Free Games and Mathematics by David Wells Page A

along which a small marble would roll in the shortest time from A to B. Bernoulli took the side of Leibniz in his dispute with Newton over the invention of the calculus so he was deliberately provocative:
     
there are fewer who are likely to solve our excellent problems, aye, fewer even among the very mathematicians who boast that [they]…have wonderfully extended its bounds by means of the golden theorems which (they thought) were known to no one, but which in fact had long previously been published by others.
     
    Newton received the challenge at 4 o'clock in the afternoon after a long day as Master of the Royal Mint but by four in the morning he had solved it. He sent his solution anonymously by an intermediary to Bernoulli who, recognising the author, famously exclaimed, ‘The lion is known by his claw’ [O’Connor & Robertson 2006 ].
    Priority disputes were an inevitable source of friction before a modern system of academic publishing was established although, according to Sal Restivo, competition between mathematicians in early Europe was not matched among either the Chinese or Hindu mathematicians [Restivo 1992 : 18–19].
    Felix Klein (1849–1925) was at the peak of his career when in 1881 Henri Poincaré (1854–1912) began publishing his work on automorphic functions. They first corresponded in a friendly fashion but Klein was stretched to keep up with his younger competitor and this plausibly contributed to the collapse of his health in 1882 and his subsequent depression: Klein's career as a leading research mathematician was largely over.
    More recently, Andrew Wiles, having decided to devote the best years of his mathematical life to Fermat's Last Theorem, famously took precautions to hide his attempt from his colleagues, even publishing occasional papers on other topics to discourage speculation that ‘he must be up to something’. He did finally ‘prove’ it but a gap was found in the proof and he had only a short time to fill it before his proof was, in effect, declared a failure. He succeeded, with the assistance of his pupil, Richard Taylor, in 1994.
    Mathematical knights are still entering the lists today. Hofstadter's sequence (p. 144) prompted a challenge by John Conway, mathematician extraordinaire and inventor of the Game of Life , for a value of n which made the absolute value of a ( n )/ n − n /2 < 1/20. Conway offered $10000. Ouch! Mistake! C. L. Mallows answered the challenge and earned himself the tactfully ‘adjusted’ prize of $1000 by showing that n was a low 1489 [Weisstein 2006 ] [Schroeder 1991 : 57–59].
     
Asking questions about
     
     
The important currents of mathematical thought develop from deep investigations of fundamental questions arising from mathematically natural phenomena.
    [Williams 1998 : xiv]
     
    Practical chess players ask questions such as, ‘What are the best first moves? What move is best in this position ? When should you double your rooks?’ They are frankly uninterested in questions such as:
    • How long is the longest possible game?
    • How many possible distinct games can be played?
    • What proportion of games should be won by white?
    • Why do the knights move the way they do?
    • What if the board were smaller, or larger?

These are what philosophers call meta-questions, questions about the game whose answers will not make you play better. They step outside the frame which is just what mathematicians – but not game players – do all the time.
    Anyone who wonders, ‘Why do some numbers have more factors than others?’ has practically invented the idea of prime numbers, and several other kinds of numbers too, and could spend the rest of their life trying to answer that one question. When we ask about we step back and take a more distant view and even step outside the original rules of the game.
    Asking questions is a crucial activity, but what questions should we ask? Which are the most effective? The easiest are ‘What if…?’ such