Key squares

 So what are the key squares for the Trébuchet?

Diagram 1

What do these key squares represent? In a clean situation, the red circles represent the squares where the white king must be in order to win the black pawn in a "guaranteed" manner.

In practice, the situation is usually more murky. What the opponent is doing might influence the effectiveness of occupying the key squares. Another term you sometimes find is critical squares. Since I intend to use that idea of a double attack in a more general sense, I prefer the term invasion squares. I think.

There are other key squares too:

Diagram 2

If the white king conquers the black pawn, then the red circles indicate where the white king must be in order to guarantee the promotion of the white pawn.

If the black king conquers the white pawn, then the blue circles indicate the key squares. The white king must prevent the black king from entering the blue squares.

Al these circles have to do with opposition. The attacking king tries to enter the circles by the technique of opposition and outflanking. 

The defending king tries to defend the key squares by maintaining the opposition and to prevent outflanking.

Corresponding squares

Key squares usually come in 3. If there is no obstacle. Why 3? That is the reach of a king. He can triple attack 3 squares at the same time.

Diagram 3

The key squares are surrounded by corresponding squares. That is where the opposition takes place. When your opponent is on square a, you must occupy the corresponding square a. The first one who must give up the opposition, cannot reach the key squares.

Experiments

The green area is where the white king must be in order to win. This is all measured with the black king at a1. That is a long way from having an influence on the pawns or the white king.

White to move

The strange thing is that the "dent" a1, b1, a2, b2 is not caused by the black king. If you place the black king on h1, the green area would be the same.

Let me change the place of the black king, keeping the white king on h3.

White to move

• When the black king is on the green squares it is a win for white
• When the black king is on the red squares it is a win for black
• When the black king is on the yellow squares it is a draw

Let me try to draw a few conclusions.

In order to win, you must accomplish the following:

• conquer the enemy pawn
• conquer the key square to secure promotion

Hence there are two battles going on. One for each series of key squares.

• The green squares indicate that black is losing when his king is too far from the key squares in comparison to the white king. Since white king is already on a key square once he conquers the black square, black cannot make a draw.
• Black can only win if he can conquer the white pawn before the white king can reach d3.
• In all other cases, where white cannot conquer the black pawn BUT is in time to reach d3, it is a draw

Apparently, one of the pawns must drop. In the next post I will investigate that mechanism further.