Confessions of a Conway Groupie

To the memory of John Conway, 1937–2020

“So let me get this straight, Mr. Propp: you plan to go to England to work with a mathematician who doesn’t even know you exist?”

It was 1982, I was a college senior applying for a fellowship that I hoped would send me to Cambridge University for a year, and the interviewer was voicing justified incredulity at my half-baked plan to collaborate with John Conway.

I’d read about Conway and his multifarious mathematical creations in Martin Gardner’s Mathematical Games column in Scientific American, and I’d become an ardent fan; I’d devoured his book “On Numbers and Games” and I’d even done some epigonic¹ work on my own, trying to extend the theory of two-player games to allow for a third player. But I hadn’t even taken the step of writing to the man, and I had to sheepishly admit as much to the interviewer.

“You sound a bit like Luke Skywalker heading off to meet Yoda,” the interviewer said. His jest made me worry that I wouldn’t get the fellowship, but he must have believed in me more than his joke suggested. I was awarded a Knox Fellowship, and later that year I went to England on a Knox, as I liked to say (enjoying the resulting homophonic confusion).

“DR. CONWAY WANTS TO TALK TO YOU”

Conway was a celebrity among mathematicians but hadn’t risen to the top academic rank at Cambridge University. Perhaps that was partly due to his refusal to draw a line between the serious and the playful the way most mathematicians do. After he’d discovered three eminently respectable algebraic structures called the Conway groups², he’d resolved that from then on he would devote himself to whatever interested him regardless of what other people thought. This resolution showed itself clearly in his subsequent output. Conway’s most profound and distinctive contribution to mathematics, his theory of surreal numbers³, was shot through with inspirations coming from the study of games, and the achievement he was best known for in the broader world was his invention of a kind of computer-aided solitaire called Conway’s Game of Life. And, unlike most mathematicians, Conway didn’t confine his research to one particular area; his breadth of interests would have smacked of dilettantism if he hadn’t made fundamental discoveries in the topics he turned his attention to. It was hard for more traditional academics to know what to make of him. When I arrived at Cambridge University, Conway was a Lecturer, not a Professor.

Later on, I realized I’d been lucky that Conway wasn’t off taking a sabbatical somewhere (perhaps in the U.S.) during the year I’d left the U.S. to work with him in England!

I found Conway in the Trinity College Common Room one day. He was (as he would remain throughout his life) happy to have a conversation with a stranger. I introduced myself and told him about the work I’d been doing on three-player games. I was looking at what are called impartial games, of which the prototypical example is Bouton’s game of Nim. In Nim, the “board” consists of one or more heaps of counters, and a legal move for a player is to take away as many counters as the player wants from a single pile. The player who makes the last move wins. What makes the game “impartial” is that every move that is available to one player is available to the other. There’s a beautiful mathematical theory of how to win impartial two-player games, and I told Conway that I wanted to extend it to three players who take turns in cyclical fashion.

If we’re watching a three-player impartial game in progress, and we freeze the action, there are four possibilities. First, the player who’s about to move (call her Natalie) could have such a strong position that, if she keeps her wits about her, she can guarantee that she’ll make the last move and win the game, regardless of what her two adversaries do. Second, the player who gets to move after Natalie (call him Oliver) might have a strategy that lets him win, no matter what. Third, the player who gets to move after Oliver (call him Percival) might have a winning strategy against the other two. Lastly, it’s possible that no player has a winning strategy — that any two players have the ability to defeat the third (leaving aside the issue of how coalitions might form or dissolve, or how agreements to share a prize might be enforced). I called these four cases N, O, P, and Q. Much of my preparatory work on the problem of classifying positions in three-player games could be summarized in the following table, whose simplicity hides the amount of work required to justify it:

For instance, the upper left entry signifies that if we have two positions, each of which is of type P, and we smoosh them together, then the resulting position is either of type P or of type Q.

I showed Conway the table and described what I was hoping to do next. What I didn’t tell him — what I didn’t have the guts to tell him — is that I was hoping to do the work with him. I mean, who was I, not even a proper graduate student, to suggest that it would be worth Conway’s time to collaborate with me? John expressed approval of my research plan, wished me luck, and encouraged me to let him know how things went. And that, I assumed, was that.

A few days later, one of my flatmates told me that while I’d been off attending classes there’d been a telephone call for me from some secretary at DPMMS (the Department of Pure Mathematics and Mathematical Statistics). When I called her back, she said “Dr. Conway wants to talk to you,” but she wouldn’t tell me more, except to say that he had seemed upset. She arranged a place and time for Conway and me to meet, and when I met him, he began by apologizing.

“This is your project, not mine,” he said, “but I couldn’t stop thinking about it, and I even did some work on it. I’m sorry. I don’t usually do things like this.”

I reassured him that I’d hoped all along that we’d work on three-player games together. He was relieved, because he said he didn’t think it was right for someone to poach someone else’s research project (especially that of a younger person).

As it turned out, guilt wasn’t the only thing bothering John; he was also frustrated mathematically. “I was able to prove all the claims in your addition table,” he said, “but I couldn’t figure out how to prove that the sum of two type-O positions can’t be another type-O position.”

“Oh yes, that’s the hard one,” I said, and I showed him how the proof went.⁴ And so my collaboration with John Conway was launched.

I hasten to say that the proof I showed John was totally in the style that I’d learned from his book, and that it didn’t use any tricks he didn’t already know. It just took more work. Consider that I’d had months to work on the theory; John had only learned about it from me a few days before. I had no doubt then (and have no doubt now) that if John had set aside a few hours for the task, as I had done earlier that year, he would’ve found the proof. But I won’t deny that it was a huge boost to my ego to know that I’d proved something about games that had stumped him the first time he tried to prove it.

WITH CLIPBOARD AND BABY

Usually we would meet at a local coffee shop called Fitzbillies. John was the scribe, filling page after page with calculations. One time, shortly after the birth of his son Oliver, we worked at his home, where I got to meet his then-wife, the mathematician Larissa (“Lara”) Queen. We entered his home through the back door, set in a featureless wall that faced a parking lot; it reminded me a bit of Bag End from “The Hobbit”. Sometimes he’d bring little Oliver to the cafe with us, and John would somehow balance the clipboard and the baby and do math while keeping his young son happy.

I wish that I’d saved some of those pieces of paper, or that I even remembered some of what was on them. The phrase “tribal markings” has stayed with me, as the name John gave a system for discriminating between positions based on how they behaved when you added k Nim heaps of size 1 to them, for k = 1, 2, 3, … . Ultimately, what sank the enterprise (or at least my enthusiasm for it) was that John’s extension of my theory didn’t seem to apply to any actual games in an interesting way. In nearly all positions in nearly all three-player impartial games, any two players can gang up on the third if they make a plan and stick to it. Years later, it occurred to me that John and I should have taken a break from delving into 3-player impartial games and taken a look at 4-player impartial games. Even though in most such games any three players can gang up on the fourth, one can look at how one two-player alliance fares against another, and we might have found something worth publishing. In the end, many years later, I did publish an article on three-player games, but it didn’t include any of the work I’d done with John in Cambridge.

I also attended a class John taught on Games, Groups, Lattices, and Loops, and while I didn’t warm to most of the topics he covered (perhaps because I didn’t put in any time playing with them on my own), I was struck by the way his ideas about games turned out to play a role in his work on lattices, codes, and packings, with connections that became even clearer in the decade that followed (see his article with Sloane listed in the References). What are the chances that a mathematician who loved games would have the luck to find that games secretly underlie other subjects he studies? It almost seemed as if mathematical reality was bending itself to his will — that he had “root access” to the Platonic realm of pure form.

Of course, I don’t really believe that. The likeliest explanation is that what attracted John to work in these areas was some subtle affinity between them — an affinity that reflected some hidden mathematical substructure they had in common. Which problems seize a mathematician’s fancy, and which ones leave a mathematician’s soul unstirred? These things are as mysterious as physical attraction, but just as some people have a physical type they’re attracted to, I think John had a “type” in the mathematical domain, so that even though his interests were broad, there is something “Conway-ish” about the problems he tackled, and he had a keen sense of smell when it came to scenting out Conway-ish problems.

PRINCETON AND ELSEWHERE

Not many years after I came to visit him in England, John moved to the U.S. and became a professor at Princeton. He traveled a lot, giving talks at conferences across the country, participating in research retreats, and even spending time at mathematical summer camps for high schoolers and middle schoolers. I went to many of Conway’s talks, some of them rather outrageous (I remember the time when, lacking a damp paper towel, he licked an overhead slide clean so that he could re-use it), but the thing that I found most striking is that he never gave the same talk twice. For him, giving a talk was an improvisatory performance, an extension of his love of one-on-one conversation.

I remember a time when I was hoping to snag Conway’s interest in a problem that struck me as Conway-ish, hoping to recreate the sort of collaboration we’d had in 1982. I drew a triangle with some lines cutting through it, something like this,

and then I asked Conway if he could add more lines so that all the small regions cut out by the lines were triangles. He thought a bit, and drew some lines:

I nodded, and then said “Do you think that if I draw any finite number of lines passing through a triangle, there’s always a way for you to to add more lines so that all the small regions cut out by the lines are triangles?” He said “Let me think about that,” and as the curiosity-bug bit into his brain, he began to draw pictures, make observations, and formulate conjectures. But he was no fool; he could see that I was deliberately trying to entice him into working on the problem, and he was too proud to want to be seen as one who is so easily seduced. He shuddered as if shaking off an unpleasant memory and said “You know, I don’t have to work on just ANY damned problem!”

The problem is still unsolved, as far as I know. (For more info, see the Math Forum webpage listed in the References.)

ATLANTA

During the past twenty years, most of my conversations with Conway took place in Atlanta, at a meeting of math-y, magic-y people held every two years called the Gathering for Gardner. (Actually the “for” is officially supposed to be rendered as the number “4”, but I find the cutesiness a bit too much.) I never had a chance to talk to Gardner himself at one of these Gatherings; he stopped attending in the late 90s, partly because he wanted to be with his wife who didn’t like travel, but mostly because he hated adulation.⁵

John, on the other hand, never minded being the center of attention, and rarely missed a Gathering. He loved to talk, and people loved to listen. He, however, didn’t always like to listen (especially as he grew older), and he seldom went to the formal talks held in the ballroom, preferring to linger in the anteroom and converse with whoever was interested in talking to him. This became a problem for me, because I didn’t want John to talk to just anyone — I wanted him to talk to me: about frieze patterns, boundary invariants for tilings, sphere packings, surcomplex numbers, group theory, knot theory, etc. John, however, was just as happy to perform magic tricks for strangers as he was to discuss our shared mathematical interests. Or perhaps he sometimes found my company dull, and was happy for the relief provided by other interlocutors eager to chat with him on other topics?

One of the last times I saw Conway was at a Gathering for Gardner in Atlanta in 2014. This Gathering was held specifically in John’s honor, and I gave a talk there on his indirect contribution to the theory of random tilings. Characteristically, he wasn’t at the talk; he preferred one-on-one conversation to attending presentations. If you were there that year, you might at one point have spied me sitting in the anteroom on the floor at Conway’s feet, and you might have thought it looked odd. Why had I adopted this undignified position? Because I had Conway’s ear, I needed to sit, there was no other chair, and I feared that if I left to get a chair, someone else would snag his attention and I’d never be able to finish the conversation.

It’s only recently occurred to me that, to the extent that Conway in later life became a less considerate person, the attention of fans like myself may have played a facilitating role. One reason people behave as well as they do is that bad behavior comes at a social price. If you’re an ordinary person, spending most of your time in a particular place, hanging out with a limited supply of people, and you’re rude to enough people for a long enough time, you’ll eventually run out of people who seek your company. But when you’re a star the way Conway was, there’s always another eager fan to shower you with attention, no matter how many people you’ve alienated. John was never unkind to me (the rudest thing he ever said to me, after I made some intelligent comment, was “You’re not as dumb as you look” and I think he meant it affectionately), but I’ve heard from a few others (women, I’m sorry to say) to whom he was not so polite, or to whom he displayed a creepy kind of attentiveness. I’ve written elsewhere about geniuses, but it now seems to me that a deeper problem has to do with how communities choose heroes, and how communities treat those heroes. I feel torn between two uncomfortably clashing beliefs: that heroes are necessary or at least inevitable, and that hero-worship damages the souls of the worshipper and the worshipped. Maybe some readers of this essay have thought more deeply about this than I have and will have useful insights.

THE END

Conway succumbed to COVID-19 in April 2020. I’m glad that while he was still alive I let him know how big a role he played in my life (something I neglected to do in the case of Martin Gardner). I think it’s fair to say that he was the Beatles of mathematics, not just because he was from Liverpool, but because so much of John’s work is so damned catchy. Just as many Lennon-McCartney songs have a memorable “hook”, many of John’s best creations have a way of sticking in the mind once you understand them, in a way that most mathematical discoveries don’t. A layperson with an interest in mathematics can get a surface appreciation of John’s work in a way that just isn’t possible for 99 percent of contemporary mathematical research. Here’s one example of an especially accessible Conway theorem (an isolated aperçu as opposed to a piece of a bigger story): If you extend the sides of triangle ABC as shown, to points A_b, A_c, B_a, B_c, C_a, and C_b (so that the various distances marked in the figure are equal), then those six new points lie on a circle.

The Conway Circle. Taken from https://mathworld.wolfram.com/ConwayCircle.html ; permission pending.

It’s hard to believe that as simple a geometric proposition as Conway’s circle theorem could have lain undiscovered for more than a score of centuries, but it did. (For a beautiful proof-without-words of this proposition, see the proof by Colin Beveridge listed in the references.)

It’s easy for songwriters to feel that all the best tunes, chord progressions, and hooks have already been used by the songwriters who came before. Likewise, if you’re a pure mathematician whose job is to create new games of pure thought, it’s easy to feel that all the beautiful simple ideas have already been thought of — that our forebears have already turned the mathematical topsoil, leaving us the more arduous task of cutting through rock in search of undiscovered gems. The main thing I learned from John is that even if the supply of beautiful yet simple mathematical truths is in some sense finite, we’re nowhere near the bottom of it. Conway’s career is an existence proof that a career like his is possible, or at least was possible up through the year 2020. I hope and believe that at least throughout my lifetime there’ll still be plenty of scope for mathematicians of his temperament to find new thought-games that somehow manage to be compellingly simple yet enduringly deep.

Thanks to Tibor Beke, Nancy Blachman, Sandi Gubin, David Jacobi, Joe Malkevich, Evan Romer, and Shecky Riemann.

Next month: The Mathematics of Irony.

REFERENCES

BAAM! (Bay Area Artists and Mathematicians) and G4G (Gathering 4 Gardner), Remembering John Conway, https://youtu.be/Ru9fX3VPR9Y

Matt Baker, “Some mathematical gems from John Conway”, https://mattbaker.blog/2020/04/15/some-mathematical-gems-from-john-conway/ .

Colin Beveridge, Conway’s circle, a proof without words, https://aperiodical.com/2020/05/the-big-lock-down-math-off-match-14/ .

John Conway, On Numbers and Games.

John Conway and Neil Sloane, “Lexicographic Codes: Error-Correcting Codes from Game Theory”, IEEE Transactions on Information Theory, Vol. IT-32, No. 3, May 1986; available at http://neilsloane.com/doc/Me122.pdf .

Donald Knuth, Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness, 1974.

MathOverflow, Conway’s lesser-known results, https://mathoverflow.net/questions/357197/conways-lesser-known-results .

James Propp, “Three-player impartial games”, Theoretical Computer Science 233 (2000), pp. 263–278; available from https://arxiv.org/abs/math/9903153.

Jim Propp, “Conway’s impact on the theory of random tilings”, talk presented at G4G11 in 2014; vodeo at https://www.youtube.com/watch?v=e_729Ehb4vQ .

Siobhan Roberts, Genius at Play: The Curious Mind of John Horton Conway.

Siobhan Roberts, “Travels with John Conway, in 258 Septillion Dimensions,” New York Times, May 16, 2020; at https://www.nytimes.com/2020/05/16/science/john-conway-math.html .

Stan Wagon, Math Forum Problem-of-the-Week 812, “A Pre-Sliced Triangle”; at http://mathforum.org/wagon/spring96/p812.html .

ENDNOTES

#1. The term “epigone” is usually an insult; who wants to be called “second-rate”? But if the tier extends beyond second-rate to third-rate, fourth-rate, etc., being second-rate isn’t so bad! And it’s no disgrace to be deemed not-as-good-as-Conway.

#2. The Conway groups are examples of algebraic structures called finite simple groups. There are several infinite families of finite simple groups and then twenty-six “bonus” finite simple groups that we call sporadic, including the three Conway found. The largest of the sporadic finite simple groups is called the Monster, and near the end of his life Conway confided that, although it was his fondest wish to understand why the Monster existed, he doubted that he would live that long.

#3. The surreal number system is an extension of the ordinary real number system that includes infinite and infinitesimal quantities as well as familiar numbers like the seventeen and the square root of two. The term “surreal numbers” was coined by Donald Knuth, whose book on the subject occupied me for many happy hours when I was in high school.

#4. In my article “Three-player impartial games”, the proposition that gave Conway trouble appears as Claim 7, and it hinges on five of the six preceding claims.

#5. I would sometimes bring students to attend the Gathering with me. One student’s father had initial misgivings about having his daughter attend some sort of strange convocation held in honor of a man who didn’t even show up. I’m guessing it reminded him of those ashrams in the U.S. that are dedicated to the teachings of a guru back in India. To him, Gardner-fandom seemed like a bit of a cult. And I can’t say he’s completely wrong.