In the movie "Good Will Hunting," an impoverished South Boston kid who scrapes by mopping floors at MIT astonishes prize-winning professors with his ability to solve--at a glance--math problems that have stumped the experts.

How likely is this scenario? Could a person with no specialized education instantaneously see his way through intellectual thickets impenetrable to the top people in the field? Even if he is a natural-born genius?

Conventional wisdom has it that science today is the province of experts bedecked with degrees and weighty with wisdom acquired through experience. It may come as a surprise, therefore, to learn that amateurs and outsiders have made substantial contributions in fields ranging from chemistry to astronomy:

* A San Diego homemaker working at her kitchen table discovered dozens of new geometric patterns that experts had thought were impossible.

* A Texas banker came up with a formal conjecture (a kind of mathematical hypothesis) that amounts to an expansion, or "sequel," to the famous Fermat's Last Theorem, which defied proof for more than 300 years.

* An electrical engineer in Anaheim discovered two new exploding stars in one night.

Amateurs can't compete with professionals when it comes to high-tech equipment, university connections, academic prestige and funding. But in some ways, their status as nonprofessionals can be a plus.

"Amazingly, lack of formal education can be an advantage," says mathematician Doris Schattschneider of Moravian College in Pennsylvania, who helped "discover" the San Diego homemaker now hailed by mathematicians for her work. "We get stuck in our old ways," Schattschneider said. "Sometimes, progress is only made when someone from the outside looks at it with new eyes."

There was nothing unusual about amateur scientists 200 years ago when science was something people did in their parlors and backyards. Science was not yet sequestered in its own private world, with obscure language and strict academic requirements barring all but highly trained experts.

"If you go back far enough, everybody was an amateur," said UCLA chemist Charles Knobler, who uses methods developed by amateur Agnes Pockels in his work (see accompanying box). "John Priestly, who discovered oxygen, was a minister." Benjamin Franklin was perhaps the ultimate amateur. The 18th century statesman not only discovered that lightning is electricity, he also invented the heat-efficient stove and bifocal eyeglasses.

However, as science has become more specialized, the occasional breakthrough by an amateur has become much more surprising.

'You Need People Who Don't . . . Have All the Wrong Assumptions'

That's one reason the work of San Diego homemaker Marjorie Rice caused such a stir.

A mother of five, Rice made a habit of getting her hands on Scientific American magazine, to which her son subscribed, before the rest of the family. She was a particular fan of Martin Gardner's long-running column, "Mathematical Games."

In July 1975, Gardner, an author and perhaps the best-known mathematical amateur of all, published a column about tiling patterns. A branch of mathematics that lives up to its name, tiling is concerned with determining what kinds of shapes can fit together perfectly without any overlaps or gaps. Since all solid matter, from brains to crystals, is made of tightly packed clusters of molecules, studying the possible ways that shapes can arrange themselves has many scientific applications.

In his column, Gardner mentioned that only certain types of pentagons could perfectly tile a plane, or flat surface, and that all of them were known--or so the mathematicians thought. After reading the piece, however, an amateur named Richard James III thought he found some pentagons that the experts had overlooked. As it turned out, he was right, and Gardner published James' results in his December 1975 column.

The minute Rice saw that, she was off and running. "It made me wonder," she said recently from her home in San Diego. "If he could find one, maybe I could." Since she had no formal training in mathematics, Rice developed her own notation, listing various combinations of sides and angles of pentagons on 3-by-5 cards.

She worked on the problem all through Christmastime 1975, drawing diagrams on the kitchen table when no one was around and hiding them when her husband and children came home, or when friends stopped by.

By February 1976 she was confident she had found new kinds of pentagons that could tile. She sent them off to Gardner. "I'd never written to anyone who wrote articles in magazines," she said, still somewhat awed.

Gardner forwarded Rice's drawings to Schattschneider, an expert in tiling patterns. Schattschneider at first was skeptical. Rice's peculiar markings seemed odd, like "hieroglyphics," the professor recalled. "I was probably condescending."

Eventually, however, Schattschneider verified the results. By October 1976, Rice had come up with 58 pentagon tilings, most of them previously unknown, and arranged them into 12 classes. Over the next 10 years, she discovered many more tiling patterns and three more types of pentagons, all of which were important contributions to the field.

"And she's still working on it," said Schattschneider recently. "That woman doesn't stop."

In 1995, when the Mathematical Assn. of America held its regional meeting in Los Angeles, Schattschneider discussed Rice's work in a lecture that she persuaded Rice and her husband to attend. At the end of the talk, Rice reluctantly agreed to be introduced. "I told the audience that we had a very special person with us today," Schattschneider said. "And everybody in the room . . . gave her a standing ovation."

Mathematicians tend to think that someone like Rice is an anomaly, Schattschneider said. "A lot of them get so wrapped up they don't even entertain the idea that someone without credentials could make a real contribution," she said. "I wanted to show that [the contributions of amateurs] can lead to something serious."

Although amateurs may not speak the professional lingo, Schattschneider said, their passionate will to know, intense concentration and fresh perspectives can make up for lack of specialized training. "Just because someone doesn't have the technical language is no reason to dismiss their work," she said.

Indeed, according to Michigan State University physiologist Robert Root-Bernstein, who has also studied the contributions of amateurs, a nonspecialist is just the person to tackle a particularly hard problem. "If the experts had the answer, it would already be out there," he said. "You need people who don't already have all the wrong assumptions."

'I'm So in Love With Mathematics,' Banker Says

Most recently, the amateur making big news has been Texas banker Andrew Beal--in part because he has put his money where his math is, offering a $50,000 prize for the solution to a problem he devised. Known as Beal's problem, it is quite similar to Fermat's Last Theorem, whose proof also carried a cash prize of about $30,000 by the time it was claimed in 1994.

Fermat's Last Theorem is deceptively simple to state for an idea that took more than 300 years to prove. Essentially, it says that although there are whole-number solutions to the equation x 2 + y 2 = z 2 (for example, 3 squared plus 4 squared equals 5 squared), there are no whole number solutions to the equation for exponents greater than 2. In other words, there is no whole number solution to the equation x 3 + y 3 = z 3 .

It took 200 pages of nearly impenetrable arguments and calculations for mathematician Andrew Wiles of Princeton to prove that Pierre de Fermat, a French lawyer who lived from 1601-65, had been right.

Beal has been working on similar problems for many years, teaching himself the field--known as number theory--as he went along.

Now he has proposed that there are no whole number solutions to the problem when the exponents are all different: for example, x 3 + y 4 = z 5 . (The problem is actually quite a bit more complicated than that.)

Beal's conjecture is taken quite seriously by mathematicians, who say it is a more general version of Fermat's Last Theorem. Beal himself is hoping to come up with a simpler proof of Fermat's Last Theorem, and he sees his own conjecture as a step on that path.

"If you could prove Beal's conjecture, then you would immediately prove Fermat," said mathematician Keith Devlin of St. Mary's College in Moraga, east of Oakland, praising Beal's work. "And let's not forget that Fermat was an amateur."

Still, while Beal has come up with a very good problem, Devlin said, he probably doesn't have the mathematical background necessary to solve it. That's one reason Beal offered the prize. The other reason is that he wants to encourage young people in math.

"I'm so in love with mathematics," said the banker, who has not studied the subject formally since high school, but educated himself. "I was inspired by the prize that was won for proving Fermat, and I'd like to inspire some other young minds."

Beal's day job is running the bank that he founded in Dallas. But he believes that amateurs like him bring to some fields a fresh perspective that professionals lack.

He said that being isolated from the professional community of like-minded thinkers can be a benefit. "When you're in a business or profession," he said, "we all start to look at things similarly. It helps to start with a clean slate."

Beal also believes that amateurs may be more motivated than many professionals. It isn't work for them; it's a hobby. "In many professions, you find the amateurs making breakthroughs because they're only in it because of their interest," he said. "They're not in it to make a living."

Moreover, amateurs don't have to worry about committing the kind of inevitable "stupid" errors that might be embarrassing to a professional worried about tarnishing a well-polished reputation.

Both Rice and Beal made their contributions in areas of mathematics that are relatively open to amateurs. In Rice's case, the visual aspect of tiling makes it accessible; Beal's work in number theory doesn't require heavy-duty mathematical machinery. Discovering new prime numbers is also part of number theory and therefore attracts a lot of amateurs.

Of course, research that needs expensive equipment and large groups of scientists--particle physics, for example--isn't accessible to amateurs. But other fields of science not only welcome amateurs, but depend on them. "Professional astronomers don't have the means to keep track of the whole sky every night," said Dan Green of the Harvard Smithsonian Center for Astrophysics. "Amateurs help out enormously. They're the unsung heroes."

Amateurs routinely discover comets, asteroids and exploding stars, and keep track of variable stars and fast-moving objects. "If a comet is discovered and you don't follow it the next night and the next, it's lost," Green said. "Professionals don't have access to telescopes all around the month, but amateurs do because they're in their own backyards."

Of course, amateurs make a lot of mistakes, he said, adding that the rate of false alarms for amateurs reporting new objects in the sky is nearly 99%. "They don't know the ins and outs," he said. "It's easy to have a glitch or a ghost image or a flaw. You have to be systematic."

Still, Green relies heavily on amateurs with solid track records, such as Wayne Johnson, an electrical engineer at Boeing in Anaheim. In 1996, Johnson discovered two supernovas--exploding stars--on a single night.

Finding a 'Diamond in the Dust' About Once a Year

Since supernovas can go off anywhere in the sky at random, they are extremely hard to spot. A typical galaxy will light up with a supernova about once every 20 to 400 years, Johnson said. Dedicated supernova hunters like him go out night after night, taking pictures of galaxies, then comparing their pictures to previous images of the same galaxy to see if something has changed.

"Galaxies are sort of like snowflakes," said Johnson, whose answering machine identifies him and his wife as Mr. and Mrs. Galaxy. "No one is like another. Amateurs call them 'faint fuzzies,' but if you look closely, you can see detail. That's the challenge. You look for something that's changing. The next time you look, there's a little diamond in the dust."

Since 1991, Johnson has bagged an average of one diamond in the dust a year.

Like Rice and Beal, he sticks to his search largely out of love. He has been looking at the sky since he was 5, and is an avid member of the Orange County Astronomy Club.

Money can't buy this kind of dedication. Only untamed curiosity, fueled by passion, seems to do the trick. Or as the late physicist Frank Oppenheimer used to say: "Understanding is a lot like sex; it's got a practical purpose, but that's not why people do it normally."

As Rice puts it: "I just couldn't understand very well what [the mathematicians featured in Gardner's column] were talking about, and I wanted to understand it." She still hasn't stopped trying.

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A Few Famous Amateurs

* Pierre de Fermat, French author of Fermat's Last Theorem, a mathematical puzzle more than 300 years old, was a lawyer. The theorem was proved in 1994 by Princeton mathematician Andrew Wiles.

* Agnes Pockels, a 19th century European homemaker with a talent for physics, discovered methods for studying single layers of molecules and "carried out first-class research in her kitchen," said Charles Knobler, chairman of the UCLA chemistry department. His own research is based on Pockels' techniques.

* Hedy Lamarr, the first actress to appear naked in a feature film, was belatedly lauded last year for her role in the 1940 invention of a technological innovation called "frequency hopping." It has been used as an anti-jamming device for radio-controlled torpedoes, and today is an integral part of cell phones.

* Documentary filmmaker Nicholas Clapp and attorney George Hedges, both from Los Angeles, discovered the ancient lost city of Ubar in southern Oman in 1992.

* Alan Hale, an astronomer from New Mexico, and Thomas Bopp, an amateur astronomer from Arizona, discovered the spectacular comet Hale-Bopp in 1995.

* Roland Clarkson, a Cal State Dominguez Hills student, discovered the longest prime number (a number that can be evenly divided only by 1 and itself) early this year. The number is so enormous it would fill a 400-page novel. In 1978, two high school students from Hayward, Calif., discovered the then-longest prime.