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       READY, STEADY, GO

       We construct and construct, but intuition is still a good thing.

      Paul Klee

      People often compare mathematics to playing chess. There certainly are connections, but when Deep Blue beat the best chessmaster the human race could offer in 1997, it did not lead to the closure of mathematics departments. Although chess is a good analogy for the formal quality of constructing a proof, there is another game that mathematicians have regarded as much closer to the creative and intuitive side of being a mathematician, and that is the Chinese game of Go.

      I first discovered Go when I visited the mathematics department at Cambridge as an undergraduate to explore whether to do my PhD with the amazing group that had helped complete the classification of finite simple groups, a sort of Periodic Table of Symmetry. As I sat talking to John Conway and Simon Norton, two of the architects of this great project, about the future of mathematics, I kept being distracted by students at the next table furiously slamming black and white stones onto a large 19×19 grid carved into a wooden board.

      Eventually I asked Conway what they were doing. ‘That’s Go. It’s the oldest game that is still being played to this day.’ In contrast to the war-like quality of chess, he explained, Go was a game of territory. Players take it in turn to place white and black pieces or stones onto the 19×19 grid. If you manage to surround a collection of your opponent’s stones with your own, you capture your opponent’s stones. The winner is the player who has captured the most stones by the end of the game. It sounded rather simple. The subtlety of the game, Conway explained, is that as you try to surround your opponent, you must avoid having your own stones captured.

      ‘It’s a bit like mathematics: simple rules that give rise to beautiful complexity.’ It was while watching the game evolve between two experts as they drank coffee in the common room that Conway discovered that the endgame was behaving like a new sort of number that he christened ‘surreal numbers’.

      I’ve always been fascinated by games. Whenever I travel abroad I like to learn and bring back the game locals like to play. So when I got back from the wild outreaches of Cambridge to the safety of my home in Oxford I decided to buy Go from the local toy shop to see what it was that was obsessing these students. As I began to explore the game with one of my fellow students in Oxford, I realised how subtle it was. It was hard to identify a clear strategy that would help me win. And as more stones were laid down on the board, the game seemed to get more complicated, unlike chess, where as pieces are gradually removed the game starts to simplify.

      The American Go Association estimates that it would take a number with 300 digits to count the number of games of Go that are legally possible. In chess the computer scientist Claude Shannon estimated that a number with 120 digits (now called the Shannon number) would suffice. These are not small numbers in either case, but they give you a sense of the wide range of possible permutations.

      I had played a lot of chess as a kid. I enjoyed working through the logical consequences of a proposed move. It appealed to the mathematician that was growing inside me. The tree of possibilities in chess branches in a controlled manner, making it manageable for a computer and even a human to analyse the implications of going down different branches. In contrast Go just doesn’t seem like a game that would allow you to work out the logical implications of a future move. Navigating the tree of possibilities quickly becomes impossible. That’s not to say that a Go player doesn’t follow through the logical consequences of their next move, but this seems to be combined with a more intuitive feel for the pattern of play.

      The human brain is acutely attuned to finding structure and pattern if there is one in a visual image. A Go player can look at the lie of the stones and tap into the brain’s ability to pick out these patterns and exploit them in planning the next move. Computers have traditionally always struggled with vision. It is one of the big hurdles that engineers have wrestled with for decades.

      The human brain’s highly developed sense of visual structure has been honed over millions of years and has been key to our survival. Any animal’s ability to survive depends in part on its ability to pick out structure in the visual mess that Nature confronts us with. A pattern in the chaos of the jungle is likely to be evidence of the presence of another animal – and you’d better take notice cos that animal might eat you (or maybe you could eat it). The human code is extremely good at reading patterns, interpreting how they might develop, and responding appropriately. It is one of our key assets, and it plays into our appreciation for the patterns in music and art.

      It turns out that pattern recognition is precisely what I do as a mathematician when I venture into the unexplored reaches of the mathematical jungle. I can’t rely on a simple step-by-step logical analysis of the local environment. That won’t get me very far. It has to be combined with an intuitive feel for what might be out there. That intuition is built up by time spent exploring the known space. But it is often hard to articulate logically why you might believe that there is interesting territory out there to explore. A conjecture in mathematics is by its nature not yet proved, but the mathematician who has made the conjecture has built up a feeling that the mathematical statement they have made may have some truth to it. Observation and intuition go hand in hand as we navigate the thickets and seek to carve out a new path.

      A mathematician who can make a good conjecture will often garner more respect than one who joins up the logical dots to reveal the truth of the conjecture. In the game of Go the final winning position is in some respects the conjecture and the plays are the logical moves on your way to proving that conjecture. But it is devilishly hard to spot the patterns along the way.

      And so, although chess has been useful to help explain some aspects of mathematics, the game of Go has always been held up as far closer in spirit to the way mathematicians actually

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