The Troitsky line is the boundary of squares showing when two knights win against a king and one pawn: if the pawn is safely blockaded on or behind the line, the knights win.
Two knights cannot force mate against a bare king: they can build the mating net, but the final position is always stalemate before it is mate. Paradoxically, giving the defender a pawn can lose the game for them, because the pawn supplies the spare tempo moves that let the knights finish the net without stalemating.
The Russian composer Alexei Troitsky spent years mapping exactly when this works. His answer is the Troitsky line: for a black pawn, the squares a4, b6, c5, d4, e4, f5, g6 and h4. If the pawn is securely blockaded by a knight on or before that line, the side with the knights wins: one knight holds the pawn, the king and the other knight herd the defending king into a corner, and the blockading knight arrives at the end to mate just in time.
There is a practical catch: some of these wins take enormously long, well beyond fifty moves in places, so over the board the fifty-move rule can rescue the defender even in a theoretically lost position. Still, knowing the line tells you instantly whether to press or to shrug and split the point.
Against a bare king the mating net always produces stalemate one move too early. An enemy pawn removes the stalemate: it still has moves, so the defender is never stalemated, and the knights gain the tempo they need to mate.
Then the win is no longer guaranteed and most positions are drawn, because the pawn promotes too quickly once the blockading knight leaves to join the mating attack. Detailed analysis gives some extra wins depending on the kings, but the line is the practical rule of thumb.
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