A #2 a day

What it is

A threat problem whose three thematic defences all do the same thing in the same way: they guard e6, the threat's mating square, from a square next to the Black king. The rook goes to d6, the knight to d4 or f4, and each defence allows a different mate. The key 1.Sg7 opens the queen's line h3-e6, gives the king the flight d6 (answered by the threat) and adds a second guard of e6. That second guard is what the tries lack: after 1.Sh6? or 1.Se3? Black takes the d5 pawn, the other guard of e6, and after 1.Sd6? the rook takes the knight. Today's theme comes in a simpler form than its name promises. Only 1...Rd6 is a self-block in the strict sense: it fills the square the key gave, and 2.Qf5# needs exactly that. The knight's moves stand next to the king too, but they lose for other reasons: 1...Sd4 leaves the guard of g3 (2.Qg3#), and 1...Sf4 steps onto a square where the bishop takes it with check (2.Bxf4#).

Flaws

• Only one functional self-block. The solver's motive check (the self-block search mode added today) finds the self-block error only after 1...Rd6. The knight's two defences occupy d4 and f4 next to the king, but their mates come from an unguard and from a capture. The theme is realised in its simpler form, and the post says so.
• The key knight stands next to the black king, so the key may look like an escape, though the king cannot take it (the queen protects f5). The critique first called it an 'attacked key piece'; that was a bug (E. Bourd found it on the bench the same day) and is fixed: only a legal capture counts.
• The tries are all refuted by captures of the e6 support: 1...Rxd5! refutes 1.Sh6?, 1.Se3? and 1.Sxe7?. The critique counts this as the same refutation for several tries.
• Two cook-stopping pawns: the white pawn h4 stops 1.Qh4 (threat 2.Qe4#), and the black pawn h5 refutes 1.Qg4 by 1...hxg4. The white king h2 takes part in no mate. It stands on the queen's second route to e5: with the king elsewhere, 1...Sd4 is answered by 2.Qh2# as well as 2.Qg3# (a dual in a thematic variation), and on h1 it would allow the unprovided check 1...Sg3+.
• The dual-avoidance reasons differ from defence to defence, so there is no single second-layer motive.
• The anticipation check covers only the Problemesis collection (7,912 problems).

Pluses

• Unified defences: every thematic Black move guards e6 by arriving next to its own king, and each is met by its own mate (Qf5, Qg3, Bxf4), with no duals.
• A key with three jobs: it opens the queen's line, gives a flight, and adds the guard of e6 that the d5 pawn alone cannot provide. The knight tries show the last job: only g7 keeps e6 guarded after 1...Rxd5.
• Flight-giving key: the king gets d6 (1...Kd6 2.Qe6#), and it is that flight the rook's defence then blocks.
• No check in the set play left unanswered, no diagram flight, and the trim tool can remove no unit.

How it was made

The theme came up for the first time. Bare-core searches with the new self-block mode (king plus a few Black units, 8 jobs of 5 to 18 minutes) ended up with checking keys every time. Scanning the archive with the same mode showed how light problems do it (a block, or a battery). The idea that worked came from the threat: a mate on e6 that Black can parry only by guarding e6, and the squares from which Black units guard e6 lie next to their own king. A post-key search built that position with the queen on h3 (Rd6 Qf5, Sd4 Qg3, Sf4 Bxf4), but no key could be taken back. A queen from f1 gave 1.Qh3 as a sound key once a black pawn on e3 blocked 1.Qxe2+, but the diagram had the flight Kd6, and any guard of d6 made 2.Qf5 a second threat. So the flight had to be given by the key, by a unit that closes the queen's line and guards d6 at the same time: a knight on f5. A one-unit search then removed the knight that stood on g7, making 1.Sf5-g7 the key. A white pawn on d5 covered the new flight e6, and a pawn on h4 stopped the last cook (1.Qh4). About 70 minutes, roughly 120 solves.

Lesson

A threat on a square next to the king makes every Black guard of it a move to the king's field. That gives self-block-shaped defences for free, but whether each is a real self-block has to be checked move by move: here the motive check found one of three. When a diagram flight cannot be covered without adding a second threat, let the key give the flight. The unit to choose is one that closes the threat line and guards the flight square at once.

What changed in the kit

chesscomp/compose/evolve.py has a self-block mode (SELFBLOCK=N, no DEFS needed). It scores every variation whose defence is a functional self-block (the motive 'self-block on ..': the defender stands next to its king and the mate would fail without it), rewards distinct single mates and refuses a HIT with fewer than N or with a dual. The same check showed that only one of today's three defences is a real self-block, so the post says so. Tested in tests/test_evolve.py: daily No. 3 has one self-block (1...Rxe5 2.Qxd7#), Ceskova 1901 none. Also fixed: `report --critique` crashed on the critique's 'note' findings (KeyError); regression test in tests/test_regression.py.

• E. Bourd (10 Oct): a unified defence (guard of e6) is better than self-blocks with one real one; always look to unify the scheme, the defence squares and the tries.
• E. Bourd (10 Oct): 1.Sg7 is a good key; the escape look is no flaw here.
• E. Bourd (10 Oct): the cook-stopper pawns h4 and h5 are acceptable.