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People started giving me problems they thought were difficult, such as integrating a function like 1/(1 + x), which hardly changed over the range they gave me. The hardest one somebody gave me was the binomial coefficient of x^10 in (1 + x)^20; I got that just in time.
They were all giving me problems and I was feeling great, when Paul Olum walked by in the hall. Paul had worked with me for a while at Princeton before coming out to Los Alamos, and he was always cleverer than I was. For instance, one day I was absent-mindedly playing with one of those measuring tapes that snap back into your hand when you push a button. The tape would always slap over and hit my hand, and it hurt a little bit. “Geez!” I exclaimed. “What a dope I am. I keep playing with this thing, and it hurts me every time.”
He said, “You don’t hold it right,” and took the damn thing, pulled out the tape, pushed the button, and it came right back. No hurt.
“Wow! How do you do that?” I exclaimed.
“Figure it out!”
For the next two weeks I’m walking all around Princeton, snapping this tape back until my hand is absolutely raw. Finally I can’t take it any longer. “Paul! I give up! How the hell do you hold it so it doesn’t hurt?”
“Who says it doesn’t hurt? It hurts me too!”
I felt so stupid. He had gotten me to go around and hurt my hand for two weeks!
So Paul is walking past the lunch place and these guys are all excited. “Hey, Paul!” they call out. “Feynman’s terrific! We give him a problem that can be stated in ten seconds, and in a minute he gets the answer to 10 percent. Why don’t you give him one?”
Without hardly stopping, he says, “The tangent of 10 to the 100th.”
I was sunk: you have to divide by pi to 100 decimal places! It was hopeless.
One time I boasted, “I can do by other methods any integral anybody else needs contour integration to do.”
So Paul puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration! He was always deflating me like that. He was a very smart fellow.
The first time I was in Brazil I was eating a noon meal at I don’t know what time—I was always in the restaurants at the wrong time—and I was the only customer in the place. I was eating rice with steak (which I loved), and there were about four waiters standing around.
A Japanese man came into the restaurant. I had seen him before, wandering around; he was trying to sell abacuses. He started to talk to the waiters, and challenged them: He said he could add numbers faster than any of them could do.
The waiters didn’t want to lose face, so they said, “Yeah, yeah. Why don’t you go over and challenge the customer over there?”
The man came over. I protested, “But I don’t speak Portuguese well!”
The waiters laughed. “The numbers are easy,” they said.
They brought me a pencil and paper.
The man asked a waiter to call out some numbers to add. He beat me hollow, because while I was writing the numbers down, he was already adding them as he went along.
I suggested that the waiter write down two identical lists of numbers and hand them to us at the same time. It didn’t make much difference. He still beat me by quite a bit.
However, the man got a little bit excited: he wanted to prove himself some more. “Multiplicao!” he said.
Somebody wrote down a problem. He beat me again, but not by much, because I’m pretty good at products.
The man then made a mistake: he proposed we go on to division. What he didn’t realize was, the harder the problem, the better chance I had.
We both did a long division problem. It was a tie.
This bothered the hell out of the Japanese man, because he was apparently very well trained on the abacus, and here he was almost beaten by this customer in a restaurant.
“Raios cubicos!” he says, with a vengeance. Cube roots! He wants to do cube roots by arithmetic! It’s hard to find a more difficult fundamental problem in arithmetic. It must have been his topnotch exercise in abacus-land.
He writes a number on some paper—any old number—and I still remember it: 1729.03. He starts working on it, mumbling and grumbling: “Mmmmmmagmmmmbrrr”—he’s working like a demon! He’s poring away, doing this cube root.
Meanwhile I’m just sitting there.
One of the waiters says, “What are you doing?”
I point to my head. “Thinking!” I say. I write down 12 on the paper. After a little while I’ve got 12.002.
The man with the abacus wipes the sweat off his forehead: “Twelve!” he says.
“Oh, no!” I say. “More digits! More digits!” I know that in taking a cube root by arithmetic, each new digit is even more work than the one before. It’s a hard job.
He buries himself again, grunting, “Rrrrgrrrrmmmmmm …” while I add on two more digits. He finally lifts his head to say, “12.0!”
The waiters are all excited and happy. They tell the man, “Look! He does it only by thinking, and you need an abacus! He’s got more digits!”
He was completely washed out, and left, humiliated. The waiters congratulated each other.
How did the customer beat the abacus? The number was 1729.03. I happened to know that a cubic foot contains 1728 cubic inches, so the answer is a tiny bit more than 12. The excess, 1.03, is only one part in nearly 2000, and I had learned in calculus that for small fractions, the cube root’s excess is one-third of the number’s excess. So all I had to do is find the fraction 1/1728, and multiply by 4 (divide by 3 and multiply by 12). So I was able to pull out a whole lot of digits that way.
A few weeks later the man came into the cocktail lounge of the hotel I was staying at. He recognized me and came over. “Tell me,” he said, “how were you able to do that cube-root problem so fast?”
I started to explain that it was an approximate method, and had to do with the percentage of error. “Suppose you had given me 28. Now, the cube root of 27 is 3..
He picks up his abacus: zzzzzzzzzzzzzzz—
“Oh yes,” he says.
I realized something: he doesn’t know numbers. With the abacus, you don’t have to memorize a lot of arithmetic combinations; all you have to do is learn how to push the little beads up and down. You don’t have to memorize 9 + 7 = 16; you just know that when you add 9 you push a ten’s bead up and pull a one’s bead down. So we’re slower at basic arithmetic, but we know numbers.
Furthermore, the whole idea of an approximate method was beyond him, even though a cube root often ca
O Americano, Outra Vez!
One time I picked up a hitchhiker who told me how interesting South America was, and that I ought to go there. I complained that the language is different, but he said just go ahead and learn it—it’s no big problem. So I thought, that’s a good idea: I’ll go to South America.
Cornell had some foreign language classes which followed a method used during the war, in which small groups of about ten students and one native speaker speak only the foreign language—nothing else. Since I was a rather young-looking professor there at Cornell, I decided to take the class as if I were a regular student. And since I didn’t know yet where I was going to end up in South America, I decided to take Spanish, because the great majority of the countries there speak Spanish.
So when it was time to register for the class, we were standing outside, ready to go into the classroom, when this pneumatic blonde came along. You know how once in a while you get this feeling, WOW? She looked terrific. I said to myself, “Maybe she’s going to be in the Spanish class—that’ll be great!” But no, she walked into the Portuguese class. So I figured, What the hell—I might as well learn Portuguese.