A #2 a day

What it is

The key piece picks its square on a line. The bishop f3 must move to open the third rank for 2.Rd3#, and it must stay on the long diagonal to keep e4 and d5. Every square on that diagonal except the corner has an error of its own. On g2 it shuts the second rank, so 1...Bg3, which blocks the third rank and gives up the guard of d2, no longer allows 2.Rd2#. Beyond d5 (c6, b7, a8) the pawn d6 can step in between it and e4 (1...d5!). On e4 or d5 the king takes it. From h1 nothing can come between the bishop and e4, and it shuts no White line.

Flaws

• Thin play: one variation, 1...Bg3 2.Rd2#, and it is already set in the diagram with the same mate. The key only adds the threat; nothing changes and nothing new appears.
• 1.Bd5? and 1.Be4? also threaten 2.Rd3#, and only the king refutes them (1...Kxd5!, 1...Kxe4!). They are not given as tries, but a solver will try them.
• One refutation for three squares: 1...d5! answers 1.Bc6?, 1.Bb7? and 1.Ba8? alike. It is one error (the line can be cut behind d5), but only 1.Bc6? is worth writing down.
• Duals after moves that do not defend: 1...Bf2 2.Rd3#/c3# (the bishop leaves c3), 1...Bd2 2.Rd3#/Rxd2#.
• A unit that only stops a cook: the black pawn h4 shuts the h-file (without it 1.Rh5! threatens 2.Rd5#). Five pawns, and c3 is guarded twice (Sa4, Sd1).
• Most black moves are unprovided in the diagram (1...Ba5, Bb4, Bc3, Bf2, d5): only 1...Bg3 and 1...Bd2 have set mates.

Pluses

• The corner has a reason the solver can check square by square: g2 shuts a White rook, c6/b7/a8 let a black pawn cut the line, e4 and d5 hang. Only h1 is clean.
• The two thematic tries fail to two different black units for two different reasons: a White line closed by the bishop itself (1.Bg2? Bg3!) and a black line-cut (1.Bc6? d5!).
• The refutation of 1.Bg2? is the key's only variation: 1...Bg3 2.Rd2# needs the second rank that the try closes. Try and variation are one idea.
• Quiet key, no flight, no checks in the set play, no promoted force; no unit can be removed with the same key and play (trim). 13 units.

How it was made

Theme from the rotation: corner key, first time on the blog. Its reference shows a Bristol into the corner, and yesterday's post was a Bristol, so the run looked for another reason. The archive's rich corner keys are line selections: Ahues (Die Schwalbe 1998, 1.Qa8!: each queen post can be cut by the knight d4), Pachl (Schach-Echo 1989-90, 1.Bh8!: four bishop tries, four refutations) and Chepizhny (diagrammes 2000, 1.Ba1!). The run first gave evolve a TRIES mode for fixed keys (see below). Evolve and two scratch searches (a random fill and a hill climb on python-chess alone) then ran around queen and bishop cores for about 50 minutes and found nothing usable: bare seeds gave no key, and every sound queen-to-h1 position took a flight (the corner queen guards h2/h3 from far away). By hand: a bishop d5 masking Rd8 was cooked by the rook's other files; a queen e4 masking Ra4 gave the king flights on the fourth rank in the diagram. The final scheme came from boxing the king on d4 with pawns and knights, so that the bishop f3 masks only the third rank (Rh3, boxed by the black pawn h4). A black knight on f1 was the first defender. It guards d2 against 2.Rd2#, but it reaches both g3 and e3, so both interpositions got the same mate and 1.Bg2? had two refutations. A black bishop on e1 does the same job with a single interposition (g3), and that made the try pure. About 60 solves.

Attempts that failed

Lesson

A corner key needs a reason for the corner, not only for the line: give every other square of the line its own error (a White line it shuts, a black cut it allows behind it, a capture) and the corner is the square without one. For a single refutation, the defender must have only one way to do it: a knight on f1 could interpose on g3 and e3, a bishop on e1 only on g3.

What changed in the kit

evolve's fixed-key mode has TRIES=N: it rewards tries by the key piece whose single refutations are distinct and answered after the key, and a HIT needs N of them, whatever the threat's line (before, this term existed only inside BRISTOL=1); the term is now one function (thematic_tries) shared with the Bristol mode, tested on E. Bourd's Bristol prototype (TRIES=2 hit, TRIES=3 not). Also the open proposal from No. 10 is done: python -m chesscomp.compose.trytable FEN [SQUARE ...] lists every move of a unit with its threat, refutations and mates in about a second (this problem was built with it), tested on today's position and its knight version.