This is an excerpt from the chapter “Lucky Numbers”, in Surely, You’re Joking, Mr. Feynman!, Edward Hutchings ed., W. W. Norton, ISBN: 0-393-31604-1.
The setting is Brazil; the narrator is Richard Feynman.
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 paper and pencil.
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. “Multiplicação!” 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.
The bothered the hell out of the Japanese man, because he was apparently 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 down 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 that 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, “12.01!”
The waiter 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 to learn 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 cubic root often cannot be computed exactly by any method. So I never could teach him how I did cube roots or explain how lucky I was that he happened to choose 1729.03.