Chess, as it has often been said, has more possible strategies than there are atoms in the universe. Yet chess is a finite game because of certain rules; for example, the game is declared drawn if a specific position is repeated three times, or if fifty moves pass without a capture or pawn move. Chess is, as well, a sequential non-cooperative game of perfect information. Thus, at least in theory, chess is amenable to what is known as rollback analysis (starting from some "final" position and working backwards, as many chess composers do) and can be solved. However, chess contains so many hidden secrets that the game will become extinct neither in our lifetimes nor in the generations to come. Solving chess problems requires more of an artist’s mind than that of a scientist because one has to sense the most promising reply without analyzing all the possibilities; in reality, what the master is performing is a modified game tree, using exactly the practices that one would find in the field of game theory.

Technically speaking, chess is a science with a solution; yet, the multitude of chess positions is so great, that to try to analyze every possible move “until the end” is not only impossible for both humans and computers but also unnecessary. Even the simple two by two tic-tac-toe game, when analyzed in full, leads to a complex tree, with forty-eight different possibilities, or “subgames.” In chess, after the second move there are over seventy-two thousand possible positions, after the third, there are nine million possible positions, so on and so forth. To try to “solve” chess by memorizing all the different outcomes to every single position is not what produces (and not what has produced in the not-so-distant past) the greatest chess minds.

Mastery of the game requires not an acute calculation but rather an intuition based on years of experience; it is most important to train and perfect one’s intermediate valuation function, which basically involves weighing up all the rational options in a given problem and evaluating each solution. The former World Champion Tigran Vartanovich Petrosian once said that chess is about feeling where each piece should go so that it is well situated to meet the many plans of the opponent, or as Vassily Vassiliyevich Smyslov explained, it is important to sense the “harmony” between the pieces, likening chess to a “spiritual experience.” Former champion Anatoly Yevgenyevich Karpov said that he “seeks beauty” in chess, and even the great Bobby Fischer expressed that the most important quality to succeed was nothing but a “genuine love for the game.” The nineteenth-century American genius Paul Morphy’s cryptic quote “help your pieces so that they can help you” is explained through the Hungarian Geza Maroczy’s remark, “it was Morphy who proved that the best in any given position is the most beautiful.” Thus, the great chess giants were able to find the most “beautiful” move in any given position, not exactly using their “natural synchronization with the heavens” as Smyslov claims, but rather a well-trained intuition. In terms of game theory, we will denote this aspect as subconscious mastery of game theory techniques and the intermediate valuation function.

In his famous book *Think Like a Grandmaster*, Alexander Kotov stresses the importance of using trees in solving difficult chess problems. Practice has shown that only a few players have mastered the technique of analysis; even highly rated players are lacking in this respect (Kotov, 1952). Kotov elaborates on what he terms the “tree of analysis.” He stresses that before calculating variations; a master should find the “candidate moves” in the position. It is the very difficulty of the analysis, that one must find the candidate moves not only at the first branch but also at all the other branches, and also the candidate moves for the opponent. However, what Kotov does not really elaborate on is that the most difficult in a position is sometimes not the calculation of variations but rather the finding itself of the one right move in the critical position. For Kotov, the art of chess is that some players like to sacrifice, while others rely on stubborn defense, and that this principle holds true even at the top levels. All this is true, since equilibrium exists in multiple ways. However, sometimes there is only one correct route in a position, and this aspect of the players’ “personalities” is more of a spectacle then the pure art, which is really the finding of the correct move through the deep inner logic of chess. To illustrate, consider the following position, taken from an example by the distinguished trainer Mark Dvoretzky:

An experienced master, when pondering this position, plainly sees that white has numerous far-advanced pieces on the Kingside, where Black’s king lies. Thus he will try to find a way to continue his attack on Black’s king. The master understands that white has numerous tempting possibilities, which makes the analysis all the harder. To the inexperienced, it is not clear what white should do, but even the best chess players will examine several moves. By constructing a game tree, one can, using his intermediate valuation function, find the solution to this difficult problem. After constructing the tree, the use of rollback is necessary to choose the most promising option.

First the master will construct the beginning nodes of the tree. He will examine several possibilities – just enough so that he is not missing anything, but just enough so that he does not become overwhelmed with variations. Here is the use of his intuition; the master is able to unconsciously discard unnecessary options. First the master will construct the branch for the first node, 22.Qh4 (defending his bishop and simultaneously attacking the one on e7, also bringing the Queen closer to Black’s king). However, instantly he sees that he doesn’t like so much his chances after 22…Bxg5 23.Qxg5 f6, as the attack is at its end and the initiative is handed over to Black. Note that he stops there, and he does not search deeper. A less experienced player would, and we show that this approach wastes time and energy. So, if he captures the knight, with 22.exd5, then Bxg5 23.d6 (or 23. Qe4 g6, 23.Be4 Qc4) 23...Qc5, with what the master deems as “mutual chances.” Maybe, he thinks, he will just capture the bishop: 22.Bxe7 Nxe7; no, let’s stop, he can sense that he has nothing here. So what about taking on e7 with the Knight? After 22…Nxe7 23.h6 (stronger than Bc1), he gets a good game after any of Black’s replies.

Assuming rationality, assigning relative “payoffs” to each variation, and using the principle of rollback analysis, it appears that if he captures the Bishop with the knight and plays 23.h6, then he gets payoffs better than anything else in the whole tree (1 and .8, respectively). Thus the game is solved and the master should choose capture the Black bishop with his knight: 22.Nxe7.

The master makes his move, the game continues, and, draw. He goes back to his hotel room that night and analyzes the game, tries to make conclusions. And he finds that, according to Dvoretzky, "all that separated White from winning this game; however, was one move – one which he did not include in our initial list of candidates. Had he been able to find it first, to sense it, we could have avoided all those complex calculations, as it would have been immediately clear which continuation was the strongest." (Dvoretzky, 2012)

22. Bg5-h4!!

Now we can expand the tree to examine Black's possibilities and see that, in fact, this move is the best solution. So, going back to f6 does not make sense; the sacrifice on c3 does not work; and on 22...Bxh4 23. Qxh4 Nf4, white can either take the exchange or continue his attack by 24. h6!

The tree is complex and contains many branches, but it can be used to solve the problem effectively. For the master, calculation is relegated into the background, for finding the right move through imagination is the most critical and is more important than knowledge of the chess “multiplication tables.”

It is interesting to note that there is another possibility in the starting position, namely one that I did not even consider, namely 22.Bh6!?. The move is very interesting, but one thing is clear: it is clearly inferior if we have found 22.Bh4. Thus, here is an example of the well-trained intuitive mind discarding such moves without even considering them. It is not important to see all the possibilities but rather to correctly choose the right one, which in fact proves the superiority of the greatest human minds over the computer’s “brute-force” calculation. As Yasser Seirawan says, “I don’t have to pretend that I’m seeing everything. I make no such pretentions. But what I am seeing, almost as instantly as the computer is calculating and reassessing everything, is the soul of the position, and I’m right there.”

**Beauty in the Endgame**

The most used in rollback are chess studies, often referred to as compositions, or “chess poems” although a beautiful game can only be considered a composition. Usually there is a certain remarkable position, which has been previously proven to produce a certain result, and it is worked backwards to get that position. Smyslov wrote that composing and solving studies helped to develop his understanding of the aesthetic elements in chess and perfected his technique in endings. We can see how complex and beautiful chess really is. For example, take the famous study composed by Richard Reti in the early twentieth century (White king on h8, pawn on c6; Black King on a6, pawn on h6). At first glance it appears that Black’s pawn cannot be stopped, while white’s pawn can be halted easily. However, every educated chess-player knows the solution: the White king goes to e5 when the Black pawn is on h4, and draw: if h3, then Kd6, and the pawns queen simultaneously; if Kb6, then Kf4, and the King catches the pawn. Thus, using a sort of intermediate valuation function, even the player who does not know the solution can figure it out: Kg7.

The Soviet chess composer A. K. Sarytschev also manifests the beautiful use of the King in a composition. Looking at the position (White King d7, pawn c7; Black king f3, Bishop h7, pawn b7. It appears that White is destined to lose. Drawing the tree in our head, we can see that c8=Q is followed by Bf5+, and when the King moves away, then Bxc8 Kxc8 b5 and wins. If Kd6, then Bf5 followed by Bc8 and wins. Should white give up hope? No, let us be patient and finish the tree. On Kc8, then b5. Kb7 loses to Bf5. Kd7, then b4, Kd6 Bf5 but now Ke5! And the King stops the pawn in time. To end, here is an original “study,” using Sarytschev’s position.