A Beautiful Mind (16 page)

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.
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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.
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Nash, who had played chess in high school,
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played both go and Kriegspiel, the latter frequently with Steenrod or Tukey.
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He was by no means a brilliant player, but he was unusually aggressive.
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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.

Nash proved that, on a symmetrical board, the first player can always win. His proof is extremely deft, “marvelously nonconstructive” in the words of Milnor, who plays it very well.
²²
If the board is covered by black and white pieces, there’s always a chain that connects black to black or white to white, but never both. As Gale put it, “You can walk from Mexico to Canada or swim from California to New York, but you can’t do both.”
²³
That explains why there can never be a draw as in tic-tac-toe. But as opposed to tic-tac-toe, even if both players try to lose, one will win, like it or not.

The game quickly swept the common room.
²⁴
It brought Nash many admirers, including the young John Milnor, who was beguiled by its ingenuity and beauty Gale tried to sell the game. He said, “I even went to New York and showed it to several manufacturers. John and I had some agreement that I’d get a share if it sold. But they all said no, a thinking game would never sell. It was a marvelous game though. I then sent it off to Parker Brothers, but I never got a response.”
²⁵
Gale is the one who suggested the name Hex in his letter to Parker Brothers, which Parker used for the Dane’s game. (Kuhn remembers Nash describing the game to him, very likely over a meal at the college, in terms of points with six arrows emanating from each point, proof, in Kuhn’s mind, that his invention was independent of Hein’s.)
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Kuhn made a board for his children, who played it with great delight and saw to it that their children learned it too.
²⁷
Milnor still has a board that he made for his children.
²⁸
His poignant essay on Nash’s mathematical contributions for the
Mathematical Intelligencer,
written after Nash’s Nobel Prize, begins with a loving and detailed description of the game.

7
John von Neumann
Princeton, 1948–49

J
OHN VON
N
EUMANN
was the very brightest star in Princeton’s mathematical firmament and the apostle of the new mathematical era. At forty-five, he was universally considered the most cosmopolitan, multifaceted, and
intelligent
mathematician the twentieth century had produced.
¹
No one was more responsible for the newly found importance of mathematics in America’s intellectual elite. Less of a celebrity than Oppenheimer, not as remote as Einstein, as one biographer put it, von Neumann was the role model for Nash’s generation.
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He held a dozen consultancies, but his presence in Princeton was much felt.
³
“We were all drawn by von Neumann,” Harold Kuhn recalled.
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Nash was to come under his spell.
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Possibly the last true polymath, von Neumann made a brilliant career — half a dozen brilliant careers — by plunging fearlessly and frequently into any area where highly abstract mathematical thought could provide fresh insights. His ideas ranged from the first rigorous proof of the ergodic theorem to ways of controlling the weather, from the implosion device for the A-bomb to the theory of games, from a new algebra [of rings of operators] for studying quantum physics to the notion of outfitting computers with stored programs.
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A giant among pure mathematicians by the time he was thirty years old, he had become in turn physicist, economist, weapons expert, and computer visionary. Of his 150 published papers, 60 are in pure mathematics, 20 in physics, and 60 in applied mathematics, including statistics and game theory.
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When he died in 1957 of cancer at fifty-three, he was developing a theory of the structure of the human brain.
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Unlike the austere and otherworldly G. H. Hardy, the Cambridge number theorist idolized by the previous generation of American mathematicians, von Neumann was worldly and engaged. Hardy abhorred politics, considered applied mathematics repellent, and saw pure mathematics as an esthetic pursuit best practiced for its own sake, like poetry or music.
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Von Neumann saw no contradiction between the purest mathematics and the grittiest engineering problems or between the role of the detached thinker and the political activist.

He was one of the first of those academic consultants who were always on a train or plane bound for New York, Washington, or Los Angeles, and whose names frequently appeared in the news. He gave up teaching when he went to the Institute
in 1933 and gave up full-time research in 1955 to become a powerful member of the Atomic Energy Commission.
¹⁰
He was one of the people who told Americans how to think about the bomb and the Russians, as well as how to think about the peaceful uses of atomic energy.
¹¹
An alleged model for Dr. Strangelove in the 1963 Stanley Kubrick film,
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he was a passionate Cold Warrior, advocating a first strike against Russia
¹³
and defending nuclear testing.
¹⁴
Twice married and wealthy, he loved expensive clothes, hard liquor, fast cars, and dirty jokes.
¹⁵
He was a worka-holic, blunt and even cold at times.
¹⁶
Ultimately he was hard to know; the standing joke around Princeton was that von Neumann was really an extraterrestrial who had learned how to imitate a human perfectly.
¹⁷
In public, though, von Neumann exuded Hungarian charm and wit. The parties he gave in his brick mansion on Princeton’s fashionable Library Place were “frequent and famous and long,” according to Paul Halmos, a mathematician who knew von Neumann.
¹⁸
His rapid-fire repartee in any of four languages was packed with references to history, politics, and the stock market.
¹⁹