GENIUS AT PLAY THE CURIOUS MIND OF JOHN HORTON CONWAY SIOBHAN ROBERTS By all outward appearances Conway was blissfully playing his way through the 1960s. Inwardly he worried that his own mathematical soul was withering away. He was doing nothing, had done nothing. He felt he didn t deserve his job. While he was piddling around, friends of his were graduating and finding no work. He felt trapped giving lectures on Foundations of Logic, which wasn t quite his thing. He also gave a lecture course on automata. As his students would find out when they continued their studies elsewhere say, at MIT Professor Conway had worked out his own original proofs and unorthodox terminology, because he wasn t as familiar as he should have been with the published literature. Typical Conway. It was easier to reinvent the wheel. He improvised as he went, hardly ever planning lectures much ahead of time (or in a haze about what he d planned; at least once he arrived for a morning lecture wearing his dinner suit from a feast in college the night before). Most lecturers would set down a sequential process: theorem 1, lemma 2, theorem 3, lemma 4. Conway would saunter in and say, Here s this theorem I ve been trying to prove let s see what we can get. This is a more honest way to lecture, showing the false starts and stuckedness that are crucial to the mathematical process. However, it s also a lot more work, for the student and the lecturer both. A student named Andrew Glass, now a Cambridge math professor, once encountered Conway in the photocopy room executing the administrative task of stapling together his sandals. Conway glanced at his student s open notebook and recognized material from his automata course. Were my lectures anywhere near that coherent?
Coherent or not, Conway s lectures were popular. An announcement appeared in the Cambridge Reporter heralding the establishment of the John Conway Appreciation Society. It specific that it did have at least one patron, and it asked: Guess who? The smitten students loved him as much for his mind as his silly high jinks, and maybe most of all for his singular hybrid of sophistication, sincerity, and lascivious showmanship. He wore sandals year-round, yes, but often he was barefoot at the blackboard, having kicked off his sandals and flung them into the radiator or the far corners of the room (hence the perpetual mending). He had a homely lecturing style, discussing abstract concepts in terms of ordinary objects such as train cars (and cats and dogs). Sometimes he brought a large turnip and a carving knife to class, and in illustrating his lesson he would transform the tuber a slice at a time from a sphere to a cube to an icosahedron, devouring the scraps as he went. A student named Edward Welbourne, now a software engineer in Oslo, had heard about Conway even before he took the Foundations of Logic course. He recalls Conway teaching mathematician Kurt Gödel s Incompleteness Theorem, the first humility theorem in mathematics. Proved in 1931, Gödel s theorem stated that in mathematics there will always be undecidable truths, statements that cannot be proved or disproved and at that, Gödel s theorem would seem to be a good theme theorem for this book and the quagmiric enterprise of remembering and telling tales that exist along a sliding spectrum of verifiability, a mirrored hall of truthiness. In a course on linear algebra, Welbourne recalls the class when Conway proved that for 2 symmetric quadratic forms, both can be simultaneously diagonalized no small feat. Doing each takes a moderately tricky piece of computation, says Welbourne. To do 2 at the same time is thus doubly tricky, like balancing a broom by its handle on one s chin while juggling, which is exactly what Conway did whilst concluding the proof. Conway quibbles that really he balanced a broom on his chin whilst simultaneously balancing a penny on the hook of a coat hanger and then, with an assured centrifugal swoop, he spun this coat hanger contraption around like a helicopter rotor.
All this appreciation for Conway couldn t have suffered, either, from his habit of starting a course by providing the entire curriculum on a single sheet of paper, full of arrows leading from one subject to the next; nor from his habit of finishing courses a class early and using that last session to speed through all 24 lectures, reprising all the proofs. These final lectures became popular, the audience swelling 30 percent beyond its usual size, attracting students from years past who stopped in for a refresher. Then there was the November that Conway finished his course 2 lectures early. He did the traditional 24-classes-in-1 on the Wednesday, which left the Friday class empty. University regulations dictated that as the professor he was obliged to show up, but he told his students that they were not so obliged, and he rather hoped they d take the hint. As he approached the lecture room that Friday, all was promisingly silent none of the raucous buzz and chatter of 200 students waiting for proceedings to commence. Thankfully his loyal followers had obeyed. He walked through the door, and thereupon the class leapt out at him screaming Surprise! The girls were in fancy party dresses, and the boys were on their worst behavior. Amidst the chaos, Welbourne inflicted a problem on his good professor, just the sort of jeu d esprit he knew Conway wouldn t be able to resist. I doubt he would be offended at the suggestion that he s somewhat scatterbrained. Observing him, it always seemed like there were ideas popping into his head all the time, though he was perfectly capable of concentrating on one idea at a time for long periods, says Welbourne, who was right in thinking his problem would appeal to Conway. It had been doing the rounds, from Belgrade to Denmark to England and beyond, and courtesy of Conway it continues. I m going to give it to you the way it was given to me. Tell me when you figure out the pattern:
Now, what comes next? The history of the puzzle dated to a recent International Mathematical Olympiad in Belgrade. The Dutch contingent sicced the puzzle on the British team, and that s how it was imported to Cambridge, according to an account in the Cambridge math journal Eureka. When I first showed this puzzle to one of my friends, the Eureka reporter said, he thought for some time and then gave an agonized cry, I ve solved it but you need a really twisted mind to think of that! I showed it to several arts students, who were all baffled, which is surprising as it requires no mathematical skills beyond counting. From my mathematical friends I got the same response as the initial one; silence and furious thinking for between two and thirty minutes followed by anguished howling. If hideous noises were heard echoing down the corridors of Newnham it was a good bet I d asked that puzzle again. Conway, however, let out none of these sounds. I could not guess it. And I could tell from the way Eddy said it that I was supposed to be able to guess it. In the end he had to tell me the answer. Once in the know, Conway was helplessly consumed with what he called The Look-and-Say Sequence. The name gives a clue for solving it. Look at the first number, 1. Say how many of that number there are one 1 and write that observation down on the next line, numerically as 11. Then look at that number and say the description: 21. And so forth. Immediately following that last lecture before Christmas, Conway flew off to Boston for a conference and he spent the entire flight fiddling with Look-and-Say sequences, trying out different starting strings of numbers. 55555 55 25
1215 11121115 He decided to deconstruct things, inserting commas around the phrases of verbal description this he called parsing. And as he explained in an article he wrote for the same issue of Eureka: The numbers in our strings are usually single- digit ones, so we ll call them digits and usually cram them together as we have just done. But occasionally we want to indicate the way the numbers in the string were obtained, and we can do this neatly by inserting commas recalling the commas and quotes in our verbal descriptions, thus: 55555,55,,25,,12,15,,11,12,11,15, He titled the article The Weird and Wonderful Chemistry of Audioactive Decay, because by the end of a flight s worth of fiddling, he had augmented the verbal metaphors with chemistry metaphors, devising the Chemical Theorem, which proved that all the numerical sequences generated by this puzzle ultimately settle into exactly 92 shorter sequences, or common atoms, as he called them. He aligned the 92 common atoms with the then 92 chemical elements of the periodic table: 3 he aligned with uranium, which is atomic number 92; 13 he aligned with protactinium, which is atomic number 91; 1113 he aligned with thorium, atomic
number 90; and so on down to hydrogen, atomic number 1, aligned with the Lookand-Say common atom 22. For instance, take the initial sequence 1 11 21 1211 111221 312211 13112221 All of those things so far are atoms but not common atoms. The next line you get is: 1113213211, or 11132.13211. When you split it like that in the middle you see it s a compound of 2 common atoms it s hafnium stannide. Hafnium, actually a real element in the world, is, according to my table, 11132. And then 12311 is tin, but in the international table the chemical symbol of tin is Sn, which comes from the Latin root stannum. So 1, after not that many moves, becomes the compound hafnium stannide! This is an example of the second theorem I proved, the Cosmological Theorem, which asserts that after a certain number of moves from the Big Bang the beginning of the sequence all the exotic elements, all the things that are not compounds of common atoms, for instance 1, disappear and everything is made of common atoms. What happens next is if you follow the left- hand portion, hafnium, and you follow the right- hand portion, stannum, they never interfere with each other. The sequence splits as a compound, and if you continue on with the Look- and- Say procedure, the 2 sides never interfere with each other.
11132.13211 311312.11131221 1321131112.3113112211 How, pray tell, did Conway notice such a thing during the ennui of a transatlantic crossing? It doesn t very much matter. I m pretty clever. The point is I did notice it. And I can prove it. Conway proved the Cosmological Theorem over Christmas, and thereafter the Look-and-Say Sequence was renamed by Eileen as The Problem That Spoiled Christmas Conway was always spoiling Christmas. He lost that first proof, which he d done with his Cambridge friend and collaborator Richard Parker, a mathematician cum computer designer, but he wasn t too upset because that proof was too long, anyway. Mike Guy did a second proof, also eventually lost. And on it went. All the while subsidiary discoveries kept coming. Conway noticed that the sequence grows in length by an approximately constant 30 percent per generation, the 20th and 21st generations being: 111312211312111322212321121113121112131112132112311321322 112111312211312112213211231132132211231131122211311123113 322112111312211312111322111213122112311311123112112322211 213211321322113311213212312311211131122211213211331121321 123123211231131122211211131221131211131231121123221112132
11322211312113211 311311222113111231133211121312211231131112311211133112111 312211213211312111322211231131122211311122122111312211213 211312111322211213211321322113311213212322211231131122211 311123113223112111311222112132113311213211221121332211211 131221131211132221232112111312111213111213211231132132211 211131221232112111312211213111213122112132113213221123113 112221131112311311121321122112132231121113122113322113111 221131221 The latter 408 digits are about 1.3 times longer than the former 302 digits. That ratio is called Conway s Constant: If you take 2 consecutive numbers, their ratio converges on or about 1.303577269.... But the number never quite settles down much like Conway himself. And case in point, to this Conway insists on adding: The really astonishing thing is that it s a root of an algebraic equation of degree 71!