Thinking in Numbers: How Maths Illuminates Our Lives (24 page)

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 forced 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.

Not that I wish to imply that playing chess is all instinct. Alacrity has its virtues (the absence of indecision for one), but they are not the virtues of the longer form. At its best, chess – like language – privileges reflection and careful thought. Looked at one way, a game of chess amounts to a long series of shifting problems, the most intriguing among them capable of making unique demands upon our imagination. As with a beautiful passage in a novel, we feel that we could happily spend almost any amount of time in its company.

Occasionally an amateur player will scour the back pages of certain newspapers (the kind whose pages end up wrapping glass vases, but never greasy chips) to savour and solve these chess problems. Illustrations of the board, its pieces frozen in print, compete for space with the week’s crossword puzzle. ‘White to play and mate in three’ or ‘Black to play and draw’ its title will announce. Quite often the position displayed is at least half cleared of its starting pieces, the game has attained its final moments. The amateur stares at the inky pieces and the smudged squares, waiting for a surge of inspiration. It is more or less the same experience as when we read some striking lines of a poem.

Many of these published chess positions have never known an actual board; they are the figment of some inventor’s imagination. Among these inventors figures one ‘Vladimir Sirin’, better known as the multilingual novelist and poet Vladimir Nabokov, for whom such compositions equated to the ‘poetry of chess’ (his 1969 anthology,
Poems and Problems
, was illustrated by eighteen of his own efforts).

Those sixty-four squares, their peculiar geometry, fascinated the great wordsmith to the point of obsession. The right-angled triangle, for instance, traced by a king’s vertical (or horizontal), and then diagonal, retreat across the board disobeys Pythagoras’s famous theorem. Off the board, in the ‘real world’, moving say three paces across and as many paces up (or down) would produce a greater diagonal distance (four and a quarter paces long) between the start and end points. The king’s ‘triangle’, on the other hand, has identical length (traversing the exact same number of squares) on every side. This optical illusion (expecting a diagonal route to require a greater number of moves than either a vertical or horizontal path) plays an essential role in the chess problem composer’s craft.