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

but not always! – feel that our conclusions are correct, that we can prove them, and the reliance of the chess player and mathematician on patterns and structure – these shared features all point to deep underlying connections between mathematics and abstract games. Let's start with perhaps the most remarkable of all – that they can both ‘in theory’ be done in the mind.
Games and mathematics can be analysed in the head…
     
    …provided we make allowances for limitations of memory and visualisation. Few of us have Euler's phenomenal memory. On the other hand, even first-school children are expected by their teachers to do simple sums by ‘mental arithmetic’ and all strong chess players can informally discuss a game they have just played with limited recourse to the board.
    Of course, you can imagine doing many activities in your head: sports psychologists recommend that champion sportsmen mentally rehearse their next high jump or ski run as an aid to better performance but that mental action is only in imagination . Euler and Koltanowski, however, were not imagining they were mentally creating mathematics and playing chess – they really were.
    There is a link here in the very language used: chess players talk of calculating possibilities and are sometimes asked, ‘How many moves ahead can you calculate ?’ A player admits he made a mistake because he calculated wrongly although no arithmetic was involved: the player calculated through a tree of possibilities: ‘If I play Qe5, that threatens mate, Black can defend with Ne8, but then I can play h6…’ and so on. What a significant form of words! Why would anyone calculate the moves in a game which have nothing to do with numbers, if there were not some mysterious connection between calculating with numbers and calculating the moves of the pieces?
     
Can you ‘look ahead’?
     
    Strong chess players can ‘look ahead’ in a position – tracing a complex tree of possible sequences of moves – in order to decide which move to make. Written
    Three blind mathematicians
    Blind mathematicians are rare, but they do exist and they have reached to the very top. (Are there blind theoretical physicists or chemists?) No doubt psychologists could learn much by studying how they think. Ironically, they have been as distinguished at geometry as at algebra, partly by exploiting their sense of touch.
    Nicholas Saunderson (1682–1739) was blinded by smallpox at the age of 12 but became Lucasian Professor of Mathematics at Cambridge University. He wrote a textbook, Elements of Algebra , and lectured on optics. He also invented the pin-board, much used today by primary school pupils, to create geometrical figures that he could feel with his fingers, a sort of Braille geometry.
    Lev Pontryagin (1908–1988) lost his sight in an accident at the age of 14. His devoted mother read books to him, wrote his notes and even learnt to read foreign languages for his sake. At 25 he entered Moscow University, blind but able to remember all his lectures. He published his first original work at the age of 19, like Euler, and went on to become one of the great mathematicians of the twentieth century.
    [O’Connor & Robertson 2006]
    Bernard Morin (1931–) has been blind since the age of 6, but became a brilliant geometer who discovered how to turn a sphere inside out – yet another ironic achievement for a mathematician who can feel but not see – and what is now called Morin's surface.
     
out in words, which is not usually how it is thought through by the player, a very short sequence might go something like this:
     
‘If I play Qe4, then Black can defend by Re8, but then I play Bc3, and if he defends by Nf8 then he has no defence against Nh5, but if he plays g6 instead then I can still play Nh5 and he's helpless. So I play Qe4…’
     
    Mathematicians may also ‘look ahead’ when examining, for example, a geometrical diagram: ‘If I draw AX parallel to BC then the triangles