Games and Mathematics by David Wells Read Free Book Online Page A

impartial games is less than it might seem.
    The extraordinary book, Winning Ways for Your Mathematical Plays , by John Horton Conway, the inventor of the game of Life, and his colleagues Elwyn Berlekamp and Richard Guy [Berlekamp, Conway & Guy 1982/2001 ] showed how some very simple games which were not impartial – but not including chess and Go which are far too complex – could also be solved mathematically.
    Conway was inspired in his early work on combinatorial games by the endgame at Go which, as we shall see, is much more mathematical than the rest of the game, so it is appropriate that one of the highlights of subsequent work on CGT was Mathematical Go: Chilling Gets the Last Point by Elwyn Berlekamp and David Wolfe [Berlekamp & Wolfe 1994 ] which reduces the final stages of any Go game to a (complex) calculation.
     
programs are about 10 kyu level or 10 grades below the basic amateur sho-dan level which any player is traditionally supposed to reach after playing ‘one thousand games’. So vastly better is the human brain at spotting patterns and developing intuition and a ‘feel’ for the game, than the most powerful of today's supercomputers.
    However, the endgame at Goin which the board is separated into regions, often relatively small, which no longer interact, is quite another matter. Using Combinatorial Game Theory , the end game of Go can be treated as a collection of solvable sub-games. We might say that the opening and middle game of Go are less mathematical than chess, but the endgame is much more mathematical. So, fortunately for the millions of players of chess and Go, although the ‘Fundamental Theorem of Combinatorial Game Theory’ says that,
     
Every game of perfect information is either unfair (one player has a winning strategy) or boring (two rational players will always bring about a draw
     
    the qualification rational is so strict and so severe that no two real human players, however skilful, can ever be totally rational, and so chess and Go are not, after all, boring.
    The games of Nine Men's Morris, Hex, chess and Go vary in complexity and subtlety, and in popularity. The hardest, chess and Go, because they offer the greatest challenge and the greatest psychological rewards in terms of beauty and elegance, are the most popular, but they are the least accessible to mathematical treatment.
    What other games might be invented in the future? A game on the lines of Nine Men's Morris could be invented by any alien civilisation on a distant planet and the chess board pattern must be discovered by intelligent life elsewhere in the universe and games may be played on it, even a game modelling two opposing armies, though they are hardly likely to invent Western chess. Hex, so simple and so non-arbitrary might actually exist ‘out there’. If it does, then it is likely that players on Planet Zorg will also have decided that playing on very small boards is uninteresting and that too-large boards are unplayable, and they may well know the same proof that white ought to win – just as we expect them to have much the same mathematics that we do. They might also have analysed the game much further than we have.
    It is in the nature of games that they have the potential to be more-or-less universal, one of many mysterious connections between mathematics and abstract games to which we now turn.

3 Mathematics and games: mysterious connections
     
    There are so many connections between games and maths that it is no wonder that many mathematicians play chess or Go or Bridge, or that so many abstract game players are into mathematics.
    The presence of rules or underlying assumptions, the fact that expert chess players can play games in their heads, just as most people can do at least some calculations in their heads and maybe visualise a cube sliced symmetrically into two parts; the fact that there are tactics and strategies for solving problems in chess and mathematics; the confidence that we – often