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tactical and strategical ideas have been discovered. Without this depth, there would be no master tournaments, no World Championships and no published collections of masterpieces, and games would never be described as beautiful .
    It has been well been said that if chess were invented today, it would never take off because players would find it much too hard to get into. It is conceivable that in the future, some powerful computer will be able to quickly analyse a new abstract game, detect a wealth of subtle tactics and deep strategies and so make it so attractive to human players that it will rival chess and Go, but it hasn't happened yet. In the meantime, even an ingenious game like Lasca, invented by world chess champion Emmanuel Lasker, only has a very modest following and a few enthusiasts.
    Strategical ideas are especially hard to pin down: there is no doubt that the square in front of a backward pawn is in some sense weak , but the meaning of this proverbial fact in your actual present position depends on the positions of the other pawns and the play of the pieces.
    Emmanuel Lasker (1868–1941)
    Very few mathematicians have been as good at abstract games as they are at mathematics and conversely. Emmanuel Lasker, World Champion from 1894 to 1921, was a bit of an exception although he spent little of his adult life actually doing maths.
    He did his higher degree under the great David Hilbert, from 1900–1902 after he had already become world champion, spent 1901 as a mathematics lecturer at the Victoria University in Manchester, England, and introduced the concept of a primary ideal which is a generalisation of the idea of a power of a prime number. His most famous paper proved the existence of ‘primary decompositions for polynomial rings’. This is today known as the Lasker–Noether theorem because Emmy Noether, the greatest woman mathematician of all time, proved in 1921 a more general version of Lasker's pioneering work.
    In 1911 Lasker invented a variant of draughts which he called Lasca and described in a booklet, The Rules of Lasca, the Great Military Game [ 1911 ]. There are two kinds of pieces, Soldiers and Officers, and pieces can be stacked in columns. It was re-published in 1973 by the German games company F.X. Schmid. Later he published a book on board games, Brettspiele der Volker [ 1925 ].
    Lasker also reached master level at bridge, learnt to play Go and wrote on philosophy – two of his works were translated into English as Struggle [ 1907 ], and The Community of the Future [ 1940 ] – and he became a friend of Einstein who wrote the foreword to his biography by Jacques Hannak. A rare polymath indeed.
     
    Not only is chess very difficult, but almost every position you reach, except for the opening and a few endings, you will have never actually reached before. Consequently, a crucial feature of chess play is the use of analogy: you must exploit your past experience by spotting analogies which ‘ring a bell’, generalising your experience and specialising your strategical understanding. Of course, analogies can easily let you down if you exploit them uncritically.
    The other side of that coin, of course, is that chess moves can be calculated, move by move, and in theory it is possible to calculate many, many moves ahead, indeed until the end of the game. In practice even the strongest human players find it hard to calculate ahead more than a few moves in most positions because the ‘tree of possibilities’ expands too rapidly, which is why judgement is so important.
    Players use their judgement to decide which lines of analysis to pursue in the first place, and then to decide whether a position they believe that they could reach – if their analysis has been flawless – is good for them or for their opponent. Players, in effect, form hypotheses about the position on the board and positions that they could reach in the future, based on a mixture of judgement and analysis of actual moves. The