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rules of their playground games and families often play their own pet versions of MONOPOLY or CLUEDO, to the surprise of visitors invited to join in.
    The chosen rules have to be consistent with each other and with the goals of the game. For example, the game ought to come to an end. It is easy to invent a game which never ends. Indeed, Max Euwe, one-time world chess champion (1935–1937) and a mathematics teacher, proved that a chess game of infinite length was possible, given the rules of his time: the rule that a game is a draw if the same sequence of moves occurs three times in succession was not sufficient to prevent the possibility.
    The Nihon Kiin , the governing body of Japanese professional Go players, has more than once in the last century, several thousand years after the claimed origins of the game, made changes to the rules to allow for anomalous situations such as the triple ko (creating an endless repetition). Moreover, no one has proved that there are no more possible anomalous situations waiting to be discovered. The ancient rules of Go could be changed yet again in the twenty-first century.
Hidden structures forced by the rules
     
    The complexity of an abstract game is created the moment the rules are laid down, instantly creating a rich miniature world, but their implications then have to be inferred which takes time and involves all three crucial aspects of mathematics: visual and mental perception, scientific exploration, and game-like calculation.
    Exploration leads – as it does in natural history and geography – to important structures and features being identified, named and classified, so that the gamedevelops its own language. These structures make abstract games playable and mathematics manageable.
    Their existence also means that there are limitations on what you can do over the chess board. Every chess player is familiar with the scene in which a player gazing at a lost position exclaims plaintively, ‘There must be a move!’ but there isn't, the position is irretrievably kaput (with the qualifications that even master players do occasionally resign in salvable positions).
    It is this dialectic between the limitations forced by the rules and the imaginative creativity of the player that makes the best abstract games so extraordinarily rich and fascinating and attracts millions of players to games such as chess and Go.
Argument and proof
     
    Proof can be a confusing idea. Scientists talk of proving this or that – that the universe started with a Big Bang, that the speed of light is constant, or that it is changing – but they actually prove nothing because it's always possible that a new chunk of evidence will be discovered next week that will undermine their theory. Scientists well know that the truth can be shocking and that they may get a personal shock tomorrow.
    In abstract games and puzzles we can prove many conclusions with total confidence and many more with a very high degree of confidence, and likewise in mathematics with the same qualifications. We'll meet examples later.
    Proof in a simple game
    Shirley Clarke wrote, describing two juniors playing a simple game, ‘They eventually “cracked it”, by convincing themselves that their strategy was a winning one, so that when they were asked, “Would you like to carry on playing?” they replied, “There's no point, it'll be boring.”’
    Figure 3.1 shows the very simple game: the players start bottom left, or any square on the left or bottom edges. The first player always moves to the right any number of squares and the second player moves towards the top edge, any number of squares. The player who reaches the top-right corner wins.
    What the young players realised after playing a number of experimental games was that the key squares all lie along the diagonal from bottom left to top right. In particular, they realised that any player who can move onto that diagonal, wins provided he or she does not latermake an error.