Games and Mathematics by David Wells

in a modified game that suddenly includes new ‘pieces’.
    Sometimes a new idea is more-or-less forced by the situation, such as the radianas a measure of angle (p. 99). Other new concepts appear when we try to formalise some rather vague intuition, such as infinity which in everyday language has many variant meanings. Galileo's paradox is just one of many that would not exist if infinity were not such a subtle concept. To use the idea in mathematics it must be tied down and stripped of most of its everyday associations to prevent mathematicians tying themselves in knots.
    Mathematicians exploring mathematical landscapes often find weird objects which seem at first sight to be of no particular significance: on deeper examination, this is hardly ever the case. Fractals are an example. The first fractals were regarded as weird monstrosities and appeared in books of mathematical recreations like strange animals exhibited in a Renaissance zoo. Today they are well understood, are recognised all over the place and have entered the mainstream.
    Often, ‘weird’ objects turn out to be specific examples of some general phenomena, just as an explorer first discovering a carnivorous flower might be initially astonished but realise later that there are entire families of such strange plants.
Increasing abstraction
     
    Asking questions about has another effect that students of mathematics know only too well: it makes mathematics even more abstract. The graphical picture of the Tower of Hanoi makes the puzzle easier to solve – it makes it trivial – but the graph itself takes some understanding: why does it perfectly represent the puzzle? Most players of the simple game of the Tower of Hanoiwould not think of the graphical representation in a million years. It takes a mathematician – and a modern mathematician at that – to ‘naturally’ think of representing the positions in a game on a graph. It is a tactic, or strategy, that is now very well known and even seems not very abstract, though it was once novel and surprising.
    The famous French mathematician Jean Dieudonné exhorted students to develop an ‘intuition for the abstract’, but that takes time and effort. John von Neumann once told a student complaining that he had not understood von Neumann's lecture, ‘You don't understand mathematics, you just get used to it!’ Not the advice you were given at school, but it does contain an important grain of truth.
    Yes,intuition is the key and it can be developed, in part, by following von Neumann's advice. Experience and then more experience leads to intuition. Just as chess players become more-or-less familiar with the miniature world of the chess board by playing many, many games, so mathematics students become familiar with the miniature worlds of group theory or differential equations experience, by playing and experimenting, calculating and speculating and checking. Students who do not ‘play around’ with such new ideas, but stick to the textbook explanation and do no more than answer a few exercises, will never develop deep intuition, and never become real mathematicians.
Finding common structures
     
    A game player would be amazed if a tactical or strategical idea that worked in chess also worked in Go (apart from the basic idea that a move which creates two threats is powerful) but mathematicians are used to finding matching features in algebra and geometry, or analysis and topology – or – you name it! The analogy is often more-or-less perfect, as when a geometrical diagram is translated into coordinates ( Figure 4.1 ).
    Figure 4.1 Mid-point of line segment
     

     
    The ‘natural’ analogue of the mid-point of the line segment AB is the ‘average’ of the two points calculated from their coordinates: ((2+8),(6+4)) = (5, 5). The analogy between the calculated weighted average of two points and their geometrical representation seems so natural that it is easy to forget that it is an analogy, and that the