A Beautiful Mind

by Sylvia Nasar

  Nash had an idea about “gravity, friction, and radiation,” as he later recalled. The friction he was thinking of was the friction that a particle, say a photon, might encounter as it moved through space due to its fluctuating gravitational field interacting with other gravitational fields.41 Nash had given his hunch enough thought to spend much of the meeting at the blackboard scribbling equations. Soon, Einstein and Kemeny were standing at the blackboard as well.42 The discussion lasted the better part of an hour. But in the end all that Einstein said, with a kindly smile, was “You had better study some more physics, young man.” Nash did not immediately take Einstein’s advice and he never wrote a paper on his idea. His youthful foray into physics would become a lifetime interest — though, like Einstein’s search for the unified field, it would not be especially fruitful.43 Many decades later, however, a German physicist published a similar idea.44

  Nash conspicuously avoided attaching himself to any particular faculty member, either in the department or at the institute. It was not a matter of shyness, his fellow students thought, but rather that he wished to preserve his independence. One mathematician who knew Nash at the time observed: “Nash was determined to keep his intellectual independence. He didn’t want to be unduly influenced. He’d talk freely with other students, but he was always worried about getting too close to other professors for fear that he’d be overwhelmed. He didn’t want to become dominated. He disliked the whole idea of being intellectually beholden.”45

  He did, however, use at least one faculty member, Steenrod, as a kind of sounding board. Temperamentally, Steenrod was an entirely different character from flamboyant, domineering types like Lefschetz and Bochner, whose lectures, it was said, were “exciting but 90 percent wrong.” Steenrod was a careful, methodical man who chose his suits and sports coats according to a mathematical formula and had a mania for thinking up highly logical, if impractical, solutions to social problems like crime.46 Steenrod also happened to be friendly, helpful, and patient. He was immensely impressed by Nash, found him more charming than not, and treated the young man’s brashness and eccentricity with amused tolerance.47

  Surrounded for the first time in his life by young men whom he regarded, if not exactly as his equals, at least as worth talking to, Nash preferred picking other students’ brains. “Some mathematicians work very much by themselves,” said one fellow student. “He liked to exchange ideas.”48 One of the students he sought out was John Milnor, the first of a number of brilliant younger mathematicians to whom Nash was drawn. Tall, lithe, with a baby face and the body of a gymnast, Milnor was only a freshman but he was already the department’s golden boy.49 In his freshman year, in a differential geometry course taught by Albert Tucker, he learned about an unproved conjecture of a Polish topologist, Karol Borsuk, concerning the total curvature of a knotted curve in space. The story goes that Milnor mistook the conjecture for a homework assignment.50 Whatever the case, he arrived at Tucker’s door a few days after with a written proof and the request: “Would you be good enough to point out the flaw in this attempt. I’m sure there is one, but can’t find it.” Tucker studied it, showed it to Fox and to Shiing-shen Chern. No one could find anything wrong. Tucker encouraged Milnor to submit the proof as a Note to the Annals of Mathematics. A few months later Milnor turned in an exquisitely crafted paper with a full theory of the curvature of knotted curves in which the proof of the Borsuk conjecture was a mere by-product. The paper, more substantial than most doctoral dissertations, was published in the Annals in 1950. Milnor also dazzled the department — and Nash — by winning the Putnam competition in his second semester at Princeton (in fact, he went on to win it two more times and was offered a Harvard scholarship).51

  Nash was choosy about whom he would talk mathematics with. Melvin Peisakoff, another student who would later overlap with Nash at the RAND Corporation, recalled: “You couldn’t engage him in a long conversation. He’d just walk off in the middle. Or he wouldn’t respond at all. I don’t remember Nash having a conversation that came to a nice soft landing. I also don’t remember him ever having a conversation about mathematics. Even the full professors would discuss problems they were working on with other people.”52

  On one occasion in the common room, however, Nash was sketching an idea when another graduate student got very interested in what he was saying and started to elaborate on the idea.53 Nash said, “Well, maybe I ought to write a Note for the Proceedings of the National Academy on this.” The other student said, “Well, Nash, be sure to give me a credit.” Nash’s reply was, “All right, I’ll put in a footnote that So and So was in the room when I had the idea.”

  Nash was respected but not well liked. He wasn’t invited to Kuhn’s room for sherry or out with the others when they went to Nassau Street to drink beer. “He wasn’t somebody you’d want as a close friend,” Calabi recalled. “I don’t know many people who felt any warmth for him.”54 Most of the graduate students were slightly odd ducks themselves, beset by shyness, awkwardness, strange mannerisms, and all kinds of physical and psychological tics, but they collectively felt that Nash was even odder. “Nash was out of the ordinary,” said a former graduate student from his time. “If he was in a room with twenty people, and they were talking, if you asked an observer who struck you as odd, it would have been Nash. It wasn’t anything he consciously did. It was his bearing. His aloofness.”55

  Another recalled, “Nash was totally spooky. He wouldn’t look at you. He’d take a lot of time answering a question. If he thought the question was foolish, he wouldn’t answer at all. He had no affect. It was a mixture of pride and something else. He was so isolated but there really was underneath it all a warmth and appreciation [for other people].”56

  When Nash did engage in one of his flights of garrulity, he often seemed to be simply thinking out loud. Hausner remembered, “A lot of us would discount a lot of what Nash said. A lot of the things he said were so far out, you didn’t want to engage him. ‘What was happening on earth when the Martians took over and there was a period of violence and why such and such.’ You wouldn’t know what he was talking about. Nash came out with things. They were unfinished and we weren’t ready to hear them. I wouldn’t want to listen. You didn’t feel comfortable with the person.”57

  His sense of humor was not only childish but odd. One former student recalled that Nash was personally responsible for getting the much-despised gown requirement at meals temporarily restored. “First,” recounted Felix Browder, who left Princeton in the fall of 1948, “he wrote a letter to Hugh Taylor, a pompous ass who was looking for an excuse, demanding that the custom be restored. After it was, nobody ate in the hall. It didn’t make John popular.”58

  He was also capable of frightening people when provoked. Occasionally, the teasing and needling would spill over into a sudden eruption of violence. On one occasion, Nash was baiting one of Artin’s students by telling him that the best way into Artin’s graces was to catch his beautiful daughter Karin.59 The student, Serge Lang, who everyone knew was painfully obsessed by his shyness around girls, threw a cup of hot tea in Nash’s face. Nash chased him around the table, threw him to the ground, and stuffed ice cubes down the back of his shirt. Another time, Nash picked up a metal ashtray stand — the kind that supports a heavy glass ashtray — and brought it down on Melvin Peisakoff’s shins, hard enough to cause considerable pain for a number of weeks.60

  In the spring of 1949, Nash ran into some trouble.61 He had acquired some strong supporters on the faculty, namely Steenrod, Lefschetz, and Tucker. Tucker was among those who believed that Nash was “very brilliant and original but rather eccentric,” arguing that “his creative ability… should make one tolerate his queerness.”62 But not everyone in the department felt that way. Some felt that Nash didn’t belong at Princeton at all. Among them was Artin.

  Slender, handsome, with ice-blue eyes and a spellbinding voice, Artin looked like a 1920s German matinee idol.63 He wore a black leather trench coat and sandals
throughout the academic year, wore his hair long, and smoked incessantly. The representative of “modern” algebra, Artin, who had been recommended by Weyl for the appointment at the institute that von Neumann eventually got, was a wonderful lecturer who admired polish and scholarship, but was famously intolerant of those who did not meet his rather fastidious standards. He was well known for screaming and throwing chalk at students who asked obtuse questions in his classes.

  Artin and Nash had clashed a number of times in the common room. Artin was always interested in talking with talented students. Yet he apparently found Nash not only irritatingly brash but also shockingly ignorant.64 At a faculty meeting in the spring, Artin commented that he could see no way for Nash to pass his generals, which the better students were expected to take at the end of their first year. When Lefschetz proposed an Atomic Energy Commission fellowship for Nash for the following year, Artin opposed it and made it clear he thought it would be better if Nash left Princeton.

  Lefschetz and Tucker overruled Artin on the subject of the fellowship.65 But they dissuaded Nash from sitting for the generals that spring and suggested that he take them in the fall instead. He was safe for the time being, but his unpopularity among some faculty members was to crop up again when he sought, two years later, to join the department as an assistant professor.

  6

  Games Princeton, Spring 1949

  JOHN VON NEUMANN, aka the Great Man behind his back, was threading his way through the crowd, nattily dressed as always and daintily holding a cup in one hand, a saucer in the other.1 The students’ common room was unusually crowded on this late afternoon in spring. A large audience, from the Institute and physics as well as math, had turned out for So and So’s lecture and was lingering over tea. Von Neumann hovered for a moment by two rather sloppily dressed graduate students who hunched over a peculiar-looking piece of cardboard. It was a rhombus covered with hexagons. It looked like a bathroom floor. The two young men were taking turns putting down black and white go stones and had very nearly covered the entire board.

  Von Neumann did not ask the students or anyone near him what game they were playing and when Tucker caught his eye momentarily, he averted his glance and quickly moved away. Later that evening, at a faculty dinner, however, he buttonholed Tucker and asked, with studied casualness, “Oh, by the way, what was it that they were playing?” “Nash,” answered Tucker, allowing the corners of his mouth to turn up ever so slightly, “Nash.”

  Games were one of the charming European customs that the émigrés brought with them to Fine Hall in the 1930s. Since then one game or another has always dominated the students’ common room. Today it’s backgammon, but in the late 1940s it was Kriegspiel, go, and, after it was invented by its namesake, “Nash” or “John.”2

  In Nash’s first year, there was a small clique of go players led by Ralph Fox, the genial topologist who had imported it after the war.3 Fox, who was a passionate Ping-Pong player, had achieved master status in go, not altogether surprising given his mathematical specialty. He was sufficiently expert to have been invited to Japan to play go and to have once invited a well-known Japanese master named Fukuda to play with him at Fine Hall. Fukuda, who also played against Einstein and won, obliterated Fox — to the delight of Nash and some of the other denizens of Fine.4

  Kriegspiel, however, was the favorite game. A cousin of chess, Kriegspiel was a century-long fad in Prussia. William Poundstone, the author of Prisoner’s Dilemma, reports that Kriegspiel was devised as an educational game for German military schools in the eighteenth century, originally played on a board consisting of a map of the French-Belgian frontier divided into a grid of thirty-six hundred squares.5 Von Neumann, growing up in Budapest, played a version of Kriegspiel with his brothers. They drew castles, highways, and coastlines on graph paper, then advanced and retreated armies according to a set of rules. Kriegspiel turned up in the United States after the Civil War, but Poundstone quotes an army officer complaining that the game “cannot readily and intelligently be pursued by anyone who is not a mathematician.” Poundstone compared it to learning a foreign language.6 The version of Kriegspiel that surfaced in the common room in the 1930s was played with three chessboards, of which one — the only one that accurately showed the moves of both players — was visible only to the umpire. The players sat back to back and were ignorant of each other’s moves. The umpire told them only whether the moves they made were legal or illegal and also when a piece was taken.

  A number of his fellow students remember thinking that Nash spent all of his time at Princeton in the common room playing board games.7 Nash, who had played chess in high school,8 played both go and Kriegspiel, the latter frequently with Steenrod or Tukey.9 He was by no means a brilliant player, but he was unusually aggressive.10 Games brought out Nash’s natural competitiveness and one-upmanship. He would stride into the common room, one former student recalls, where people were playing Kriegspiel, glance at the boards, and say offhandedly but loudly enough for the players to hear, “Oh, white really missed his opportunity when he didn’t take castle three moves ago.”11

  One time, a new graduate student was playing go. “He managed not just to overwhelm me but to destroy me by pretending to have made a mistake and letting me think I was catching him in an oversight,” Hartley Rogers recalled. “This is regarded by the Japanese as a very invidious way of cheating — hamate — poker-type bluffing. That was a lesson both in how much better he was and how much better an actor.”12

  That spring, Nash astounded everyone by inventing an extremely clever game that quickly took over the common room.13 Piet Hein, a Dane, had invented the game a few years before Nash, and it would be marketed by Parker Brothers in the mid-1950s as Hex. But Nash’s invention of the game appears to have been entirely independent.14

  One can imagine that von Neumann felt a twinge of envy on hearing Tucker tell him that the game he was watching had been dreamed up by a first-year graduate student from West Virginia. Many great mathematicians have amused themselves by thinking up games and puzzles, of course, but it is hard to think of a single one who has invented a game that other mathematicians find intellectually intriguing and esthetically appealing yet that nonmathematical people could enjoy playing.15 The inventors of games that people do play — whether chess, Kriegspiel, or go — are, of course, lost in the mists of time. Nash’s game was his first bona fide invention and the first hard evidence of genius.

  The game would likely not have appeared in a physical manifestation, in the Princeton common room or anywhere else, had it not been for another graduate student named David Gale. Gale, a New Yorker who had spent the war in the MIT Radiation Lab, was one of the first men Nash met at the Graduate College.16 Gale, Kuhn, and Tucker ran the weekly game theory seminar. Now a professor at Berkeley and the editor of a column on games and puzzles in The Mathematical Intelligencer, Gale is an aficionado of mathematical puzzles and games. Nash knew of Gale’s interest in such games since Gale was in the habit, during mealtimes at the Graduate College, of silently laying down a handful of coins in a pattern or drawing a grid and then abruptly challenging whoever was dining across the table to solve some puzzle. (This is exactly what Gale did when he saw Nash for the first time after a fifty-year hiatus at a small dinner in San Francisco to celebrate Nash’s Nobel.)17

  One morning in late winter 1949, Nash literally ran into the much shorter, wiry Gale on the quadrangle inside the Graduate College. “Gale! I have an example of a game with perfect information,” he blurted out. “There’s no luck, just pure strategy. I can prove that the first player always wins, but I have no idea what his strategy will be. If the first player loses at this game, it’s because he’s made a mistake, but nobody knows what the perfect strategy is.”18

  Nash’s description was somewhat elliptical, as most of his explanations were. He described the game not in terms of a rhombus with hexagonal tiles, but as a checkerboard. “Assume that two squares are adjacent if they are next to each other in
a horizontal or vertical row, but also on the positive diagonal,” he said.19 Then he described what the two players were trying to do.

  When Gale finally understood what Nash was trying to tell him, he was captivated. He immediately started to think about how to design an actual game board, something that had apparently never occurred to Nash, who had been toying with the idea of the game since his final year at Carnegie. “You could make it pretty, I thought.” Gale, who came from a well-to-do business family, was artistic and a bit of a tinkerer. He also thought, and said as much to Nash, that the game might have some commercial potential.

  “So I made a board,” said Gale. “People played it using go stones. I left it in Fine Hall. It was the mathematical idea that counted. What I did was just design. I acted as his agent.”

  “Nash” or “John” is a beautiful example of a zero-sum two-person game with perfect information in which one player always has a winning strategy.20 Chess and tic-tac-toe are also zero-sum two-person games with perfect information but they can end in draws. “Nash” is really a topological game. As Milnor describes it, an “n by n ” Nash board consists of a rhombus tiled with n hexagons on each side.21

  The ideal size is fourteen by fourteen. Two opposite edges of the board are colored black, the other two white. The players use black and white go stones. They take turns placing stones on the hexagons, and once played the pieces are never moved. The black player tries to construct a connected chain of black stones from the black to black boundary. The white player tries to do the same with white stones from the white to white boundary. The game continues until one or the other player succeeds. The game is entertaining because it is challenging and appealing because it involves no complex set of rules as does chess.