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«From the beginning, he compelled me by preciseness of his moves, to go along the theoretical way in which he was perfectly confortable. But as we ventured off the beaten tracks, my amazement went increasing facing with the illogicality of my opponent's game, which fell through in approximately forty moves.» «I had understanded the absurdity of his book learning, that was not founded on basic concepts and depended only on memory exercises. Also, having this experiment, in the following game, I made a deliberate inversion of moves as early as the beginning, on which the adversary, led astray, played the theoretical move. My weak move, not immediately countered, became strong and I won the game in ... 13 moves.» «I found out from this encounter a lesson. I understood equally this subtile and unrecognized Alekhine's thought, expressed in his book : "Two hundred games", seeing only two phases in a chess game :» Robert J. Fischer was right saying : "Chess is life". Like life we may certainly consider that a chess game has a deep unity and we are mainly interested by this idea in the present chapter. On the other hand, it is interesting and usual, in an analysis attempt, to break down life and similarly a chess game in many successive periods. It is exactly what we want to do on the next chapter, in an unusual way!
1o) Does exist a mathetical solution to chess game? For a mathematician, Chess is only a game among million variants that it is possible of imagine; really the chess game is too singular, too special (Why 64 squares and not 180? Why one Queen in each camp and not three? Why two players and not seven? etc ...). In fact : the mathematical research is interested by great theories where it is possible to use powerful methods of investigation. Hence, in mathematical ways, "the Theory of Games" is only a part (i.e. a branch) of greater research fields. It is true, computing more easily investigates chess. Now computers play as well as grandmasters and in the near future, it is completely evident, will appear new chess engines playing at a higher up level than any human chess player. In reality, we observe a fascinating confrontation between engines and human players, both playing better and better. But nevertheless we maintain that (in any case still for a very long time) :
1o) Chess deepness and numbers In all real topics people is confronted with particular numbers. For example 2, 8, 30 (number of diagonals), 32 (number of pieces), 64 (number of squares) are numbers strongly connected with the chess game. It is quite clear that this numbers are little ones. Like you know, computers are able to process complex problems concerned by great and very great numbers, intervening as parameters. We name all this ones "computer numbers". This notion is not particularly accurate, but nevertheless is very interesting. Let us consider a very great computer number N. Then N + 1, N + 1000, 2N, 10N are also computers ones ... But it is not evident that N x N and especially NN are also computer numbers, because both last numbers are incomparably greater than previous ones!
TAB 1 :
This illustration allows us to begin grasping a little more how many moves are, theoretically speaking, at the disposal of a chess player when he has to play. 2o) Study of a famous example Like an example, let us examine now the famous following game, where we use breaf annotations coming from the book : «Le Guide des Échecs» - Traité complet Nicolas Giffard & Alain Biénabe, ©-Éditions Robert Laffont, SA Paris 1993, p.698.
ALEKHINE - EUWE
1.d4
d5
It is particularly interesting to know Alekhine's thoughts facing this configuration. According to A. Baratz («Testament d'Alekhine» p.35) : «The sixth black move results in significant complications ; more usual is here the manoeuvre : 6...Nbd7, 7.Nxc4 etc...» In fact this comment is now quite obsolete, 6...e6 being currently the more popular answer. Nevertheless, at the present time, it is still very difficult to know what is the better move between both classical sixth Black retorts. «Alekhine values the situation after the sixth black move in a purely strategical point of view, because it is impossible to be able to calculate all tactical possibilities present on the chessboard.» Let us follow again the unfolding of this game, with the indication of precise number of theoretical occurrences at the disposal of each player when he goes playing :
3o) What means TMN and CMN On this particular game you may observe how many important and in the same time quite regular is the number of occurrences at the disposal of each player. By commodity we will use the following terminology : Another surprising fact is that typically in this game Black TMN is nearly always greater than White one ; a priori this singularity give a good representation of the modest Euwe's advantage in this opening (Activity of black Queen and Bishops, whereas black Rooks control open files etc...) But, facing Alekhine's precise game, Euwe is unable to increase significantly his very slight advantage and, at the stage of the game we are watching, the position is approximately balanced. Notice also that the prophylactic 19th black move is not really appropriate and that better seems to be 19... Qd5 with a threat on g2.
4o) Go on our investigation Now we pursue the calculation of the TMN in the particular context of this endgame.
1o) More about Alekhine-Euwe game The final position (see the next diagram) shows, without any doubt, the triumph of Alekhine's strategy, based on the increase in value of King-side white pawns, with the support of all white pieces and, in particular, an ideally active King. Subtle and deep are moves like : 32.Bd3!, 34.Kg4!, 35.Bb1 etc...
DIAG 3
:
We think that the TMN concept is formally introduce here for the first time. It is really a great surprise for us to discover its deep meaning. Before presenting conclusions of all this study we wish make here some remarks concerning the beautiful Alekhine-Euwe game we had followed in the present chapter. During the first part of this game (opening and beginning of the middle game) the Black TMN, like we have mentioned this fact, is singularly greater than the White one, testifying a relative Black initiative. But Alekhine's play is always very exact and Max Euwe is frankly incapable of increase his very little avantage in a significant way. Another main interest of the attentive examination of this endgame is to reveal the progressive collapse of Euwe's position, after 30...Ne7 and even afterwards 35... Ke8?! We point out how clearly the weak black TMN show then a bad coordination of black pieces and especially the fatal influence of absolute pin and check situations. 2o) Chess mumbers transcend Computing ones In a computing statement of a complex and deep problem it is absolutely essential to distinguish two very different point of view. In the first one, says Absolute or Theoretical statement people wish obtaining an integral solution wich is then clearly formulated in a mathematical way. In the second one, says Relative statement or Practical statement, you are searching only for a very good approximation or simulation. It is not the same thing at all! At the present time best theoretical results concerning chess are represented by the 6 men Nalimov Endgame tablebases and other research of the same kind. Now we can maintain : 3o) Toward a mathematical conjecture The study of many chess games, like the present one, allows us to propose the following estimation. A chess game, if you neglect grandmaster draws and beginner games, consists, on average, of approximately a little more than 40 moves. In fact, in a more radical way, it is easy to count only such games!... Moreover, for each half-move the TMN is on average more than 32 = 25. Hence, we are led to consider the astronomical number, given as an acceptable estimation :
E = (32x32)40 = (25x25)40
= 2400
E = 2400 Hence we think E = 2400 is a chess number but is not currently and, still for a long time, a computing one! By way of comparison, think to the following numbers : less than 259 * * * NEW CHESS THEORY :
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