Two tries quickly suggest themselves to the solver: 1.Ra8?, threatening to move the wK off the a-file and deliver mate, and 1.Qh4?, aiming for 2.Qh1#. Both are refuted by opening the h8-queen: 1...Bh7! neutralises the first, and 1...Nxf6! the second.

One may also consider moving the knight on d2, for instance 1.Nf3, to set up 2.Rd1#, but Black again replies 1...Nxf6, unpinning the bPb2, so that 2.Rd1+ can now be met with 2...b1=Q. This failure, however, hints at the key: if wBf6 retreats along the long diagonal, then a move of the knight on d2, clearing the way for Rd1#, becomes a genuine threat.

The halfway measures 1.Be5? and 1.Bd4? are dismissed at once, failing to 1...f6! and 1...e5! respectively. The solution, indeed, is to withdraw the bishop all the way: 1.Bc3!. As noted above, this threatens 2.Nf3 (alternatively, 2.Nde4 or 2.Nf1) followed by 3.Rd1#. Black can defend only by opening a line for the h8-queen — yet each such attempt simultaneously shuts another of her lines, handing White a new road to mate:

1...Ne8 opens the h8–c3 diagonal but closes h8–a8, allowing 2.Ra8! followed by mate with the wK stepping to b8 or b7.
1...Bh7 opens the h8–a8 diagonal but closes h8–h1, so 2.Qh4! with the threat of 3.Qh1# is decisive; 2...g2, intending to guard h1, runs into 3.Qe1#.
1...Nf6 opens the h8–h1 diagonal but closes h8–c3, enabling 2.Na4!, after which 3.Bxb2# is inevitable.

A move that opens one line while closing another for the same line-moving piece is known as a valve. Here, three black pieces operate like valves, their interplay weaving an exquisite cycle as the bQ’s lines open and close in turn.

Mate in 2