by Wells, David
Nine Men's Morris is a world of possibilities – created entirely by its rules – but these possibilities are far richer and more complex than the rules themselves. And the rules don't tell you what possibilities follow from them. That is for you, the player, to discover – and it can take a very long time. With tactical sequences come very elementary proofs: you can prove by analysing the many possibilities that in certain positions one player will win whatever the second player does. We can also ask questions about the game: what is the shortest possible game? The longest possible? What if the size of the board were changed?
Hex
The Danish mathematician Piet Hein invented the game of Hex which he originally called Con-Tac-Tix in 1942 when he was a student at the Niels Bohr Institute of Theoretical Physics. He marketed it under the name of Polygon and later it become very popular in Denmark as a pen and paper game. Hein had been thinking about the Four Colour Theorem which says that a plane map can be coloured using only four colours so that no two regions with a common boundary are the same colour, and this led him to think of chains of regions.
In 1948 it was independently reinvented by a young mathematician at Princeton called John Nash, recently famous as the hero of the film A Beautiful Mind (2001): he was a brilliant mathematician who became schizophrenic but many years later, astonishingly, recovered and was awarded the Nobel Prize for Economics. In 1952 the games company, Parker Bros, manufactured Nash's game as Hex.
Figure 2.5 Empty Hex board, 11 by 11
The board is a ‘parallelogram’ of hexagons (Figure 2.5), usually a rhombus, the two pairs of opposite sides being of different colours, one for each player. 11 by 11 is the commonest size but it can be played on any size at all or indeed on other shapes: it has been played on a map of States of the United States of America, with the border divided into four parts.
The rules are extremely simple: the board starts empty and players take turns to place a piece of their colour on one of the empty hexagons and the winner is the player to make a chain which connects their two sides of the board.
Hex seems impossible to master. Although the rules are so simple, there are 121 first moves on a board 11 by 11, so a move-by-move tree analysis of all the possible starting sequences is prohibitive. We can say that one player must win a completed game because only a chain of one colour can completely stop a chain of the other colour being formed, as John Nash realised. He also proved in 1949 that the first player must win with best play but unfortunately he gave an existence proof which tells you nothing at all about the winning strategy.
Also unfortunately, the first player has a very large advantage so most games are played today with the ‘swop’ rule: after the first move has been played, the second player has the option of ‘swopping’ places with the first player and accepting the move played as his first move. (This has the disadvantage of reducing the number of reasonable first moves; playing on a larger board would presumably reduce the first-move advantage.)
John Nash favoured a larger board 14 by 14 which makes the game harder but also allows greater opportunity for strategies to appear: on very small boards, calculation alone will help you to play well and strategy hardly exists. (This is a general feature of board games: tactics dominate on small boards, strategy appears as the board gets larger.)
Hex has become a favourite game among AI enthusiasts. The small 7 by 7 Hex game has been ‘solved’ by computer analysis, meaning that a winning strategy has been found: the first player plays in the centre of the board to take advantage of symmetry.
Even more than Nine Men's Morris, Hex is a highly mathematical game. In fact, it has even been used to prove Brouwer's fixed point theorem in pure mathematics. We can illustrate this theorem with two maps, one a small version of the other, lying on top of one another and completely overlapping (Figure 2.6).
Figure 2.6 Overlapping maps and common point
PQ corresponds to AB. The theorem says that there is a common point, X, representing the same location on both maps. (This would still be so if the top map were crumpled up into a ball instead of lying flat, but it would not be so if part of one map lay off the edge of the other.)
One way to see this is to create this sequence of maps in which each pair of maps has the same relation as the previous pair so that the maps get smaller and smaller and tend towards a limiting point which will be solution to the problem (Figure 2.7).
Figure 2.7 Sequence of repeated maps
The original theorem is much harder to prove but David Gale, a fellow student with John Nash, did it by showing that it is equivalent to the theorem that the game of Hex cannot end in a draw.
The miniature world of Hex is far more complex than Nine Men's Morris so there are more tactical possibilities and strategical ideas. The board appears empty at the start but that, as always, is a kind of illusion. Although the rules and the board are amazingly simple, they still force into being a world of complex possibilities: they don't create a world out of nothing, but out of very little, and these potential patterns and possibilities existed the moment the game was invented.
Cameron Browne has written the first complete book on Hex tactics and strategy, Hex Strategy: Making the Right Connections [Browne 2000]. His use of technical terms such as bridges, templates, spanning paths, ladders, groups, chains, edge defence, and so on, suggests the wealth of concepts that aid the experienced player. (Browne has also published a more general book on Connection Games.)
To discover these tactical and strategical ideas, however, is not easy. Many years ago while puzzle editor of Games & Puzzles magazine I wrote a review of TwixT, a proprietary variation of Hex in which pegs are connected by knight's moves, and made the mistake of presenting an annotated game without consulting the inventor, Alex Randolph, and on the basis (it became clear) of far too little experience. I had only explored TwixT very superficially. The result was embarrassing, a letter by return of post explaining why my analysis was feeble, mistaken and wrong.
My illustrative game was obviously not elegant let alone beautiful but Hex is sufficiently complex to be both, though no-one has yet, to the best of my knowledge, published a collection of X's Best Games of Hex – but who knows, the time may come. In the meantime, interested readers might like to try the Hex problems in Cameron Browne's book.
Chess
‘Chess is the image of war’ [Keene 2006] but with the blood and gore stripped out to leave only a mental and psychological struggle. The early Persians listed ten virtues enjoyed by the game of chess. The first, that it nourished the mind, the tenth that it mixed war and sport [McLean 1983: 113].
Chess is far more complex than Hex, let alone Nine Men's Morris. There are six different types of pieces and their movements are far richer. The starting arrangement has been determined by history and is highly arbitrary, as are the rule for castling and the modern rules about repetition of moves, so it is extraordinarily difficult to understand and impossible to understand completely.
No one has even tried to prove mathematically either that the first player should win or that it should be drawn with best play or that some specific opening is winning for white, or black.
Players analyse actual sequences of moves as far as they can but deep understanding of tactics and strategy and a creative imagination are necessary to play well. Because the ideas are so subtle and can only be learnt through experience it has taken the greatest players centuries to develop the repertoire of tactical and strategical concepts which now fill thousands of books. Some of the most powerful ideas in the opening were invented, or discovered, as late as the 1950s and 1960s and new possibilities are still being developed, while computer analysis is revealing the secrets of many obscure end-games.
Players learn through experience to spot tactical patterns and develop strategical insight. They conjecture and generalise, then test their theories through play. Reading books, playing over old master games and playing with other players, preferably stronger than yourself, is the best way to
pick the brains of better players and speed up the process of developing a subtle and deep intuitive understanding of the game. Like golfers or tennis players or footballers (those games have some abstract features also) developing your chess insight can be a lifelong task.
The extraordinary subtlety of chess explains a curious feature: the vast number of chess variants that have been invented set against the very few that attract serious attention or are played by millions. It is extremely easy to vary any abstract game by simply asking, ‘What if…?’ What if chess were played in three dimensions? What if it were played on a larger board? With pieces with different moves? With two kinds of knights? With the first player making 1 move the second player making 2 moves, the first player 3 moves, and so on? These and scores of other variants have all actually been played. The last is called Progressive Chess and is great fun. It has the added advantage that it usually doesn't last long.
The problem with (almost) all these variations is depth. Chess has been played for hundreds of years and over these very long periods deep 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 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 like his own idiosyncratic invention, but if it turns out to be sound then the invention becomes a discovery of an effective plan. The rules of an abstract game may be very simple, yet their implications may be unfathomable. Our human creativity includes the ability to illuminate the obscure – but also to create the obscure.
The same turns out to be true of mathematics though with a subtle twist. If a mathematician believes he has discovered some new theorem, but it turns out to be untrue, then naturally he thinks of the erroneous proof as something he invented, which didn't work, like a mechanical device that was built to perform a certain function but failed to do so. But if it turns out to be true, then there is a strong feeling that it has actually just been discovered. This is how Richard Hamming puts it, referring to his own mathematical research:
When I try to examine my own beliefs…I find that if the result seems to be important then I found it, but if it seems to be rather trivial, then I created it!
