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Each summer six math whizzes selected from nearly a half-million American teens compete against the world’s best problem solvers at the International Mathematical Olympiad. Steve Olson followed the six 2001 contestants from the intense tryouts to the Olympiad’s nail-biting final rounds to discover not only what drives these extraordinary kids but what makes them both unique and typical. In the process he provides fascinating insights into the science of intelligence and learning...
Each summer six math whizzes selected from nearly a half-million American teens compete against the world’s best problem solvers at the International Mathematical Olympiad. Steve Olson followed the six 2001 contestants from the intense tryouts to the Olympiad’s nail-biting final rounds to discover not only what drives these extraordinary kids but what makes them both unique and typical. In the process he provides fascinating insights into the science of intelligence and learning and, finally, the nature of genius.
Brilliant, but defying all the math-nerd stereotypes, these teens want to excel in whatever piques their curiosity, and they are curious about almost everything—music, games, politics, sports, literature. One team member is ardent about both water polo and creative writing. Another plays four musical instruments. For fun and entertainment during breaks, the Olympians invent games of mind-boggling difficulty. Though driven by the glory of winning this ultimate math contest, they are in many ways not so different from other teenagers, finding pure joy in indulging their personal passions. Beyond the the Olympiad, Olson sheds light on many questions, from why Americans feel so queasy about math, to why so few girls compete in the subject, to whether or not talent is innate. Inside the cavernous gym where the competition takes place, Count Down uncovers a fascinating subculture and its engaging, driven inhabitants.
On July 4, 1974, a bus carrying eight U.S. high school students wound through the narrow medieval streets of Erfurt, East Germany.
The students were all a bit nervous. In those days of heightened Cold War tensions, few Americans ventured beyond the Iron Curtain. Just that morning, after an all-night flight from New York City, the students had endured a brusque round of questioning by the East German border police. As they stepped out of the bus in the center of Erfurt, beneath the spires of the cathedral where Martin Luther preached his first sermons, they felt both isolated and highly visible.
They were nervous for another reason. These high school juniors and seniors were the first team from the United States ever to compete in an International Mathematical Olympiad. In 1974 the Olympiad was already fifteen years old; the first one had been held in 1959 in Bucharest, Romania. But throughout the 1960s the United States had been reluctant to field an Olympiad team.
The Olympiad is a competition for individuals in which gold, silver, and bronze medals are awarded. But unofficially the teams always have added their individual scores and compared themselves country against country. In this informal contest the Olympiad had been dominated by teams from the Soviet Union and eastern Europe. Even as more teams from western Europe began to compete—Finland in 1965 (finishing last), Great Britain, Sweden, Italy, and France (also finishing last) in 1967—the U.S. mathematics community had no desire to pit America’s best high school students against the world’s best. “A lot of people were dead set against it,” says Murray Klamkin, a former Olympiad coach who now lives in Edmonton, Canada. “They thought a U.S. team would be crushed by all those Communist countries.”
In 1971 the mathematician Nura Turner, from the State University of New York at Albany, wrote an article that began to change people’s minds. She pointed out that several state-level competitions, established mostly since the 1950s, had laid the groundwork for American participation at the international level. She admitted that a U.S. team might be humiliated in its initial attempts but argued that Americans were tough enough to bounce back. “We certainly must possess here in the USA the strength of character,” she wrote, “to face defeat and the capability and courage to then plunge into systematic hard training to compete again with the desire to strive for a better showing.”
In 1974 the major U.S. mathematical organizations finally agreed to send a team. Two years earlier the Mathematical Association of America had instituted a national exam designed to identify the best high school mathematicians in the country. In the spring of 1974 the association named the top eight finishers on the exam as the members of the U.S. Olympiad team.
Eric Lander, who is now one of the world’s preeminent geneticists and the director of the Broad Institute of Harvard University and Massachusetts Institute of Technology, was a member of the team that first year. It was his senior year at Stuyvesant High School in Manhattan, and Lander was captain of the school’s math team. “Math team was great,” he says. “About thirty kids met each morning for an hour before school in a fifth- floor room of Stuyvesant High School, and the captain of the team was responsible for running the session. This was before you had databases full of math problems, so the captain of the math team, upon his ascension to office, came into possession of what we called ‘the shopping bag.’ It contained mimeographed sheets of problems and strips of problems and records of the city math contests for a long time. So the captain of the team would pull problems out of the bag and be responsible for leading the group.”
When most people think about math competitions, they probably envision a roomful of kids struggling to perform complex calculations faster than the next person. But most of the problems in high-level competitions have very little to do with calculations. Solving these problems requires a sophisticated grasp of mathematical ideas, so that familiar concepts can be extended in new directions. The mathematical procedures everyone learns in school aren’t enough. Becoming an excellent problem solver demands creativity, daring, and playfulness. A math competition is more like a game than a test—a game played with the mind.
The structure of an International Mathematical Olympiad reflects the nature of the problems. The size of the teams has changed over time. In the early years each team had eight members; since 1983 they have had six. But the format has stayed the same. On the first day of the competition all of the Olympians reeceive a sheet of paper containing three problems, and each competitor, working individually, has four and a half hours to make as much progress on the problems as he or she can. The next day they have the same amount of time to solve three additional problems.
But the competition doesn’t begin when the competitors arrive in the Olympiad city, because the assembled team coaches first have to decide which problems will be on the exam. In Erfurt the teams had five days to tour the city and get to know one another.
“It was fascinating—the single team we most resembled and got along with were the Russians,” says Lander. “So we hung out with the Russians a lot and got into all sorts of mischief.
We were in East Germany, and the Russians figured at that point that they owned East Germany, so they weren’t going to get in trouble. I remember very well going up to the top of the dormitory at the school where we were staying, and the Americans and Russians throwing water balloons down on the street. The Russians might not do it back home, but they could do it in East Germany.”
On July 8 the eighteen teams competing in the Sixteenth International Mathematical Olympiad gathered at a local university to take the exam. All the worries about the U.S. team’s abilities had been for naught. Lander and his teammates finished second—just a few points behind the Soviet Union.
_ This book is first and foremost the story of the Forty-second International Mathematical Olympiad, which took place in 2001 on the campus of George Mason University in Fairfax, Virginia, right outside Washington, D.C. The event has grown substantially since 1974. Nearly 500 kids from eighty-three countries competed in the Forty-second Olympiad, compared with about 125 in 1974 (and compared to the 150 or so who competed in 1981, the only previous Olympiad held in the United States).
The Soviet team has splintered into teams from Russia, Latvia, Kazakhstan, and other former republics. Teams from South America and Africa—Argentina, Brazil, Colombia, Paraguay, Peru, Uruguay, Venezuela, Morocco, Tunisia, and South Africa—now compete. So do teams from East Asian countries such as Macau, Hong Kong, and the Philippines.
As one might expect, the competitors at the Forty-second Olympiad had their cultural differences, most notably the more than fifty languages that were spoken. But in general the Olympians were remarkably compatible. Most knew at least a little English, since English has become the language in which most of the world’s higher-level mathematics is conducted. A soccer game immediately sprang up in the courtyard of the dormitory complex where they were staying and continued on and off for the duration of the event. All of the competitors could share CDs and hand-held video games, compare national qualifying exams, and lament the poor quality of the food offered in the college cafeteria.
Into this talkative, energetic, competitive mass of young mathematicians the U.S. team fit perfectly. Its members were fairly typical of those who had been on past U.S. teams. Five had just graduated from high school; one would begin his sophomore year that September. Three had spent at least part of their childhood in the San Francisco Bay area, two were from New Jersey, and one was from outside of Boston. Three participated in other team sports and were fairly athletic; the other three limited their athletic endeavors mostly to Ultimate Frisbee. All had been participating in math competitions at least since middle school.
If you had met the members of the U.S. team in a cafeteria or library or on the street, you wouldn’t think there was anything special about them. They talked quickly and intensely among themselves, sometimes about math but usually about other subjects.
They were rabidly interested in games of all sorts. They liked music, pizza, and movies.
But these kids were special. They were the products of one of the most intense selection processes undergone by any group of high school students. More than 15 million students attend public and private high schools in the United States, and nearly half a million take the first in a series of exams that culminates in the selection of the U.S. Olympiad team. The six individuals who emerge from that process are the best mathematical problem solvers of any American kids their age. Even someone who knew as much mathematics as they do would not have the benefit of the rigorous training the Olympians undergo.
What is it about the members of an Olympiad team that makes them such superb problem solvers? Some people would ascribe their talents simply to genius, saying that their accom- plishments are so remarkable as to be beyond understanding.
This use of the word “genius” as a label for the inexplicable has a long history. In classical Rome genius was the spirit associated with each individual from birth who shaped that person’s character, conduct, and destiny. People sacrificed to their genius on their birthday, expecting that in return the guiding spirit would provide them with worldly success and intellectual power.
In the modern world the term often retains a hint of the supernatural.
To call someone a genius is to imply that he or she is somehow distinct from normal human beings, with apparent access to experiences or thoughts that are denied to others. Genius from this perspective can seem to be, in the words of Harvard professor Marjorie Garber, “the post-Enlightenment equivalent of sainthood.”
This way of thinking can skew even the most levelheaded analysis. In describing the achievements of the physicist Richard Feynman, Cornell University mathematician Mark Kac once made what has become a well-known distinction: There are two kinds of geniuses, the “ordinary” and the “magicians.” An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better.
There is no mystery as to how his mind works. Once we understand what they have done, we feel certain that we, too, could have done it. It is different with the magicians. . . . The working of their minds is for all intents and purposes incomprehensible.
Even after we understand what they have done, the process by which they have done it is completely dark.
Kac’s distinction is beguiling, but it’s really just a modern restatement of the Roman belief in spirits. Are the workings of some minds really incomprehensible? Or do great achievements rely on straightforward extensions of everyday thinking and imagining?
Can profound advances in the arts and sciences be analyzed in such a way as to reveal their origins? Or are some realms of ex- perience shut off from us forever, hidden behind the tantalizing veil of “genius”?
The varied meanings of the word complicate efforts to answer these questions. In modern parlance the term is often debased.
People say that a politician is a genius at wooing voters.
Newspapers label successful football coaches sports geniuses. Interior decorators, advertising writers, land developers, and country and western singers are all hailed as geniuses.
In middle and high schools, “genius” is usually a term of derision.
The word is used to taunt someone who is good at math or a dedicated writer or simply more interested in schoolwork than the average student is. Even as adults, many people would feel uncomfortable being labeled a genius. The word seems an unwanted burden, a harbinger of unfulfilled expectations.
The kids on a U.S. Olympiad team would not consider themselves geniuses. They have become incredibly adept at solving immensely difficult mathematical problems. In that sense, they are prodigies, in that they have attained very high levels of performance at a young age. But they certainly are not geniuses in the sense that Homer, Archimedes, Shakespeare, Rembrandt, Newton, Mozart, or Einstein are so considered.
Nevertheless, the members of an Olympiad team do share the attributes of genius in one respect: they employ the same intellectual tools that history’s great creators have. They use insight, talent, and creativity to produce original solutions to baf- fling problems. They exhibit the competitiveness, breadth, and sense of wonder that enable them to achieve at levels inconceivable to most people. By watching the Olympians solve mathematical problems, it’s possible at least to glimpse the qualities that have produced humanity’s greatest triumphs.
_ Besides being about extraordinary achievements, this book is about mathematicians, a group that has received much attention in popular culture recently. Mathematicians have been the protagonists of hit movies (Good Will Hunting, A Beautiful Mind) and have figured prominently in well-received plays and novels (Proof, Uncle Petros and Goldbach’s Conjecture). Princeton mathematician Andrew Wiles, who solved a famous mathematical problem called Fermat’s last theorem in 1994, was even the inspiration for a musical in 2001 called Fermat’s Last Tango.
This attention has been a mixed blessing. More than a few of these entertainments have made mathematicians out to be fools, nerds, or madmen. “Many recent works of mathematical fiction portray mathematicians as insane,” says Alex Kasman, a mathematician at the College of Charleston in South Carolina, who maintains a Web site that reviews hundreds of fictional works involving mathematics. “Certainly there are mathematicians with mental illnesses, just as there are people of other professions with mental illnesses. But the high correlation of the two in fiction both supports and generates an unfair stereotype in the general population that there is some deep connection between the two. When I was watching A Beautiful Mind, and the character of John Nash was suffering terribly from his mental illness, I heard a woman behind me say, ‘I’m glad I’m not a genius.’”
Other stereotypes plague works of fiction featuring mathematicians, says Kasman. Occasionally mathematicians are depicted as flamboyant and eccentric, like the “chaos theorist” played by Jeff Goldblum in the movie Jurassic Park. In other cases they are boring and repressed, like the husband (who also ends up deranged) in William Boyd’s novel Brazzaville Beach.
Rarely do moviegoers or novel readers encounter mathematicians with whom they might enjoy a conversation at a party. “I suppose no author wants to write about people who are ordinary,” Kasman says. “So it’s not surprising that very few fictional mathematicians are just ordinary people who like mathematics.
But because most people do not personally know any mathematicians, they form their opinions of them based on these works of fiction. My experience, on the other hand, suggests that mathematicians are as normal as the people in any other profession.”
One response to the stereotyping of mathematicians is to observe that scriptwriters and popular novelists stereotype all professions, even their own. But mathematicians have been absorbing abuse for a long time. In the Greek drama The Birds, written by Aristophanes in the fifth century b.c., a geometer named Meton arrives at a city founded by the Athenian Makedo and announces that he intends “to survey the plains of the air for you and to parcel them into lots.” The populace denounces him as a “quack and imposter,” beats him, and drives him from the city. In the novel Emma, published in 1815, Jane Austen asks whether a linguist, a grammarian, or “even a mathematician” could fail to appreciate the ardor of newfound love.
In American secondary schools, the stereotype of kids who are good at mathematics is somewhat different. They are seen as social misfits, physically uncoordinated, interested only in mathematics and other geeky subjects. Sometimes this stereotype turns up in television shows and movies as the badly dressed, awkward, computer-programming male who can’t find a girlfriend.
The kids on an Olympiad team defy these brutally unfair stereotypes. Not all of them are interested in computers, science, or Star Trek. Some even claim to be not very good at mathematical calculations, at least compared with other Olympians. In fact, many of their traits initially seem antithetical to mathematics.
They have deep insights into the problems they are solving. They are blindingly creative. They perceive the beauty in abstract mental constructs with an almost religious passion. And they are able to combine those traits in such a way that each trait builds on the others (though in this book I examine a different trait for each team member and each Olympiad problem).
None of the Olympians fits comfortably into the stereotype of a mathematician. Each can be understood—and appreciated—only as an individual.
_ Finally, this is a book about mathematics—about its complexities, its unreasonable effectiveness, its stark and breathtaking beauty. Many people believe that higher-level mathematics is conducted on a plane separate from normal thought, using concepts and logic that they could never hope to understand. To many mathematicians this belief seems misguided. They see mathematics as a smooth continuum from the numbers and shapes everyone learns in grade school to the frontiers of mathematical research. In many professions, acolytes need to make sudden leaps of achievement or skill, as when someone flies an airplane for the first time or teaches a class of boisterous students.
Mathematics is not one of those professions.
A book about art has to include some reproductions of artwork, and a cookbook has to have recipes. By the same token, a book about problem solving should contain a few mathematical problems. For many people, the automatic reaction upon turning a page and seeing a geometric diagram or an equation will be “Oh no, not math!” That reaction is perfectly understandable. It arises from the boring mathematics classes most of us had to endure in school, the common belief that “I was never any good at math,” and the widespread conviction that only the gifted few can hope to understand mathematics.
The six Olympiad problems in this book probably should be seen as extended examples rather than as core parts of the story.
You don’t have to understand the problems in detail to appreciate the skills that distinguish the Olympians. And readers who skip or skim over the problems will be in good company. When the English zoologist Sir Solly Zuckerman was asked once what he did when he came across mathematical formulas in scientific papers, he replied, “I hum them.”
But anyone who can calculate a loan payment or a batting average is capable of understanding the problems described in this book. Olympiad problems are designed to involve only the mathematics that people learn in high school. They don’t require a knowledge of subjects usually learned in college, such as calculus.
Coming up with solutions to the problems is very challenging.
The reason the Olympiad is generally considered the world’s hardest mathematical competition is that high school students have relatively few tools with which to solve the problems, compared with older students who know more mathematics. Still, many of the solutions the Olympians devise are relatively easy to describe. For the three problems given on the first day of the Olympiad, the chapters of this book provide relatively complete solutions, with a few supporting details given in the appendix.
For the second three problems, which are more complex, the chapters provide a general description of the solutions, with a somewhat more detailed treatment in the appendix.
Working through one or more of the problems may take some time (though discussions of international relations, political gerrymandering, or the science of dieting are often more complicated), but the effort will be rewarded. As James Newman wrote in his classic anthology The World of Mathematics, “There are few gratifications comparable to that of keeping up with a demonstration and attaining the proof. It is for each man an act of creation, as if the discovery had never been made before.”
Just as anyone can marvel at a great painting, a sublime piece of music, or a thunderous slam dunk without being a painter, composer, or basketball player, so anyone can appreciate the power and beauty of elegant mathematical problems and solutions.
They are products of the human mind, as mysterious and inspiring as are all acts of creation.
Copyright © 2004 by Steve Olson. Reprinted by permission of Houghton Mifflin Company.
Introduction 1 Part i The Path to the Olympiad 1 Inspiration 15 2 Direction 38 Part ii Attributes 3 Insight 59 4 Competitiveness 80 5 Talent 98 6 Interlude: An Afternoon to Rest 122 7 Creativity 126 8 Breadth 149 9 A Sense ofWonder 167 Part iii Results 10 Triumph 187 11 Epilogue 199 Appendix: Solutions and Commentaries 201 Sources 211 Acknowledgments 227 Index 231