Hans Christian Von Baeyer
How Fermi Would Have Fixed It

At twenty-nine minutes past five, on a Monday morning in July of 1945, the world's first atom bomb exploded in the desert sixty miles northwest of Alamogordo, New Mexico. Forty seconds later, the blast's shock wave reached the base camp, where scientists stood in stunned contemplation of the historic spectacle. The first person to stir was the Italian-American physicist Enrico Fermi, who was on hand to witness the culmination of a project he had helped begin.

Before the bomb detonated, Fermi had torn a sheet of notebook paper into small bits. Then, as he felt the first quiver of the shock wave spreading outward through the still air, he released the shreds above his head. They fluttered down and away from the mushroom cloud growing on the horizon, landing about two and a half yards behind him. After a brief mental calculation, Fermi announced that the bomb's energy had been equivalent to that produced by ten thousand tons of TNT. Sophisticated instruments also were at the site, and analyses of their readings of the shock wave's velocity and pressure, an exercise that took several weeks to complete, confirmed Fermi's instant estimate.

The bomb-test team was impressed, but not surprised, by this brilliant bit of scientific improvisation. Enrico Fermi's genius was known throughout the world of physics. In 1938, he had won a Nobel Prize for his work in elementary particle physics, and, four years later, in Chicago, had produced the first sustained nuclear chain reaction, thereby ushering in the age of atomic weapons and commercial nuclear power. No other physicist of his generation, and no one since, has been at once a masterful experimentalist and a leading theoretician. In miniature, the bits of paper and the analysis of their motion exemplified this unique combination of gifts.

Like all virtuosos, Fermi had a distinctive style. His approach to physics brooked no opposition; it simply never occurred to him that he might fail to find the solution to a problem. His scientific papers and books reveal a disdain for embellishments in preference for the most direct, rather than the most intellectually elegant, route to an answer. When he reached the limits of his cleverness, Fermi completed a task by brute force.

To illustrate this approach, imagine that a physicist must determine the volume of an irregular object - say, Earth, which is slightly pear-shaped. He might feel stymied without some kind of formula, and there are several ways he could go about getting one. He could consult a mathematician, but finding one with enough knowledge and interest to be of help is usually difficult. He could search through the mathematical literature, a time-consuming and probably fruitless exercise, because the ideal shapes that interest mathematicians often do not match those of the irregular objects found in nature. Or he could set aside his own research in order to derive the formula from basic mathematical principles, but, of course, if he had wanted to devote his time to theoretical geometry, he wouldn't have become a physicist.

Alternatively, the physicist could do what Fermi would have done-compute the volume numerically. Instead of relying on a formula, he could mentally divide the planet into, perhaps, a large number of tiny cubes, each with a volume easily determined by multiplying the length times the width times the height, and then add together the answers to these more tractable problems. This method yields only an approximate solution, but it is sure to produce the desired result, which is what mattered to Fermi. With the introduction, after the Second World War, of computers and, later, of pocket calculators, numerical computation has become standard procedure in physics.

The technique of dividing difficult problems into small, manageable ones applies to many problems besides those amenable to numerical computation. Fermi excelled at this rough-and-ready modus operandi, and, to pass it on to his students, he developed a type of question that has become associated with his name. A Fermi problem has a characteristic profile: Upon first hearing it, one doesn't have even the remotest notion what the answer might be. And one feels certain that too little information exists to find a solution. Yet, when the problem is broken down into subproblems, each one answerable without the help of experts or reference books, an estimate can be made, either mentally or on the back of an envelope, that comes remarkably close to the exact solution.

Suppose, for example, that one wants to determine Earth's circumference without looking it up. Everyone knows that New York and Los Angeles are separated by about three thousand miles and that the time difference between the two coasts is three hours. Three hours corresponds to one-eighth of a day, and a day is the time it takes the planet to complete one revolution, so its circumference must be eight times three thousand, or twenty-four thousand miles-an answer that differs from the true value (at the equator, 24,902.45 miles) by less than four percent. In John Milton's words:

so easy it seemed
Once found, which yet unfound most would have thought

Fermi problems might seem to resemble the brainteasers that appear among the back pages of airline magazines and other popular publications (Given three containers that hold eight, five, and three quarts, respectively, how do you measure out a single quart?), but the two genres differ significantly. The answer to a Fermi problem, in contrast to that of a brainteaser, cannot be verified by logical deduction alone and is always approximate. (To determine precisely Earth's circumference, it is necessary that the planet actually be measured.) Then, too, solving a Fermi problem requires a knowledge of facts not mentioned in the statement of the problem. (In contrast, the decanting puzzle contains all the information necessary for its solution.)

These differences mean that Fermi problems are more closely tied to the physical world than are mathematical puzzles, which rarely have anything practical to offer physicists. By the same token, Fermi problems are reminiscent of the ordinary dilemmas that non-physicists encounter every day of their lives. Indeed, Fermi problems, and the way they are solved, not only are essential to the practice of physics; they teach a valuable lesson in the art of living.

How many piano tuners are there in Chicago? The whimsical nature of this question, the improbability that anyone knows the answer, and the fact that Fermi posed it to his classes at the University of Chicago have elevated it to the status of legend. There is no standard solution (that's exactly the point), but anyone can make assumptions that quickly lead to an approximate answer. Here is one way: If the population of metropolitan Chicago is three million, an average family consists of four people, and one-third of all families own pianos, there are two hundred and fifty thousand pianos in the city. If each piano is tuned every ten years, there are twenty-five thousand tunings a year. If each tuner can service four pianos a day, two hundred and fifty days a year, for a total of one thousand tunings a year, there must be about twenty-five piano tuners in the city. The answer is not exact; it could be as low as ten or as high as fifty. But, as the yellow pages of the telephone directory attest, it is definitely in the ball park.

Fermi's intent was to show that although, at the outset, even the answer's order of magnitude is unknown, one can proceed on the basis of different assumptions and still arrive at estimates that fall within range of the answer. The reason is that, in any string of calculations, errors tend to cancel out one another. If someone assumes, for instance, that every sixth, rather than third, family owns a piano, he is just as likely to assume that pianos are tuned every five, not ten, years. It is as improbable that all of one's errors will be underestimates (or overestimates) as it is that all the throws in a series of coin tosses will be heads (or tails).The law of probabilities dictates that deviations from the correct assumptions will tend to compensate for one another, so the final results will converge toward the right number.

Of course, the Fermi problems that physicists face deal more often with atoms and molecules than with pianos. To answer them, one needs to commit to memory a few basic magnitudes, such as the approximate radius of a typical atom or the number of molecules in a thimbleful of water. Equipped with such facts, one can estimate, for example, the distance a car must travel before a layer of rubber about the thickness of a molecule is worn off the tread of its tires. It turns out that that much is removed with each revolution of the wheels, a reminder of the immensity of the number of atoms in a tire. (Assume that the tread is about a quarter-inch thick and that it wears off in forty thousand miles of driving. If a quarter-inch is divided by the number of revolutions a typical wheel, with its typical circumference, makes in forty thousand miles, the answer is roughly one molecular diameter.)

More momentous Fermi problems might concern energy policy (the number of solar cells required to produce a certain amount of electricity), environmental quality (the amount of acid rain caused annually by coal consumption in the United States), or the arms race. A good example from the weapons field was proposed in 1981 by David Hafemeister, a physicist at the California Polytechnic State University: For what length of time would the beam from the most powerful laser have to be focused on the skin of an incoming missile to ignite the chemical explosive in the missile's nuclear warhead? The key point is that a beam of light, no matter how well focused, spreads out like an ocean wave entering the narrow opening of a harbor, a phenomenon called diffraction broadening. The formula that describes such spreading applies to all forms of waves, including light waves, so, at a typical satellite-to-missile distance of, perhaps, seven hundred miles, a laser's energy will become considerably attenuated. With some reasonable assumptions about the temperature at which explosive materials ignite (say, a thousand degrees Fahrenheit), the diameter of the mirror that focuses the laser beam (ten feet is about right), and the maximum available power of chemical lasers (a level of a million watts has not yet been attained but is conceivable), the answer turns out to be around ten minutes.

Trying to keep a laser aimed at a speeding missile at a distance of seven hundred miles for that long is a task that greatly exceeds the capacity of existing technology. For one thing, the missile travels so rapidly that it would be impossible to keep it within range. For another, a laser beam must reflect back toward its source to verify that it is hitting its target (a process comparable to shining a flashlight at a small mirror carried by a running man at the opposite end of a football field so that the light reflected from the mirror stays in one's eyes).

The solution of this Fermi problem depends on more facts than average people, or even average physicists, have at their fingertips, but for those who do have them in mind, the calculation takes only a few minutes. And it is no less accurate for being easy to perform. So it is not surprising that Hafemeister's conclusion, which predated the President's 1983 StarWars speech (research on laser weapons began two decades ago), agrees roughly with the findings of the American Physical Society's report entitled "Science and technology of Directed Energy Weapons," which was the result of much more elaborate analysis. Prudent physicists-those who want to avoid false leads and dead ends - operate according to a long-standing principle: Never start a lengthy calculation until you know the range of values within which the answer is likely to fall (and, equally important, the range within which the answer is unlikely to fall).They attack every problem as a Fermi problem, estimating the order of magnitude of the result before engaging in an investigation.

Physicists also use Fermi problems to communicate with one another.When they gather in university hallways, convention-center lobbies, or French restaurants to describe a new experiment or to discuss an unfamiliar subject, they often first survey the lay of the land, staking out, in a numerical way, the perimeter of the problem at hand. Only the timid hang back. deferring to the experts in their midst. Those accustomed to tackling Fermi problems approach the experiment or subject as if it were their own, demonstrating their understanding by performing rough calculations. If the conversation turns to a new particle accelerator, for example, they will estimate the strength of the magnets it requires; if the subject is the structure of a novel crystal, they will calculate the spacing between its atoms. Everyone tries to arrive at the correct answer with the least effort. It is this spirit of independence, which he himself possessed in ample measure, that Fermi sought to instill by posing his unconventional problems.

Questions about atom bombs, piano tuners, automobile tires, laser weapons, particle accelerators, and crystal structure have little in common. But the manner in which they are answered is the same in every case and can be applied to questions outside the realm of physics. Whether the problem concerns cooking, automobile repair, or personal relationships, there are two basic types of responses: the fainthearted turn to authority - to reference books, bosses, expert consultants, physicians, ministers - while the independent of mind delve into that private store of common sense and limited factual knowledge that everyone carries, make reasonable assumptions, and derive their own, admittedly approximate, solutions. To be sure, it would be foolish to practice neurosurgery at home, but mundane challenges-preparing chili from scratch, replacing a water pump, resolving a family quarrel-can often be sorted out with nothing more than logic, common sense, and patience.

The resemblance of technical problems to human ones was explored in Robert M. Pirsig's 1974 book, Zen and theart of Motorcycle Maintenance, in which the repair and upkeep of a machine served as a metaphor for rationality itself. At one point, the protagonist proposed to fix the slipping handlebars of a friend's new BMW motorcycle, the pride of a half-century of German mechanical craftsmanship, with a piece of an old beer can. Although the proposal happened to be technically perfect (the aluminum was thin and flexible), the cycle's owner, a musician, could not break his reliance on authority; the idea had not originated with a factory-trained mechanic, so it did not deserve serious consideration. In the same way, certain observers would have been skeptical of Fermi's analysis, carried through with the aid of a handful of confetti, of a two-billion-dollar bomb test. Such an attitude demonstrates less, perhaps, about their knowledge of the problem than about their attitude toward life. As Pirsig put it, "The real cycle you're working on is a cycle called 'yourself."'

Ultimately, the value of dealing with the problems of science, or those of everyday life, in the way Fermi did lies in the rewards one gains for making independent discoveries and inventions. It doesn't matter whether the discovery is as momentous as the determination of the yield of an atom bomb or as insignificant as an estimate of the number of piano tuners in a Midwestern city. Looking up the answer, or letting someone else find it, actually impoverishes one: it robs one of the pleasure and pride that accompany creativity and deprives one of an experience that, more than anything else in life, bolsters self-confidence. Self-confidence, in turn, is the essential prerequisite for solving Fermi problems. Thus, approaching personal dilemmas as Fermi problems can become, by a kind of chain reaction, a habit that enriches life.