Troitsky line

King and two knights cannot force checkmate against a lone king. Just before completing the mating net, the defender can be left stalemated. An extra defending pawn changes the position because it supplies a legal move and may also block one of its own king's escape squares.

The Troitsky line gives a sufficient winning condition for two knights against a pawn. One knight is assumed to blockade the pawn securely. In algebraic notation, where files are named a through h and ranks 1 through 8, the classical boundary runs through a4, b6, c5, d4, e4, f5, g6, and h4. If the pawn is blockaded on the line or on the side farther from its promotion square, the knights theoretically win regardless of the kings' initial locations.

Why the pawn helps the attacker

One knight stays in front of the pawn while the king and other knight drive the defending king toward a mating area. At the correct moment the blockade is released. The defender must move the pawn, stalemate is avoided, and the blocking knight can join the mating net.

The original line does not guarantee a win within the fifty-move rule. This rule allows a draw claim after fifty moves by each player without a capture or pawn move. Some winning positions take longer, so the defender may reach that limit before checkmate. A more restrictive second Troitsky line is used to study which pawn placements allow the process to finish within that limit.