< ... Not the best plan, but the best move. The concept of a plan is totally foreign to my theory. Or maybe it is better to say that my theory discredits such a concept. Complex systems develop spontaneously in nature. So to find the best move in a given position we need to consider the following four factors: 1) Material (m). A very simple and well-known calculation 9-5-3-3-1. Divide your sum by the opponent’s sum. Usually m=1. It is analogous to the mass of a system. 2) Mobility (p). The number of legal moves in a given position. P1 is the mobility of our pieces, P2 is the mobility of the opponent’s pieces. P1/ P2 = p. Mobility is analogous to the kinetic energy of a system. 3) Expansion factor. It is defined by calculating the center of gravity of a given position. Take the starting position. White has 8 pieces on the 1st rank, and 8 on the 2nd rank: 8*1+8*2=24, divided by the number of pieces, 24/16=1.5, so the center of gravity lies “between” the first and second ranks. In every position there are two expansion factors, White’s and Black’s. An important number is the Greek Delta, which denotes a change or variation of a quantity, and the difference between them. The expansion factor is analogous to the potential energy of a system. 4) Packing density. There are several different types of packing density: by all pieces and pawns; by king and pawns; by king, pawns and knights (short-range pieces); by pawns only; and local density at certain sections of the board. The most informative densities are those by short-range pieces. Imagine a position: Kg2, Nf3, and pawns on f2-g3-h2. The area of the smallest rectangle that contains these pieces is 6. There are 5 pieces, 5/6=0.83, very dense and therefore a safe construction!

Notice that I only need four factors to describe any position. Two of them, the expansion factor in the form of a center of gravity; and packing density, are original, or, at least, I never seen them suggested by anyone else. In most positions it is enough to know m and p. If m=1, p is the main factor we need to identify, which algorithm of the three known should be applied. If p>1.25, it’s “Tal,” p<0.8, it’s “Petrosian” (note that 1.25*0.8=1), and if p belongs to (0.8, 1.25), it’s “Capablanca.”>

Fascinating stuff, even if I myself remain skeptical. Any mathematicians/physicians (Gypsy?) care to comment?

There is more material on the web site

http://eng.e3e5.com/

I'd give the king strength, since he too influences the battle. I'd give him weight analogous to his mobility - roughly the same as a minor piece.

Packing density seems silly. If he wants to prevent looseness of pieces, then he should be more concerned about protecting them rather than packing them. You can have tightly packed pieces that are still unprotected, and in fact this immobilizes them a bit.

Expansion just seems like an extension of mobility. We wouldn't fight for space that can't be used.

Really only material and mobility seem meaningful, and they're also somewhat related.

I'd like to see software that only calculates mobility. Usually, it will be materialistic since more pieces generally leads to more mobility. This would be interesting, although I'm sure it would get smashed by the top chess programs.

<MS: And how would you describe talent for chess?>

<AS: There are essentially two types of chess talent. The superior one ­ intuitive, and the `usual' one ­ better memory, better calculating speed. People like Capablanca, Smyslov, and Karpov somehow learned where to properly place their pieces all by themselves and at a very early age. Troubled by mediocre memories, which prevented them from memorizing zillions of intermediate positions in the process of calculation, they nevertheless were the world's strongest players for many years. They don't calculate as much as visualize images. The objective of such players is simply to choose the most aesthetically pleasing image of a position, which is often an ideal, and then to find the regrouping maneuver, which requires short-range calculation.

The most brilliant players representing another kind of chess talent are Tal and Kasparov, who have a photographic memory and are unusually fast at calculation. They digest a lot of variations and consider many more critical positions than intuitive players; however, their evaluation is much rougher. But it is not all hopeless for them, as one is able to learn strategic play. Also geniuses of the intuitive type are naturally lazy. They don't work much. This is not a reproach ­ it's their greatest problem, the consequence of their merits. While intuitive geniuses observe desirable positions and don't see any reason to improve, men like Tal and Kasparov gradually expand their evaluation function, accumulate more and more critical positions with refined evaluations, and slowly approach the inborn clarity of vision of intuitive players. That's why the great Capablanca lost a match to a workaholic like Alekhine, and the 23-year-old Kasparov defeated the mature champion Karpov, which was, in my opinion, an absolutely impossible task. Kasparov is the greatest player in the history of chess. I am a big fan of Capablanca, but Kasparov is the greatest.>

<MS: What do you think of Fischer?>

<AS: The position of the bishop on g7 is strategically suspect, it is playable, but not against a natural intuitive genius! Kasparov feels Indian positions with his fingertips, but did not risk playing the KID against Karpov until their 4th match. And when Garry did not lose, he confirmed his absolute dominance over Karpov. It became clear that Karpov's attempts to regain the title would never succeed.>

<MS: Do you like Kramnik's play?>

His primary point is that chess, as a self-sufficient complex system which acts according to the known laws, can be analyzed using scientific methods as any other dynamic system. Chess has all the qualities of a complex system (the abundance of elements, hierarchy and a so-called "chaos zone", where it is impossible, in principle, to predict the direction that the system would develop in certain positions).

The main part of his "post-non-classical" chess theory is a move-search algorithm, a method for finding the strongest move in any position.

This is the starting position for analysis:

1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 a6 6.Bg5 e6 7.f4 Qb6 8.Qd2 Qxb2 9.Rb1 Qa3 10.e5 dxe5 11.fxe5 Nfd7 12.Ne4 Qxa2 13.Rb3 Qa1+ 14.Kf2 Qa4

click for larger view

Intuitively sensing that the position demands a sacrifice, Tal played here <15.Bb5!?>. Soon afterwards they found <15.Nxe6!> as the strongest move in this position.

" My first acquaintance with A.A. Shashin took place in the spring of 2003, at a very difficult point in my career. The year 2002 had been one of the most unfortunate in my entire 15 years of professional activity. My rating had dropped almost 80 points, to a level (humbling for me) of 2679; my world ranking fell to twentysomething, and I hardly knew what to do next. Fate had it that I should then meet this amazing person, coach, and physicist by training (and perhaps by calling): San Sanych Shashin, as I freely started to call him. It is difficult to overestimate the amount of support he gave me.

In a purely chess sense, the question logically arises: could a Soviet master with thirty years’ experience as a trainer, and master of his own attitude in the search for a decision-making algorithm, talk about chess as an equal with someone who until recently had ranked in the Top Ten in the world, and still teach him something?

I reply: both yes and no.

“No,” because it is difficult to remake an already fully formed player, 26 years old and successful as a result of his own talent and his chess understanding.

But “yes,” because any crisis is an opportunity for growth – and I, discovering the conclusions of S.S.’s theory at precisely that moment, significantly enriched and broadened my horizons: while selecting a move, I often successfully employed the ideas he presented to me. The fact that the period from July 2003 to July 2004 was the most successful of my entire career, I owe in great part to our kitchen-table discussions. I returned triumphantly to the Top Five, along the way winning practically every tournament I participated in." (Alexander Morozevich, Foreword)

An excerpt from the book at

That's an interesting parallel. Packing density and the expansion factor (analogous to the potential energy) are original concepts introduced by Shashin.