# There is no winning move in chess

A mathematician is not necessarily a good chess player. But for a mathematician a little involved in the mathematics of games and especially in decision trees, the vocabulary used to comment games seems absurd. In fact the words usually used should be ... inverted!
Is there, for example, an acceptable concept of "good move"? The answer is unfortunately negative. There are only "bad moves"!
To begin with, one has to classify the types of positions in the game. Obviously, there are positions where one side has lost. They can be called W- (white side is lost) and B- (black side is lost). But a precise definition of "lost game" is needed. Without entering into the workings of the formalism of the trees, the game is lost to the player P when, whichever move he plays, there is at least one replica move of the opposing player that maintains the player P in a "lost game" position. This definition is necessarily recursive. Of course, a "final" position where a player plays - for example - with a king and a queen against a single king is "lost" for the king alone. But the recursion mechanism operates at infinity, and therefore any position falls into one of the three categories W-, B-, and equal (=). An equal position can only be defined a contrario. A position is equal if none of the players has lost, ie if it is neither W- nor B-. This is true for all the positions of the game, as complex as they are, and therefore even for the starting position. There is no intermediate concept, nor a 4th category! In strict logic, there is no "advantageous", "favorable", "difficult" position ...
The set of losing positions of a player (of a camp) is called a "losing kernel", or simply a "kernel".
That being established, the moves can be classified in turn. There are exactly six categories of moves. Here P represents W or B, depending on whether the trait is for white or black, and O represents the color of the player's opponent. The possibles moves are:  ==, OO, PP, O=, OP and =P.

1. Moves labeled == occur when none of the players is in a kernel, and when the move played does not change anything. This is probably the case with the most frequent opening moves, and this is probably the most common type of move in early and mid-game, especially by good players. It's a neutral move.
2. Moves labeled =P occur when none of the players is in a kernel, but the move selected by a player puts him in a lost position (in the kernel). This is the case of obvious faults, but also of many moves which are possibly very difficult for us to identify as fault. It is a losing move.
3. Moves labeled O= occur when the opponent is in a kernel, but the move played saves the opponent from this situation. It is a move renouncing to the gain.
4. Moves labeled OP occur when the opponent is in a kernel, but the move played not only saves the opponent from this situation, but also puts the player in a lost position. It is a big mistake, a blunder.
5. Moves labeled OO occur when the opponent is in a kernel, and the move played keeps the opponent in a lost position, hence in the kernel. These moves are played by players who have a decisive advantage and keep it. This is sometimes obvious, but of course it is extremely difficult, even for an excellent player to distinguish it at once from a ==. This is a positive neutral move.
6. Moves labeled PP occur when the player is in a kernel, and the move played can't have any other effect than to keep him in that kernel. These moves are played by the players who have a decisive disadvantage and necessarily maintain it. This is sometimes obvious, but of course it is extremely difficult, even for an excellent player to distinguish it at once from a ==. It is a negative neutral move.

There is no 7th category of move! There is no "good move" and no "winning combination"!
It is funny to note that the games comments often emphasize "winning moves", although this notion does not exist. If a super-intelligence alien saw a game, he would always answer something like, "It's not a winning move, it's just one of the OO move that the player played since his opponent has given him the pleasure of playing a =P move. It was not he who played well, it was his opponent who gave him a winning path by placing himself in the kernel. "
In the same way, it is amusing to note that the "problems" presented to chess amateurs do not in fact aim to find a winning move, but rather to find a unique positive neutral move (OO) among a vast set of moves renouncing to the gain (O =) or worse (OP). It is not a question of finding a good move that would move the balance of the game, but rather of avoiding bad moves that would compromise the favorable imbalance present in the problem definition. Farewell then, from a technical point of view, to the "genial" move, the "fabulous" combination, the "diabolical" sacrifice. They are at best only positive neutrals. But fortunately for our limited minds, the personal satisfaction gained when finding a positive neutral move is already ... magical.

## A simple equivalent to the game of chess

Here is a description of a game that from a formal point of view has the same properties as the game of chess. This game is called the "Three-Six", because it involves only 3 squares and only 6 possible moves. The game opposes two players: white and black. The playing surface contains 3 squares: a gray square in the center, a white square close to the white player and a black box close to the black player. A token is placed on the gray square at the beginning of the game. The two players can alternately play, starting with the white player. The players can:

• do nothing (it is a == if the token is on the gray square, a PP move if the token is on the square of the player's color, and a OO move if the token is on the colored square opposite to that of the line player)
• bring the token back one or two squares towards himself. For the white player, he can bring the token from the black square to the gray square (move O =), bring the token from the gray square to the white square (move = P), or bring the token from the black square to the white square (OP move). The black player is in the symmetrical situation.

The game ends when both players agree. The game is then declared null if the token is on the gray box, it is declared lost by the white side if the token is on the white square, and it is declared lost by the black side if the token is on the black square. In reality, there are slight differences in relation to the game of chess, regarding the draw by repetition, the stalemate situation, the modalities of nullity mutual agreement, etc ... Of course in the game "Three-Six ", a good player always chooses to do nothing. Moving the token is always the act of a weak player. But for extraterrestrials with an absolute intelligence, the observation of our chess game would astonish them. Why, then, would they say, do these humans play the trivial "Three-Six" game in such unnecessarily complex terms?