Mate in 3

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.