the beaten paths. And it's not always at the top of the mountain. It
might be in a crack on the smoothest cliff or somewhere deep in
the valley."
    In the afternoon, when he heard Root's voice at the door, the
Professor came out of his study, no matter how absorbed he was
in his work. Though he had always hated to have his "thinking"
interrupted, he now seemed more than willing to give it up for
my son.
    Most days, however, Root simply delivered his backpack and
went off to the park to play baseball with his friends, and the Professor
would retreat dejectedly to his study.
    So the Professor seemed particularly cheerful when the weather
turned rainy and he was able to help Root with his math homework.
    "I think I'm a little smarter when I'm in the Professor's office,"
Root told me. There were no bookshelves in the little apartment
where we lived, so the Professor's study, with its stacks of volumes
lining the walls, seemed magical to him. The Professor would
sweep aside the notebooks and clips and eraser shavings on his
desk to make space for Root, and then he would open the textbook.
    How is it possible for a professor of advanced mathematics to
teach a child in elementary school? The Professor was especially
gifted, he had the most wonderful way of teaching fractions and
ratios and volume, and it occurred to me that all parents should be
giving this kind of help to their children.
    Whether it was a word problem or just a simple calculation, the
Professor made Root read it aloud first.
    "353 × 840 =...
    "6239 ÷ 23 =...
    "4.62 + 2.74 =...

    "A problem has a rhythm of its own, just like a piece of music,"
the Professor said. "Once you get the rhythm, you get the sense of
the problem as a whole, and you can see where the traps might be
waiting."
    And so Root read in a loud, clear voice: "I bought two handkerchiefs
and two pairs of socks for ¥380. Two handkerchiefs and
five pairs of socks cost ¥710. How much did each handkerchief
and each pair of socks cost?"
    "So, where do we start?" asked the Professor.
    "Well, it seems pretty hard."
    "You're right. This is the trickiest one in your homework today,
but you read it well. The problem consists of three sentences. The
handkerchiefs and socks appear three times each, and you had the
rhythm just right: so many handkerchiefs ... so many socks ... so
many yen; handkerchiefs ... socks ... yen. You made a boring
problem sound just like a poem."
    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