Games and Mathematics by David Wells Page B

simplification.
     
Abstract games and checking solutions
     
    In this respect many mathematical recreations are as close to abstract games as to mathematics. Just as the rules of chess show a degree of arbitrariness that usually precludes us from proving our conclusions other than by analysing a tree of possibilities, so some mathematical objects – pentominoes are an example – seem so arbitrary that we may wonder whether we shall ever understand them deeply enough to replace trial-and-error methods by much shorter proofs.
    We saw that the game of Nine Men's Morris has been completely solved but only by using a computer to check possibilities, not by using deep insights into the structure of the game to create a short proof. No doubt the programmer used as many short cuts as possible, but he could not do more because – as far as we know – the game simply does not possess any deep mathematical structure. The rules of the game, the movements of the pieces, and the size of the board (in all their different variations) are too arbitrary and too little patterned to force deep structure. The shape of the board is also somewhat arbitrary like the boundaries of many boardgames that give peculiar roles to corner and edge squares (an especially significant feature of Go).
    This is one reason why these game are so fascinating: their structures can be understood scientifically through experience but they can never be reduced to mathematical proofs, so the possibilities for subtle insights and deep intuition are endless and even the very strongest players play imperfectly.
    It is true that certain special cases can be treated mathematically. As we saw, Elwin Berlekamp and David Wolfe have written Mathematical Go: Chilling Gets the Last Point on the endgame at Go, but the book is only possible because as the game proceeds the board (as we noted earlier) is naturally divided into mutually separate regions of still-active play which can be analysed by combinatorial game theory. Endgame success does not promise to spread to the opening and middle game.
    Every property and feature of such abstract games is forced by the rules but there may be no means of reducing their complexity to the relatively simple patterns that mathematicians can handle, and even if you occasionally prove aglobal proposition about a game, for example that Hex should be a win for the first player, most smaller-scale local propositions remain unprovable.
How do you ‘prove’ that 11 is prime?
     
    We can illustrate what we mean with the above question. How would we prove that 11 is prime? Well, 11 is odd so it is not divisible by 2 and 11 + 1 = 12 is a multiple of 3 and 11 – 1 = 10 which is a multiple of 5, and 11 < 14 which is 2 × 7. So 11 is prime.
    It's obvious, is it not? Yes, indeed, but the fact is that we have only confirmed this obvious fact by checking the possibilities. Given that it is so obvious, the check is not especially short, but there is no simpler proof which would make checking unnecessary.
    Pretty much the same is true if we wanted to check that 97 is prime. Suppose that we realise that all its prime factors are less than or equal to √97 which is under 10 so that we only need to check 2, 3, 5 and 7 as possible prime factors. The check is simple enough and much faster than testing all the primes up to 50, but it hardly qualifies as a snappy proof.
    The same is true, however, of more powerful tools for checking the primality of far larger numbers. Each specific tool is too powerful to be efficient for small numbers: then a point is reached at which the tool is worth using, but eventually it gives out by becoming too slow for all practical purposes and a more complex and powerful tool is sought.
Is ‘5 is prime’ a coincidence?
     
    These thoughts suggest the question, ‘Are there coincidences in mathematics?’ For example, all number puzzle buffs know that 6 = 1 × 2 × 3 = 1 + 2 + 3, and that 153 = 1 3 + 5 3 + 3 3 , so it is one of