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Walking around the hastily built wooden barracks that housed the soul of the atomic bomb project in 1943 and 1944, a scientist would see dozens of men laboring over computation. Everyone calculated. The theoretical department was home to some of the world's masters of mental arithmetic, a martial art shortly to go the way of jiujitsu. Any morning might find men such as Bethe, Fermi, and John von Neumann together in a single small room where they would spit out numbers in a rapid-fire calculation of pressure waves. Bethe's deputy, Weisskopf, specialized in a particularly oracular sort of guesswork; his office became known as the Cave of the Hot Winds, producing, on demand, unjustifiably accurate cross sections (shorthand for the characteristic probabilities of particle collisions in various substances and circumstances). The scientists computed everything from the shapes of explosions to the potency of Oppenheimer's cocktails, first' with rough guesses and then, when necessary, with a precision that might take weeks.

...

When he (Feynman) started managing groups of people who handled laborious computation. he developed a reputation for glancing over people's shoulders and stabbing his finger at each error: "That's wrong." His staff would ask why he was putting them to such labor if he already knew the answers. He told them he could spot wrong results even when he had no idea what was right something about the smoothness of the numbers or the relationships between them. Yet unconscious estimating was not really his style. He liked to know what he was doing. He would rummage through his toolbox for an analytical gimmick, the right key or lock pick to slip open a complicated integral. Or he would try various simplifying assumptions: Suppose we treat some quantity as infinitesimal. He would allow an error and then measure the bounds of the error precisely.

...

When Bethe and Feynman went up against each other in games of calculating, they competed with special pleasure. Onlookers were often surprised, and not because the upstart Feynman bested his famous elder. On the contrary, more often the slow-speaking Bethe tended to outcompute Feynman. Early in the project they were working together on a formula that required the square of 48. Feymnan reached across his desk for the Marchant mechanical calculator

Bethe said, "It's twenty-three hundred."

Feynman started to punch the keys anyway. "You want to know exactly?" Bethe said. "It's twenty-three hundred and four. Don't you know how to take squares of numbers near fifty?" He explained the trick. Fifty squared is 2,500 (no thinking needed). For numbers a few more or less than 50, the approximate square is that many hundreds more or less than 2,500. Because 48 is 2 less than 50, 48 squared is 200 less than 2,500-thus 2,300. To make a final tiny correction to the precise answer, just take that difference again-2-and square it. Thus 2,304.

Feymnan had internalized an apparatus for handling far more difficult calculations. But Bethe impressed him with a mastery of mental arithmetic that showed he had built up a huge repertoire of these easy tricks, enough to cover the whole landscape of small numbers. An intricate web of knowledge underlay the techniques. Bethe knew instinctively, as did Feynman, that the difference between two successive squares is always an odd number, the sum of the numbers being squared. That fact, and the fact that 50 is half of 100, gave rise to the squares-near-fifty trick. A few minutes later they needed the cube root of 2 1/2. The mechanical calculators could not handle cube roots directly, but there was a look-up chart to help. Feynman barely had time to open the drawer and reach for the chart before he heard Bethe say, "Thats 1.35. " Like an alcoholic who plants bottles within arm's reach of every chair in the house, Bethe had stored away. a device for anywhere he landed in the realm of numbers. He knew tables of logarithms and he could interpolate with unerring accuracy. Feynman's own mastery of calculating had taken a different path. He knew how to compute series and derive trigonometric functions, and how to visualize the relationships between them. He had mastered mental tricks covering the deeper landscape of algebraic analysis-differentiating and integrating equations of the kind that lurk dragonlike in the last chapters of calculus texts. He was continually put to the test. The theoretical division sometimes seemed like the information desk at a slightly exotic library. The phone would ring and a voice would ask, "What is the sum of the series 1 + (1/2)^{4} + (1/3)^{4} + (1/4)^{4} + . . . ?"

"How accurate do you want it?" Feymnan replied.

"One percent will be fine."

"Okay," Feymnan said. "One point oh eight." He had simply added the first four terms in his head-that was enough for two decimal places. Now the voice asked for an exact answer. "You don't need the exact answer," Feynman said.

"Yeah, but I know it can be done."

So Feynman told him. "All right. It's pi to the fourth over ninety."

He and Bethe both saw their talents as labor-saving devices. It was also a form of jousting. At lunch one day, feeling even more ebullient than usual, he challenged the table to a competition. He bet that he could solve any problem within sixty seconds, to within ten percent accuracy, that could be stated in ten seconds. Ten percent was a broad margin, and choosing a suitable problem was hard. Under pressure, his friends found themselves unable to stump him. The most challenging problem anyone could produce was: Find the tenth binomial coefficient in the expansion of (1 + X)^{20}. Feynman solved that just before the clock ran out. Then Paul Olum spoke up. He had jousted with Feynman before, and this time he was ready. He demanded the tangent of ten to the hundredth. The competition was over. Feynman would essentially have had to divide one by pi and throw out the first one hundred digits of the result-which would mean knowing the one-hundredth decimal digit of pi. Even Feymnan could not produce that on short notice.

**Reproduced (scanned) from:**

GENIUS. Richard Feynman and modern physics. J Gleik. First Published in Great Britain in 1992 by Little Brown and Company (1992). Chapter 4 (Los Alamos).

"A superb biography..As in Chaos, Gleick manages to convey the fact that scientific communities are indeed human, collections of struggling, fallible beings driven by ambition, jealousy and other base motives, while conveying also the real love of knowledge and insatiable curiosity of the best of them" THE AUSTRALIAN