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stronger the player, the more likely that their hypotheses are – usually! – more-or-less! – sound.
    Figure 2.8 White wins
     

     
    So the analysis of particular chess positions is both highly mathematical and analytical, and highly scientific, as well as imaginative. In Figure 2.8 , (Tartakower-Stumpers, Baarn 1947) there is no practical doubt at all that White,to play, wins by N-e4 threatening to trap the Black queen by Rb3 and Nc3 or Nc5. The analysis is as sound a proof as any in mathematics, and much simpler than most. In Figure 2.9 , (Klein-Tartakower, match 1935) however, although it is considered that White should have a larger advantage with four pawns against three than with three against two, which is usually drawn, there is no simple way to prove what the result ought to be.
    Figure 2.9 An uncertain outcome
     

     
    Once we start to draw conclusions based not on strict analysis but on concepts such as strong squares or a weak king position or a superior pawn structure , or two bishops against two knights, then we are no longer talking about game-like concepts and we can no longer share our conclusions so easily or communicate them so effectively. The idea of a strong square is a scientific concept, not a mathematical one, and a part of the player's scientific understanding. On the other hand, some chess problems are completely mathematical, such as the task of moving a bishop so that it visits every square of the board ( Figure 2.10 ) in as few moves as possible: 17.
    Figure 2.10 Bishop path round 8 by 8 board
     

     
    The solution, however, like a typical knight tour, is a mixture of pattern and absence of pattern and was constructed by a combination of smart tactics and strategy and trial-and-error [Novcic 1986 : 65].
    These examples highlight another feature that chess shares with Hex and Go and mathematics, the use of notation. Match and tournament games are recorded and the best are published with annotations. Players also use notation when talking about games without a board: ‘I should have played f4 at once, then if you take, e5 is strong. Nh7 is terrible and if you pin me with Re8 then Qg4 is a double threat.’
    When chess is played on a physical board, it is possible to carelessly (or sometimes with malice aforethought) place a piece so that it is half on one square, half on another. When played mentally, using notation, this is not possible, though you could mumble so badly that your opponent cannot clearly hear what you are saying!
    Mathematics, of course, has its own notations and language, as well as figures, diagrams and illustrations.
    Every legal possibility on the chess board is forced by the rules which created the subtlety and richness of all the greatest games of chess ever played, from the masterpieces of the world champions from Steinitz to Fischer to Kasparov, to all the games played by the kibitzers at your local club with all their mistakes and blunders. (When the rules have been changed – they have changed several times since chess was born in ancient India – the tactical and strategical possibilities have changed too.)
    These thoughts prompt the question, ‘Are such features of the game of chess invented or discovered?’ Did the weak squares in a chess position exist long before they were ‘discovered’? Was the Sicilian defence invented by an ingenious Italian, or did it already, as it were, exist as a possibility and he just discovered it? Was the famous smothered mate discovered? Or invented? Plausibly, both: the game of chess and its many variants were invented, they are certainly man-made objects, but their features – as forced by the rules – still had to be discovered in practice, a process that took many centuries and which still continues today.
    Chess and other abstract games are created by men (and occasionally by women) but their creation has implications which are very hard to grasp. When a player has an original idea for an opening move, it may seem