Mate in 2: queen to h8 (b2·3)

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Corner mate

Driven into the corner, the king runs out of squares.

The corner mate catches a king driven into the corner, where it has only three neighboring squares — all of them taken away or blocked. The corner is the worst home for a king, because the attacker needs to control the fewest squares there. Herd the king toward the corner with checks and never let it slip back toward the center. Cut off the escape routes before the final check, and the mate practically assembles itself.

Only move

Everything else loses — find the only move that holds.

An only move is a position where everything loses except one single continuation. You cannot play "by hand" here: run through fundamentally different defenses — run away, block, trade, counterattack — and discard the ones that fail. The hallmark of these puzzles is that the obvious moves lose quickly while the solution often looks strange. First understand exactly what the threat is — then the one saving move reveals itself.

Mate in 2

A forced checkmate in two moves.

Mate in two: your move, any reply, then the mating blow. Start with the most forcing first moves — checks and sacrifices that shrink the opponent's options to almost nothing. Against every single reply you must have a mate ready, so check them all without exception. The common failure is analyzing the pretty defense and forgetting the boring one.