strategy - Where are the Wazirs?

A wazir is a fairy chess piece that moves like a rook, but can go only one square.

                                                              

We wish to place a number of wazirs on a 9 $\times$ 9 chessboard so the following conditions are satisfied

1. Each wazir is being attacked by at least one other wazir.


2. Each empty square is being attacked by at least one wazir.

What is the minimum number of wazirs we need to place to satisfy the conditions above?

Answer

I have a solution with

25 wazirs:

Step by step:

We can cover all border squares with 8 pairs of wazirs whose attacking squares do not overlap. Note that each wazir covers two border squares, which is the maximum because it's not possible for a piece to be orthogonally adjacent to 3 border squares on a 9x9 board. So there is no way to cover the entire border with fewer wazirs.

Also note that this arrangement covers the maximum amount of non-border squares as well. Every wazir covering two border squares covers exactly one non-border square, unless it is located one step diagonally from a corner (in which it covers 2). Every square one step diagonally from a corner is in use, so the wazirs cover the maximum possible amount of non-border squares.

There are 17 dark squares left, so we need a minimum of 5 wazirs to cover them all.

Then we need four more wazirs to cover the remaining 12 light squares.