John Horton Conway

~.:;:::,~: ""od,"oo of Am,d" TaIkingaG ood Game

Quick now-what day of the week did pert, whom I shall leave nameless, in­ December 4, 1602 fall on? Sorry, time's stalled a secret switch under his desk. If up. You have to give the answer (Satur­ one of his bosses entered the room he day) in less than two seconds to compete would press the button and switch his with Professor John Conway ofPrinceton computer screen from its 'Life' program University. Conway enjoys mentally cal­ to one of the company's projects." culating days of the week so much that Conway says that "Life" arose out of he has programmed his computer so he "the aim to find a system in which you cannot log on until he does ten randomly­ can see what happens in the future .. .. I selected dates in a row. He usually does always thought you ought to be able to ten dates in about 20 seconds. His best design a system that was deterministic, time is 15.92 seconds. Conway says "the and his invention of a theory of numbers but unpredictable." ability to do these lightning mental cal­ that has its origins in games. Conway's Although he has co-authored with culations is very important to me. You've enchantment with games is reflected in Berlekamp and Guy Winning Ways for no idea how fast you have to think to do the title of one of his papers, "All Games Your Mathematical Plays, the two-volume them. The reason I do it is because it gives Bright and Beautiful." In Conway's classic on games, he asserts that he is not me a kick. The adrenaline spills all over theory of numbers, every two-person very interested in playing actual games. you, and when you're thinking that game is a number! Don Knuth, the noted He claims that "I can't play chess, I know quickly, it's really nice." computer scientist, was so taken with the rules, but you would be amazed at how Conway, at age 56, is one ofthe world's Conway's new theory of numbers that he badly I play. That's not the thing that turns most original mathematicians and is a wrote Surreal Numbers, a novel that ex­ me on. I' m interested in the theory of it, member of the prestigious Royal Society plains the theory for students. especially if it's simple and elegant. I re­ of London. He is in the middle of a sec­ ally like to consider the simpler games, ond career as professor of mathematics "Life" like checkers. I used to play checkers with at Princeton University with his second "Life," Conway's most famous game cre­ my first wife, and she always used to beat family. It was a great coup for Princeton ation to date, burst on the scene in 1970 me. Perhaps I would win one game in ten mathematicians when they lured Conway when Martin Gardner brought it to the at­ or twenty, and I was trying very hard ... . I away from Cambridge University in 1986. tention of hundreds ofthousands ofread­ had a similar experience with my daugh­ We are visiting with him today (Novem­ ers ofhis "Mathematical Games" column ter, playing the game called Reversi ." ber 29, 1993) to gain a few insights into ofScientific American magazine. "Life's" Conway's mathematical abilities, es­ his work and what makes him tick. popularity was quick and far-reaching. Its pecially his rapid calculating skills, were Conway has made substantial contri­ great popularity spawned "Lifeline," a evident as a little boy. He says that "my butions to several branches of mathemat­ newsletter for "Life" enthusiasts, which mother found me reciting 2, 4, 8, 16, 32, ics: set theory, number theory, finite was published for many years. 64, .. . -the powers of2 when I was four." groups, quadratic forms, game theory, and Gardner later wrote that his "column When he was eleven, he told the head­ combinatorics. He is best known, in a on Conway's 'Life' fOims was estimated master of his grammar school in popular sense, for his work on the theory to have cost the nation millions ofdollars Liverpool that "I want to go to Cambridge of games, especially the Game of "Life" in illicit computer time. One computer ex­ and study mathematics."

Reprinted from Spring 1994, pp. 6-9 1 2 The Edge of the UniverAe: Celebrating 10 YearA of Math Horizons

me up anyway." He also remembers be­ During our visit, Conway tells us about ing taken into an ancient air raid shelter his new system for clarifying the myster­ and having lighted cigarettes applied to ies of knot theory. He brings in two un­ his skin.- Ouch! He eventually got to dergraduate students to join him and the Cambridge, but Conway says that "from interviewer in a special "square dance" ages 11 - 13, with the onset ofadolescence, using two colored ropes that do indeed puberty, and all of that, I didn't do terri­ serve to explicate what he calls the bly well. I started to hang around with a "theory of tangles." He recalls working bunch oflay-abouts. I had a hell ofa time out a good part of his theory of tangles when I was a high school student. while still a high school student. "Teachers and my parents were getting At one point in our discussion, he concerned about me," he remembers. "I brings out a few dozen tennis balls to il­ was given several good talkings to by lustrate a problem in sphere packing. various people, and by age 16, I started Packing the maximum number ofspheres going to classes again and started being in a given space, especially higher dimen­ on top again." sional spaces, has been one of Conway's passions for several years. The sphere Conway illustrates sphere packing. The Real Me packing problem in eight-dimensional Conway indeed got on top again to the space is very important to transmitting point that he won a scholarship to Cam­ data over telephone lines. He tells us that What was it about mathematics that bridge. He clearly remembers his train trip twenty-four dimensional space is wonder­ attracted Conway so strongly? "I can't to Cambridge, and being rather intro­ ful for "there is really a lot of room up recall what started it," he says. "It's prob­ verted, quiet, and shy at the time. "I was there among those packed spheres." ably just the fact that I was good at it, and on the train when I said to myself, 'You His interest in sphere packing led to his that was that. If you regard it as a com­ don't have to be like this anymore. No­ writing, with Neil Sloane, the book en­ petitive subject, then to stand out and beat body at Cambridge knows you.' I had titled Sphere Packings, Lattices, and the other kids was fun. stepped out ofthe world I was previously Groups. He is quite proud of a recent re­ "When I was a teenager, I thought a in . So I decided to tum myself into an view of the book which describes it as lot about the different departments of extrovert, and I did. I decided I was go­ "the best survey of the best work in the knowledge, in some sense, and I know ing to laugh with people, and make fun best fields ofcombinatorics written by the what turned me on to math was this feel­ ofmyself. I got there, and that's what hap­ best people. It will make the best reading ing of objectivity. Consider other things pened. For quite a long time, I felt like a by the best students interested in the best you might do, like law. Then you're bas­ fraud. I said to myself, 'This isn't the real mathematics that is now going on." He is ing your life on essentially arbitrary de­ me.' And then it ceased to be acting. Ev­ so proud of the review that he has dis­ cisions that have been taken by individu­ ery now and then, I still feel shy on occa­ played the "best" parts of it in large let­ als, or by the way society has developed sions, but not very often." ters on one ofhis walls. Ofthe many nice as a whole. I can't develop much interest Anyone who watches Conway bounce reviews he has had of his work, he says in that. .. . I like the idea that with philo­ around a classroom or organize a knot this one is the "best." sophical, mathematical, and scientific theory square dance would agree that the Tennis balls (sphere-packing), colored questions, there's a chance of communi­ introvert is long gone. ropes (knot theory), and counters on a cating with beings on other planets, so to Conway's Princeton office is an envi­ checker board (the game of "Life") all speak. There's a certain universality that ronment that clearly would appeal to chil­ reflect Conway's intense need to make definitely is central to mathematics. dren ofall ages from two to one hundred. things simple. He claims that "lots of "When I was young, things were quite Pleasantly cluttered with books, bric-a­ people are happy when they've under­ difficult. It was quite a rough district we brac, and mathematical models hanging stood something. And I'm usually not. lived in, and some terrible things hap­ from the ceiling and walls, it bears a strik­ I'm only happy when I've really made it pened." Conway remembers being beaten ing resemblance to a classroom in a pro­ simple. Moreover, I don't understand so up by older boys "because I hadn't cho­ gressive elementary school. He even has many deep things as a lot of other people sen the right professional soccer team. a home-built "quaternion machine" hang­ do. But I'm interested in getting a still Sometimes it didn't matter whether I ing on one wall, which he gleefully oper­ deeper understanding of some simpler chose the right one or not, they would beat ates for visitors. things." JOHN HORTON CONWAY - TALKING A GOOD GAME 3

The Splash Conway became a mathematical star in 1969 when he discovered a new simple group, now called the Conway Group. At the time he was a junior faculty member at Cambridge University who was de­ pressed because "I had been known as a quite-bright student at Cambridge, but I had never produced anything of any sig­ nificance. I was feeling guilty and was almost suicidal. In addition I was married with four little girls, and we had very little money. "About then John Leech discovered a beautifully symmetric structure in 24-di­ John Conway in his Princeton office. mensional space. It was believed that a special group corresponded to Leech's about that childish game, whereas previ­ you're not really entitled to make any structure. I decided to take a crack at find­ ously I would have sort of looked around judgment.. . . The argument that says we ing it since I knew a bit about groups. So and wondered what my colleagues were haven't found a proof for over 300 years I set up a schedule with my wife. We thinking. " is not so fascinating. We're not all that agreed that I would work on the problem Last June, Andrew Wiles, one of clever. Many of the other things he wrote on Wednesday nights from six until mid­ Conway's Princeton colleagues, made an down lasted for 200 years .... When I die, night and on Saturdays from noon until even bigger splash with his announcement I might knock on Fermat's door and see midnight. I started work on a Saturday, ofa proof of Fermat' s Last Theorem. (See what happened. He'd be an interesting and made progress right away. At a hal f the premiere issue of Math Horizons for guy to talk to. I've often toyed with the hour past midnight on that first Saturday, the story on Wiles.) Conway worked in idea of talking to people from the past." I came out with the problem solved! I had number theory for several years and has Archimedes and Euler are two other found the group! been fascinated with the work of Wiles mathematicians with whom Conway "That did it. I immediately got offers and the ensuing excitement. He says "I would like to chat. He adds that " I to lecture on my new group all over the have slightly different opinions about the wouldn't like to talk with either Newton world. I became something of a math­ Fermat problem from most people. What or Gauss, because neither of them seems ematical jet-setter. My discovery really I think is that it's quite likely that Fermat to be my kind of person." was a big splash." proved it, not just that he believed that Conway also has a secret to pass on he'd proved it. ... There was no reason for about producing mathematics. He advises So Much for Guilt him to deceive anybody. There's not much us to "keep several things on the board, Conway went on to explain another plus point in writing something, writing a note or at least on the back burner, at all that followed his discovery. "I suddenly to yoursel f, and telling a lie in it. It times .... One ofthem is something where realized that it was a good idea not to feel wouldn't work. you can probably make progress .. .. If you guilty; feeling guilty didn't do any good. "And now you're faced with a real work only on the really deep interesting Guilt just made it impossible to work ... . problem. Even if Fermat deceived him­ problems, then you're not likely to make So I decided for myself that from then on self, what was his proof? What was that much progress. So it's a good idea to have I wasn't going to work on something just proof? Until you can solve that problem some less deep, less significant things, because I felt guilty. If I was interested by exhibiting something that was avail­ that nevertheless are not so shallow as to in some childish game, I would think able to Fermat and would fool Fermat, be insulting." • Photographs by Carol Boxtel: 4 The Edge of the Univenre: Celebrating 10 YeaN! of Math Horizons

II Here is a little example. The five circles in diagram G I indicate the live The Game of Life" cells in the first generation. Those marked i will die in the next generation due to isolation (Rule 1). The cells with the black dots are empty cells that will become live in the next generation (Rule 2) . The rules of Conway's Game of "Life" has 4 or more live neighbors, then it is Diagram G indicates what the second were chosen, after experimenting with crowded out and dies. 2 generation looks like. The cell marked c many possibilities, to make the behavior 2. On the other hand, if a dead (unoccu­ will die ofcrowding (Rule 1). To help you of the population both interesting and un­ pied) cell has exactly 3 live neighbors, draw the pattern for the third generation predictable. The genetic laws are remark­ then it is a birth cell; a counter is placed we have used the marks i and· as before. ably simple. The game is played on a on it in the next generation. You should continue to draw the pictures chessboard, an infinite chessboard, where for several successive generations. each cell has eight neighboring cells. The game begins with some arrangement of counters placed on the board (the live cells) as the initial generation. Each new generation is determined by the follow­ ing rules: 1. Consider a live cell. If it has 0 or I live neighbors, then it dies from isola­ tion. If it has 2 or 3 live neighbors, then • it survives to the next generation. If it

If you play the Game of "Life" with various initial generations you will find that the population undergoes unusual and unexpected changes. Patterns with no ini­ tial symmetry become symmetrical. Some initial configurations die out entirely (al­ though this may take a long time), others become stable (still lifes), while others oscillate forever. Conway originally conjectured that no finite initial pattern can grow without limit. But he was wrong. There is a "gun" that shoots out "gliders" and a "train" that moves along but leaves a trail of"smoke." If you draw several more generations ofthe above example you will understand that it is called "glider" because it is a glide-reflection of an earlier generation (which one?). To learn more about the Game of "Life," including the glider gun, see Mar­ tin Gardner's Wheels, Life and Other Mathematical Amusements (1983), which contains three chapters on "Life." You may want to write a program to play John 'Horned' (Horton) Conway "Life" on your computer. •