Helpmate in 3: (a) diagram (b) Move f4 to e4

In the diagrammed position, the bishop on b8 is clearly poised to deliver mate. If the final move is to be Bxe5# with Black’s bishop on e4, White must first find a way to control the flight squares f3, g4, and g5. Well, a white rook on g3 just does the trick! So, if the rook on e8 could magically be whisked away to g3, Black could play Bf3–e4, allowing White to finish with Bxe5#. As it happens, there’s only one way to reroute the rook from e8 to g3 in two moves, yielding the elegant solution: 1.Qc7 Rxe3 2.Be4 Rg3 3.Qe5 Bxe5#. The black queen steps aside to let the rook through, then reverses course with a graceful switchback to facilitate the coup de grâce.

In part (b), with Black’s king on e4 instead of f4, the mantle of delivering the final blow passes to the white rook, while the white bishop takes on the vital task of controlling the flight squares. The solution—once again featuring the black queen’s switchback—is analogous: 1.Qe6 Bf4 2.Rd3 Be3 3.Qe5 Rxe5#, illustrating the Orthogonal-Diagonal Transformation (ODT), a thematic framework in which strategic effects that unfold along orthogonal lines in one phase reappear along diagonals in another, and vice versa.