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like his own idiosyncratic invention, but if it turns out to be sound then the invention becomes a discovery of an effective plan. The rules of an abstract game may be very simple, yet their implications may be unfathomable. Our human creativity includes the ability to illuminate the obscure – but also to create the obscure.
    The same turns out to be true of mathematics though with a subtle twist. If a mathematician believes he has discovered some new theorem, but it turns out to be untrue, then naturally he thinks of the erroneous proof as something he invented, which didn't work, like a mechanical device that was built to perform a certain function but failed to do so. But if it turns out to be true, then there is a strong feeling that it has actually just been discovered. This is how Richard Hamming puts it, referring to his own mathematical research:
     
When I try to examine my own beliefs…I find that if the result seems to be important then I found it, but if it seems to be rather trivial, then I created it!
    [Hamming 1998 : 649]
     
Go
     
    Go is an extremely ancient oriental game much played in Japan, China and Korea. The best players in Japan, equivalent to our chessgrandmasters, are well-paid professionals who earn as much as top Western golfers. Their games are followed by devoted fans and their services sought by eager businessmen for whom a high amateur grade is a status symbol.
    The standard board which is 19 lines by 19 lines (smaller boards, 13 by 13 or 9 by 9, are used when teaching beginners) starts empty, and players place a single stone, either black or white, alternately, on an intersection of two lines, with the object of surrounding territory or their opponent's stones.
    In Figure 2.11 , the players are sketching out positions in the corners of the board: in Go, the edges are used as a defence, and in the opening the corners where you are protected by two edges are crucial.
    Go has a far larger board than chess but much simpler ‘moves’. The result is another rich game that lies near the boundary of whatever the human mind can master. It might seem that Go is a ‘mathematically’ simple game because its rules are so simple, but no, the board is so large and there are so many possible moves at each turn that Go players rely even more than chess players on strategical judgement and intuition, as well as tactical calculation.
    The Japanese Rules of Go
    This is a simple version. The official rules complete with explanatory notes are at: www.cs.cmu.edu/∼wjh/go/rules/Japanese.html on which this epitome is based.
    Two players compete on a board, to see which can take more territory.
    The board is a grid of 19 horizontal and 19 vertical lines forming 361 intersections. A stone can be played on any unoccupied intersection.
    The board is initially empty.
    The players can alternately play one move at a time, one player playing the black stones, his opponent the white stones.
    After a move is completed, a group of stones belonging to one player exists as long as it has a horizontally or vertically adjacent empty point, called a ‘ liberty ’.
    If, after a player's move, a group of his opponent's stones has no liberties, the player removes all these stones, as prisoners .
    A shape in which the players can alternately capture and recapture one opposing stone is called a ‘ ko ’. A player whose stone has been captured in a ko cannot recapture in that ko on the next move.
    A group of stones which cannot be captured by the opponent is alive. A group which can be captured is dead.
    Empty points surrounded by the live stones of just one player are called ‘ eye points ’. Other empty points are called ‘ dame ’. Stones which are alive but only possess dame are said to be in ‘ seki ’.
    Eye points surrounded by stones that are alive but not in seki are called ‘ territory ’, each eye point counting as one point of territory.
    When a player passes his move and his opponent passes in succession,