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

                                                              enter image description here

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?

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  • 1
    $\begingroup$ @PotatoLatte Try this: chess.com/analysis $\endgroup$ – jafe Jul 5 at 11:46
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    $\begingroup$ lichess.org/editor allows to set up a position, and, if you click on "Analysis Board" you can also insert moves and link to the game. $\endgroup$ – shoopi Jul 5 at 14:14
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    $\begingroup$ (+1) Any reason why you chose the name wazir? $\endgroup$ – TheSimpliFire Jul 5 at 19:53
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    $\begingroup$ @TheSimpliFire en.wikipedia.org/wiki/Wazir_%28chess%29 :) $\endgroup$ – jafe Jul 5 at 20:08
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    $\begingroup$ Hah, wazir. That's a funny name :) $\endgroup$ – Mr Pie Jul 6 at 3:07
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I have a solution with

25 wazirs:
enter image description here

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.
enter image description here

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.
enter image description here

Then we need four more wazirs to cover the remaining 12 light squares.
enter image description here

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    $\begingroup$ What a pity you have to break the symmetry with that one piece, as you can't put it in the centre. Lovely solution though. $\endgroup$ – Jaap Scherphuis Jul 5 at 12:27
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    $\begingroup$ this is minimal :) $\endgroup$ – Oray Jul 5 at 12:40
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    $\begingroup$ Very nice argument, I like the diagrams. I was hoping somebody would be able to present something like this. $\endgroup$ – hexomino Jul 5 at 15:11
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Here is an attempt which needs

27

wazirs:

enter image description here

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A perhaps more elegant way to do

27 wazirs

than @Glorfindel's solution is:

Put wazirs on every square on the b, e, and h files. Alternatively, the 2nd, 5th, and 8th rank. You get the idea.

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