Games and Mathematics by David Wells Page A

use of coordinates to represent geometrical figures by analogy was one of the great milestones in the history of mathematics.
    Such translations are nothing like the translation of a novel or poem into another language – imprecise, arguable and never perfect – but exact and very effective. Given a suitable translation, you can often transform one probleminto another which is solved more easily, illustrating how important it is to represent a problem in the best manner.
    Mathematics can also be used – for reasons that remain mysterious – to represent natural phenomena, which is why the hard sciences, the most successful, are all profoundly mathematical.
The interaction between mathematics and sciences
     
    Mathematics started as simple models of the real world – counting objects and measuring – but even the most abstract mathematics often turns out to model some part of reality, a mystery which no one has solved and to which we shall return when we look at mathematics in science.
    When the ancient Greeks studied the conic sections they had no idea that Kepler, Galileoand Newton would use them to explain the motions of the planets and falling bodies: the Greeks thought the planets moved in simple circles! When mathematicians obtained the square roots of negative numbers while solving quadratic and cubic equations, they were mystified – they did not know they would eventually be used by electrical engineers, among many other uses.
    Chess and Go have entirely lost touch with the real world conflicts they model but mathematics has never lost touch with reality without which it would be incomparably less rich, and would perhaps have stagnated totally – just one of the many differences between mathematics and abstract games.

5 Proving versus checking
     
    The solution to Euler's Bridges of Königsberg is as neat as it is simple but for that reason it is now little more than an historical curiosity, only suitable for elementary puzzle books and long since left behind by much more challenging topological problems.
    The knight tours that Euler investigated have the advantage of often being ‘pretty’ and perhaps suggesting some underlying structure which, however, is very hard to find: so hard that they have not led, unlike the Bridges of Königsberg, to a flourishing field of mathematical research. They have usually been found by a combination of ingenuity, a certain amount of mathematical argument, and trial and error, often aided today by computers.
    Such puzzles are typical of many mathematical recreations but not typical of mathematics as a whole, where we expect to do better than merely find a solution by trial and error or check results by brute force calculation. Indeed, we could say that several mathematical recreations remain so because they have not proved amenable to deeper analysis.
The limitations of mathematical recreations
     
    Recreations tend to emphasise the synthetic construction of solutions, and proofs that show that constructions are adequate – for example, by displaying a set of pentominoesfilling a certain shape – while deeper questions often seem impossible to answer. For example, what formula will predict the number of arrangements of the 12 pentominoes which will fill a rectangle 5 m by n ?
    It is hard enough to calculate the answer for particular rectangles by brute force using a computer program: a formula for the number seems hopelessly out of reach, not least because the shapes of the pentominoes are so ‘irregular’. Pentominoes fit more or less awkwardly into corners or against edges, somepentominoes fit each other like a key in a lock but others don't, and there is just so little pattern in them.
    We cannot say that the 12 pentominoes are totally arbitrary because they are, after all, forced by the condition that each consists of 5 identical squares joined along complete edges – but their combinations do seem pretty arbitrary and not amenable to elegant calculation or