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Earlier, I introduced the beautiful game of Swiss chess.

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These are animated gifs. You may have to click on them to make them move.

Unlike the peace-loving Swiss, we are interested in "domination", the art of threatening or occupying every square of the board using a given set of pieces, four Swiss queens for this puzzle.

But how to persuade four pacifist queens to do such a crude and undemocratic thing?

Luckily, Swiss queens have one well documented weakness: they will do anything to achieve symmetry, especially mirror symmetry.

With that in mind our road to domination should be clear:

Place four Swiss queens on the board such that the position is mirror symmetric and every square is either occupied or threatened.

For reference here are the modified rules of movement in Swiss chess:

affected pieces: king, knight, bishop and queen (moves like a rook or bishop) from squares where the standard pieces have maximum mobility (king: 8 squares, knight: 8 squares, bishop: 13 squares, queen: 27 squares) Swiss pieces are no different from standard pieces. on other squares additional destinations are added to keep mobility at the set level. Exceptions: king and knight in or directly next to the corner have only 7 possible squares, queens on the edge have fewer destinations because of the overlap between rook and bishop patterns

 king: a1 -> a2 a3 a4 b1 b2 c1 d1
       a2 -> a1 a3 a4 b1 b2 b3 c2
       a3 -> a1 a2 a4 a5 b2 b3 b4 c3
       a4 : same as a3 shifted

knight: a1 -> a2 a3 b1 b2 b3 c1 c2
        a2 -> a1 a3 a4 b1 b4 c1 c3
        a3 -> a1 a2 a4 a5 b1 b5 c2 c4
        a4 : same as a3 shifted
        b2 -> a1 a3 a4 c1 c4 d1 d3
        b3 -> a1 a2 a4 a5 c1 c5 d2 d4
        b4 : same as b3 shifted

 bishop: a1 -> a2 a3 a4 b1 b2 c1 c3 d1 d4 e5 f6 g7 h8
         a2 -> a1 a3 a4 a5 a6 a7 b1 b3 c4 d5 e6 f7 g8
         a3 -> a1 a2 a4 a5 a6 a7 b2 b4 c1 c5 d6 e7 f8
         a4 -> a1 a2 a3 a5 a6 a7 b3 b5 c2 c6 d1 d7 e8
         b2 -> a1 a3 a4 a5 c1 c3 d1 d4 e1 e5 f6 g7 h8
         b3 -> a1 a2 a4 a5 a6 a7 c2 c4 d1 d5 e6 f7 g8 
         b4 -> a1 a2 a3 a5 a6 a7 c3 c5 d2 d6 e1 e7 f8 
         c3 -> a1 a5 a6 b2 b4 d2 d4 e1 e5 f1 f6 g7 h8
         c4 -> a1 a2 a6 a7 b3 b5 d3 d5 e2 e6 f1 f7 g8

For a procedural rule of bishop movement: observe that no matter where the bishop is positioned there is at least one pair of (traditional bishop) directions comprising the maximum of 7 squares. We complement those 7 by another 6 on the remaining two directions making a 45 degree bend if we would otherwise run off the board. The exact placement of the 6 squares is governed by the following: If the bishop is on the main diagonal we allocate them symmetrically, 3 on each side. Otherwise, determine the longer side of the bounding box of the 7 "free" squares. The 6 other squares will span this axis minus the farthest away square.

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  • $\begingroup$ @bobble So many rules... (Done.) $\endgroup$
    – Albert.Lang
    Mar 4, 2021 at 17:31
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    $\begingroup$ Speaking of "so many rules" ... your rules for where a "Swiss bishop" (hence a Swiss queen) can move seem, on the face of it, pretty arbitrary. Is there a short, principled way to explain which squares it can move to, or is it just a matter of "just look it up in this list, that's all there is to it"? $\endgroup$
    – Gareth McCaughan ♦
    Mar 4, 2021 at 17:33
  • $\begingroup$ Touche @GarethMcCaughan. There is indeed method in this madness. I've updated the post and tried to explain. $\endgroup$
    – Albert.Lang
    Mar 4, 2021 at 17:59

2 Answers 2

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I think this arrangement works

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with mirror symmetry on the diagonal.

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We can also have rotational (180° only, though) and mirror symmetry at the same time:

    . . . . s . . . 8
    . Q . . . . . . 7
    . . . . . Q . . 6
    . . . . . . . s 5
    s . . . . . . . 4
    . . Q . . . . . 3
    . . . . . . Q . 2
    . . . s . . . . 1
    a b c d e f g h

Please note that only four squares really use Swiss rules (labeled s in the diagram). All others are covered by plain chess moves.

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