Thinking in Numbers: How Maths Illuminates Our Lives

by Daniel Tammet

  I think back to the menu. For each course there are just a few options. A good opportunity to put my imaginary mother to the test. But will the model and the mother agree? Recalling the trios of starters, mains, and desserts, I assign to each a probability. I assess not only each of the individual courses but also the potential combinations across them. For example, two-thirds of the mains have meat: for this reason I downgrade the odds of my mother selecting pâté as a starter (unless, that is, she were to opt for a second course of fish). Assuming she begins with the good intentions of a salad, I plump for the caramel cake as her choice of dessert.

  My imaginary mother decides to steer a middle route between the pâté and the salad. When the waiter looms, his voice announces the roast vegetable soup. I inhale the satisfying sweetness as it passes my chair.

  Now the beef rises even higher up my mental shortlist. But when the table has been cleared, and the waiter next comes, I find my stare returned by the glassy eye of a baked cod. Bits of the fluffy flesh scatter around her plate and down her top as she eats.

  Finally, we arrive at dessert. There is no room for doubt in my mind. My model mother’s fondness for chocolate has been corroborated many times before. But not today. My actual mother finishes her meal with a bowl of exotic fruit.

  Outside the restaurant, she takes my arm in hers. She wants to show me the street on which she grew up. She grips my arm tight as we take the short walk side by side. Of her life before me I know next to nothing, only one or two titbits, gathered here and there. I ask whether it is true that before starting a family, she worked as a secretary. ‘Leave off,’ she laughs. ‘I typed addresses onto envelopes.’ She still has postcodes on the brain.

  ‘Name a town,’ she says suddenly.

  Bethlehem, I think. ‘St Ives,’ I say.

  ‘St Ives,’ she repeats and it sounds twice as long when she says it. ‘TR26.’

  We turn onto the street, a stone’s throw from the Palace of Westminster. My grandfather worked for the local brewery here, delivering the beer by horse and cart. The block of flats was home to several of the workers’ families. A single washroom, with enamel bath, did for everyone. We cannot see inside, however. The building has become a base for the homeless; some of the windows are boarded up. Before we leave, I pull out my camera and take a photo.

  I think of the girl who would grow up to become my mother. Who was the future woman that she imagined for herself? Did she dream of having a loving partner, a big house and children who always smiled at her? In her mind’s eye, would she be well-educated and well-travelled, always generous, patient, and kind? Did she imagine that every cherished moment would be remembered forever; every grievance instantly forgotten?

  And thinking of this girl I feel at once immensely happy and immensely sad. I feel as I feel when I think of myself.

  Talking Chess

  To win at chess is simple: victory belongs to the player who makes the last-but-one mistake.

  Whoever it was that first came up with this line spoke much truth. The strongest players operate neither like machines nor angels; their superiority lies in accomplishing a better class of error.

  A winning mistake would presumably owe nothing to sloppiness, incuriosity or a yellow belly. It would be far closer to the lucky slip of an artist’s brush or writer’s pen, one that suddenly infuses a picture or page with unforeseen possibility. I am thinking of the (perhaps apocryphal) story of the painter who, hot-tempered at his many failed attempts to perfect a detail of his portrait, tossed his sponge with disgust at the easel and thereby achieved just the desired effect. Or that of the book printer who inadvertently won Herman Melville lavish accolades for the phrase ‘soiled fish’ (in place of the intended ‘coiled fish’ in reference to an eel).

  I will not risk stretching this argument any further. Creativity is obviously so much more than an unexpected gesture here and there. But talent in chess bears this similarity with other creative pursuits, in tolerating error, as if the grandmaster – like the great artist – is the one who truly explores possibility’s outer limits. Or, as a character in Joseph Conrad’s Lord Jim intones, ‘To the destructive element submit yourself and with the exertions of your hands and feet in the water make the deep, deep sea keep you up.’

  Chess is a perfect arena for just such an exerted exploration of the possible. Its chequered sea is very deep indeed. The mathematics behind the game’s complexity are staggering. An initial move by each player will create one of four hundred legal positions. A second move by each: seventy-two thousand. There are some nine million possible set-ups after the players’ third move; 288 billion after their fourth. Back in 1950, the mathematician Claude Shannon calculated the possible number of forty-move games, a figure henceforth known as the ‘Shannon Number’. He estimated as about thirty the potential number of viable moves at every turn. In this way he arrived at a total number (1 followed by 120 zeroes) that easily exceeds the number of atoms in the observable universe.

  For all its immensity, chess is a finite game. It is therefore at least conceivable that a machine might one day be programmed with the knowledge, deep down in its nodes, of every possible sequence of moves for every possible game. No combination, however ingenious, would ever surprise it; every board position would be as familiar as a face. Like checkers, which was solved by computer scientists in Canada in 2007, we would finally discover how chess, played perfectly by both sides, ends.

  This perfect game of chess – the immaculate order and configuration of its moves, the exquisite ballet of its pieces in their precisely timed roles – is imprinted on the imagination of every player. In his innermost being, every player carries some notion of the divine game. For one, it begins with the two-square forward march of the white king’s pawn, to which the L-shaped spring of black’s queenside knight replies. Six moves later, the white queen arrives on a4 – a side square – only to be promptly rebuffed by black’s bishop. No, no, says a second player: the game begins with a white knight – either one – to which black responds in kind. The central pawns advance in pairs. But another player disagrees, citing the fact that a piece would need to be sacrificed by white after eleven moves – the exchange of his queen for a rook. For still others the white pawns creep up the far sides of the board like ivy, or the black king crouches incessantly behind his queen, or all four bishops dance the diagonals till precisely half of all the original pieces remain.

  Who, then, prevails in this Platonic ideal of chess? Each player confesses a secret faith. A triumph for white in forty-three moves, or in forty-one if black’s sixth move disturbs a pawn. Or else a win for black, after a marathon two hundred and twenty-seven moves, the surviving white piece finally confiscated by the king. But such are the visions of romantics; a large majority seem resigned to the probability of a draw.

  A small number of enthusiasts (a handful of masters, too) claim to have adjudicated the question in favour of one or the other side, outlining a ‘system’ for the player to follow. Understandably these systems have been subjected to many critiques. In his books and articles, Weaver Adams, who won the U.S. Open Championship in 1948, argued that with a first-move king’s pawn advance ‘white ought to win’. In his own matches, however, Adams seemed to have perversely better luck with black. A certain Dr Hans Berliner agrees with Adams concerning white’s predestined triumph, but differs in his choice of the clinching initial move. According to Berliner, white must employ the queen’s pawn instead.

  Within our lifetimes a definitive solution, using the fastest computers, might indeed emerge. But there is still a very, very long way to go. So far algorithms have resolved every lawful chess position containing at most six pieces (including the two kings). The compiled data has yielded more than a few surprises. Numerous endgame positions long considered mere draws, we know now beyond all shadow of doubt, can in fact produce a winner. Among the most recent analysis, which has moved on to the study of endgames of seven pieces, researchers have hit upon an astonishing for
ced win for white – provided that his concentration could somehow hold out over five hundred and seventeen flawless moves!

  In the end maybe no solution of chess exists, or at least none that we could ever extract in the time allotted to our universe. Any complete solution might even reveal itself to be simply beyond the pale of our imagination: knights that forage for pawns for no apparent reason, bishops that take turns occupying consecutive squares, or rooks that slide up and down, and left and right, ninety-nine times in a row.

  But of course chess would not be chess without its mystery, or its players’ mistakes. Men, like chessmen, are made from crooked timber. With his mistakes, the beginner (what players call a ‘patzer’) immediately blows his cover. He brings his queen out too quickly, exchanges too many pieces too soon, moves his pawns in such a way that their formation finishes by resembling Swiss cheese. But the problem is more one of quantity than quality. The patzer loses not because he makes too many mistakes, but because he makes too few – a mere handful of classic blunders. He does not last long enough to make more! Traps abound, and he duly falls into one or another. Cannier, moderate-strength players (such as club champions) make more mistakes than beginners. Sidestepping the early pitfalls, they grant themselves far greater scope in which to stumble.

  Stronger play demands more than the avoidance of blunders. The player has to learn to make his own mistakes, which is far more difficult than it sounds. He has to stop mimicking the moves he reads in books and magazine columns, and which he does not really understand: even the best moves can turn bad when played on the wrong square or at the wrong instant. He has to root out his most cherished errors, the kind that he plays with the frequency of a tic. In short, he has to clear his head and think and feel and suffer for himself. Only then can he hope to obtain the slightest grip upon the game.

  All of this amounts to nothing less than that nebulous attribute we call personality. It is the indescribable quality that seems to bring the pieces on the board to life. As with the brushstrokes of a gifted painter, we can identify a strong player from the moves, including the mistakes, which he makes. The observer traces in the movements of the pieces the movement of the player’s thought. What we call his mistakes are also the expressions of a deep and personal understanding of such-and-such position, which is – like every human understanding – imperfect. Out of this personal understanding comes both his biggest howlers, and his finest moves.

  No master of the game, I would suggest, had more personality than the Soviet champion, ‘the Magician of Riga’, Mikhail Tal. Not a few of his games have gained the status of the masterpiece. At its best, Tal’s play betrayed a courage that bordered on insouciance. He was always in the thick of the action, inviting complications. Of this proclivity for problems, he once remarked, ‘You must take your opponent into a deep dark forest where 2 + 2 = 5, and the path leading out is only wide enough for one.’

  In the depths of this forest even he sometimes lost his way. During one simultaneous exhibition against a score of Americans, the grandmaster took the fight to a plucky and talented twelve-year-old. At a crucial moment Tal surrendered his queen in exchange for the initiative, but the sacrifice eventually proved unsound and he went on to lose. With a shrug of his shoulders, the former World Champion gamely shook the boy’s hand, before continuing to ply his trade among the other boards.

  Tal played instinctively. Faced with the game’s intractable complexity, he always followed his nose. He would feel his way around the squares of the board, for feeling is also a kind of thinking. In his autobiography, there is a small but marvellous anecdote of this intuitive approach. Tal describes a contest with the grandmaster Vasiukov at the Championship of the USSR. By dint of adventurous play, the two men had reached a highly entangled position. Tal says he hesitated a long time over his next move. His path to victory, he sensed, began with the sacrifice of his knight, but the immense number of possible variations unnerved him. Head in his hands, he meditated on one after another without result. Mental chaos ensued. All of a sudden, from out of nowhere, an amusing couplet by the children’s poet Chukovsky broke surface in his mind. ‘Oh, what a difficult job it was. To drag out of the marsh the hippopotamus.’

  Tal was at a loss to know by what process of connection his mind had suggested the poem’s lines. But now the thought gripped him: how exactly would a man pull such an animal from a marsh? As the spectators and journalists looked on, the grandmaster contemplated any number of hippopotamus-rescuing methods: jacks, levers, helicopters, and ‘even a rope ladder’. Here, as well, his calculations came to no avail. ‘Well, just let it drown,’ he said to himself at last, in a fit of ill temper. Tal’s head cleared immediately and he resolved to trust his instincts and play. The next morning the newspaper report read, ‘Mikhail Tal, after carefully thinking over the position for forty minutes, made an accurately-calculated piece sacrifice.’

  Before we leave Tal, one word about his chess education. The young Mikhail’s development proved vertiginous. He first learned to play at the age of eight, watching the games of patients at a hospital where his father worked. The boy did not shine, far from it. His youthful style was simply one more against which the older players gained their points. Only from the age of twelve did he apply himself seriously. A local master took him under his wing. Within two years the teen had qualified for his national championship; a year later he finished ahead of his trainer. The next year, aged sixteen, he won his country’s chess crown and the title of master.

  Such rapid acquisition is reminiscent of the facility with which we learn our mother tongue. Only four years separated the beginner Tal from his first national championship; only four years – as a rule – separate the baby from fluent speech. In both cases, an adult’s guiding hand makes all the difference. Left to his own devices, neither the baby nor the beginner can hope to make much progress. Linguists say that a child learns his language by exposure to highly structured input; his parents address him more slowly, more questioningly, in short sentences that verge on curtness. The chess player, in a similar fashion, learns best from a coach’s counsel; he is shown many patterns and combinations of moves that characterise expert play.

  Wittgenstein observed that language, like chess, is a game governed by rules. Knowing how to use a word, he said, is like knowing how to move a chess piece. From a small number of these initial rules, immense complexity is spawned. Gossip on a street corner can rival in complexity any game of chess. This is because the number of concatenations of words that form meaningful sentences approaches infinity. Speakers (and writers) are always coming up with original sentences, like chess masters with their novel moves. And like any player worthy of winning, the speaker (and again the writer, to some extent) anticipates the other’s response. He modifies his speech to accord with his anticipation. Not only does he know what he can say, he knows what he should say, and – perhaps even more interestingly – what he should not. A skilled conversationalist has this knack for knowing which avenues to explore and which to avoid. Similarly, certain chess moves, in certain situations, while perfectly legal, are considered taboo. I have heard of a grandmaster who once described his opponent’s early capture of a pawn with his bishop as being ‘low culture’. The move boasted the advantage of quick material gain, but at the expense of the formation and co-ordination of his pieces. Such moves are rarely propitious to a good game.

  I am reminded of a scene from the 1951 novel The Master of Go by Japanese author Yasunari Kawabata. The story’s narrator, a passionate amateur of the game (Go is an ancient and highly strategic board game), plays on his pocket magnet set during a train ride home. Opposite him counters a tall American tourist, on whose knees the gold leaf board rests for the duration of the journey. The Japanese narrator prevails quickly, game after game. ‘It was as if I were throwing a large but badly balanced opponent in a wrestling match.’ Throughout, he notices a sort of thoughtlessness, a lack of personal investment, in the American’s play. For the tourist, he
imagines, playing Go was ‘like having an argument in a foreign language learned from grammar texts.’

  Taking this thought to its natural conclusion, Kawabata goes so far as to declare the game’s subtleties as being inaccessible to foreigners. What he means by this, I think, is that a good game (whether of Go or chess), like a good conversation, requires a certain sensibility born of complete immersion. I am thinking of that attention to form which elevates a phrase or move beyond the merely functional. Even in translation, Kawabata’s own elliptic writing style can often puzzle the non-Japanese reader. Many details, which to his countrymen appear straightforward, can pass entirely over our heads. Why, for instance, does the narrator in the novel mention that his Go board is decorated with gold leaf? Might it be a reference to illumination? Or victory? Should we read in it the suggestion of Go as an art? We can only hazard a guess as to its meaning.

  Grandmasters, of course, are immersed in their game. Some drown, sunk by an insanity brought on by virtually infinite complexity. Most, however, find their own fullest and richest means of expression in the combinations of moves they deploy. These players do not so much think about chess, as think in chess, just as we think in language. I read an account of one master who recalls each day’s events as though they were moves on a chessboard. He remembers going for an afternoon swim as, say, King’s knight to F6, while a restaurant meal with his wife comes back to him as a descent four squares of the Queen’s rook. These associations appear to him as unremarkable as they are spontaneous.

  We can also see this spontaneity, the spontaneity of speech, in what we call ‘blitz’ or ‘rapid’ chess. In its most popular form, both players have just one minute in which to dispense all their moves. Pieces slide frantically from square to square; agile hands swat over and over the button on their clock. At a rate of one move per second or thereabouts, whole games, upward of forty moves, are frequently accomplished. In spite of having no time to think, the players in these games often attain surprisingly high standards.