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Another way of playing with this theme: Instead of the a-file being the royal determinant file, let f1/f8 squares be royal determinant squares. If equidistant, then neither is royal. Equidistance is defined as spaces away, counted either diagonally or orthogonally. Otherwise, the king closer to the f1/f8 square is the royal king. If one king is eliminated, the other becomes royal.