Corresponding squares 4

 There are 3 systems that are used to describe correspondent squares. The reason for that is that you have to deal with two problems:

• ambiguous squares
• blocked squares

A square can be blocked for two reasons:

• it is covered by an enemy pawn
• it is occupied by your own pawn

I have no complete description of these 3 systems. I think this is how some people call them:

• 8-square system
• quadratic system
• triangular system (not sure of the name. Is it the same as the T-system from Zinar?)

The 8-square system is said to be the most complete. If there is no room to use all squares because some squares are blocked, the quadratic system is used. If there are ambiguous squares, the T-system is used. At least, that is what I suppose.

I experimented with unifying the 8-square system and the quadratic system. At first glance that worked. But it is not battle tested.

Let's go a bit deeper.

This is the position:

White to move

7k/1p6/1P2p3/1P2P3/4P1p1/6P1/8/K7 w - - 0 1 

In an ideal world, the 3x3 system works. How does that look like?

White to move

The 3x3 is centered around the digit 5. The choice where to place the 5 is related to the chain (maybe not the best name). The chain is the shortest path from invasion square to invasion square. Like this:

White to move

There is some triangle between two squares of the chain and the 5.

The 3x3 square extends as follows:

White to move

More of it:

White to move

The premise is that the 9 squares are really different. If the black king is on a 7, you really must step with the white king on a 7.

The 8-square is equal to this 3x3 (cubic?) system. With the difference that one square is blocked. How does the 3x3 system copy to black? Remember, the base is the chain!

White to move

The correspondent squares are build around the shortest path from one invasion square to the next invasion square.

White to move

If black reaches the chain. white must step on the chain too, and well exactly on the square with the correspondent number. It works like opposition. Black can't maintain the opposition, and must choose towards which invasion square he wants to go. Then white goes towards the other invasion square. So it is all about reaching the chain while maintaining the opposition.

White to move

Square 5 is in contact with two squares of the chain in the middle. Square 1 is in contact with two squares of the chain at the end.

You can see that black has two squares with a 1. These squares (e8 and f8) are ambiguous. They have an  equal distance to the squares of the chain.

To be continued. . .