Free Games and Mathematics by David Wells

with the Tower of Hanoifor a while and you ‘get the hang of it’, you feel there's a pattern there before you can follow it on every move, until finally the penny drops and you feel pleased with yourself. The chaos of the original play is replaced by structure, and you get an aesthetic ‘kick’, as you do from the proof that it can be solved in 2 n − 1 moves. The ‘kick’ from playing the Tower of Hanoi successfully is limited, however, to beginners and children. To serious game players it is just too simple.
    So, plausibly, are Nine Men's Morrisand Hex. As we have seen, no-one has yet published a book of some world champion Hex player's Greatest Games but all the best games of all the great chess and Go players have been printed. Indeed all the games played in all the great international chess tournaments and matches have been published, sometimes in several languages, and often with annotations which help the amateur to understand what is going on. Chess is so complex, so subtle, and so hard to understand that without expert assistance most players would simply not have more than a slight idea what was happeningin a typical grandmaster game – but that is where the greatest examples of the beauty and elegance of chessappear.
    Moves can be surprising, even astonishing; they can develop a theme in an apparently inexorable sequence, or they can be so bizarre as to look at first sight like a mistake; they can be overwhelmingly powerful, or subtle and delicate. All these features contribute to the feeling that leads chess players – and Go players also – to use the language of art to describe individual moves, or sequences of moves or even the ideas themselves.
    We said ‘the greatest examples’ but even the patzers in the local chess club can enjoy making moves which, at their own level, have the same delightful features, just as amateur mathematicians can enjoy any number of mathematical ideas and theorems and illustrations without being experts on the theory of sheaves.
Science and games: let's go exploring
     
    What happens when you play around with a puzzle or game? Playing around is not the same as playing seriously, but rather a sort of loose exploration or experiment which helps to develop a feeling for what is happening. As this feeling develops you think perhaps of more serious experiments or tests that will push your exploration ahead scientifically . This points to another link with science: it is easy to draw conclusions that are false and which further experiment will show to be false, by counter-example.
    We shall look at science and mathematics in Chapters 11 and 14 .

4 Why chess is not mathematics
     
Competition
     
    For all their resemblances, abstract games and mathematics are far from identical. Games, for a start, are competitive but everyday arithmetic is not, though even this difference should not be exaggerated.
    Leonardo of Pisa, known as Fibonacci(1170–1250) was challenged by John of Palermo at the court of the Holy Roman Emperor Frederick II to solve a set of mathematical problems. Fibonacci solved them and won. Tartaglia (1500–1557) was challenged by Fior, a student of the famous Scipione del Ferro, to a fight not with swords but with cubic equations, each proposing 30 cubics for solution. Tartaglia solved all of Fior's in less than two hours while Fior struggled. Tartaglia later accepted a challenge to debate Cardan's assistant, Ferrari, in Milan with cubics and quartics, but the quartics were too much for him. At the end of the first day he crept out of the city and lost by default.
    Vieta (1540–1603) was challenged to uphold the glory of French mathematics at the court of Henry IV against the sneers of the Dutch ambassador by solving an equation of degree 45. He succeeded by recognising it as a disguised trigonometrical identity.
    The list goes on. Johann Bernoulli (1667–1748) challenged the mathematicians of Europe to solve the brachistochrone problem: to find the curve