Free Games and Mathematics by David Wells Page B

as, ‘What if we asked Euler's question about a different map? What if the Tower of Hanoipuzzle had four pegs? What if we added up not the counting numbers, but their squares?
    Other questions can start How or How Many or When or Which or Where…How can you draw a circle that goes exactly through three given points? In how many parts can a circle be cut by 5 straight lines? How many prime numbers are there? When will a prime number be the sum of three squares? Which numbers have the most prime factors? Which infinite series have sums which are irrational?
    These questions can often be answered provisionally by experiment. Draw a different map for the ‘Bridges of Königsberg’ or try the Tower of Hanoi with four pegs and see what happens. Draw a circle and cut it up. Make a calculation, investigate a sample case – a dozen sample cases. Check on primes that are the sum of three squares, collect some data, get some idea of what is going on . Mathematicians spend a lot of time on such activities – that's why mathematics has such a strong experimental and scientific side.
    Finally, however, we come to the Why questions. ‘Why?’ is often – usually – almost always! – a deeper question and harder to answer, not least because we are looking for proof . Ideas of proof have notoriously changed over the centuries. Today, many proofs by eighteenth-century mathematicians seem flawed, and ideas of problem solving have changed too: we now expect more than just An Answer.
Metamathematics and game-like mathematics
     
    David Hilbert wanted to formalise all of mathematics, as if mathematics could be entirely reduced to a logical game. He failed, but his program did promote metamathematics in which mathematics is turned in on itself, to examine its own foundations, the logic of its proofs, and the soundness of its assumptions – a perfect example of asking questions about mathematics.
    (Since pictures and diagrams, however useful to many mathematicians, are used sparingly in most published professional papers, as illustrations, and have a reputation for being potentially misleading, it is no surprise that metamathematics is extremely verbal, or that it is has links to the theory of computer programming, another very game-like activity.)
    There are students of chess and other abstract games who specialise in asking questions about – they can join the International Society for Board Games Studies and contribute to its journal Board Game Studies – but they do not have to be chess or Go players to do so.
Changing conceptions of problem solving
     
    The American Mathematical Monthly commenced in 1894 as a journal ‘Devoted to the solution of problems in pure and applied mathematics’, and the problems and solutions published much resembled textbook exercises.
    In 1932 the problems were split into Elementary and Advanced and the latter section ‘[sought] problems containing results believed to be new, or extensions of old results…’ There is a hint here that a “result” might not be the end of a problem, that it might not be the last word, though in practice solutions were simply submitted and published as before.
    Towards the end of the 1960s, however, a new section started called ‘Research Problems’, later renamed simply, ‘Unsolved Problems’. It was edited by Richard Guy from 1970 and in the odd numbered years included updates describing the progress made on the problems published. Over a period of a century or so the idea of a problem had changed from being little more than a textbook exercise to being a public challenge, open to development [Wells 1993 ].
Creating new concepts and new objects
     
    Chess players create new concepts as they understand the game better but these are a part of their analysis and don't change the rules and don't change thegame. When mathematicians pick out, let's say, the prime numbers and label them, they are changing the nature of the game and the prime numbers become an entity