Earlier, I introduced the beautiful game of Swiss chess.
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.