Troitsky line: endgame with two knights

Troitsky line

Definition

The Troitsky line is a classic endgame concept that identifies, for each pawn file, the furthest square to which a defending pawn may be advanced and still allow the stronger side (with two knights and a king) to force a checkmate against the defending king. If the pawn is blockaded on or behind this line by one of the attacking knights, then with correct play the side with two knights can win. If the pawn has advanced beyond the line, the position is generally a theoretical draw (with occasional exceptional wins depending on piece placement).

The line is named after the Russian endgame composer Alexey Troitsky, who formulated it in the early 20th century. Modern tablebases have largely confirmed its correctness, refining some edge cases.

The squares of the Troitsky line

Conventionally given for a pawn advancing from the 2nd toward the 8th rank (i.e., a White pawn), the Troitsky line consists of the following squares:

• a4, b4, c5, d6, e6, f5, g4, h4

Interpretation:

• a- and h-pawns: 4th rank
• b- and g-pawns: 4th rank
• c- and f-pawns: 5th rank
• d- and e-pawns: 6th rank

For a pawn advancing in the opposite direction (a Black pawn going toward the 1st rank), mirror the ranks across the middle of the board (e.g., a4 ↔ a5, c5 ↔ c4, d6 ↔ d3, etc.).

Usage in chess

The Troitsky line is used to evaluate endgames of KNN vs KP (king and two knights versus king and pawn). It tells you whether it is worth steering into such an endgame, or whether you should aim to establish a blockade with a knight on a specific square before attempting to drive the opposing king to a mating net.

Practical steps for the stronger side:

1. Blockade the pawn with a knight on the Troitsky-line square for that file.
2. Coordinate your king and the other knight to restrict the defending king.
3. Drive the defending king toward a corner (often—but not always—the corner opposite the pawn’s file).
4. At the right moment, allow the pawn to move (or force it) to avoid stalemate and complete the mate with the two knights.

Strategic significance

Two knights cannot force mate against a bare king; the defender needs a pawn to avoid stalemate at the critical moment. The Troitsky line quantifies how far that pawn may be advanced and still allow the win. Central pawns are most dangerous for the defender to advance (the line rises to the 6th rank on d/e), while rook and knight pawns must be kept further back (4th rank on a/b/g/h).

In practical play, this knowledge guides whether to trade into KNN vs KP and how urgently to establish the knight blockade. If the defender’s pawn has already crossed the line, the stronger side should generally avoid the trade, as the endgame is usually drawn with correct defense.

Examples

• c-pawn case (pawn advancing upward): If the defending pawn is on or behind c4 and the attacking side can plant a knight on c5 to blockade it (e.g., pawn on c4, blockade on c5), the position is theoretically winning for KNN vs KP, given time to mobilize the king and the second knight.
• d-pawn case: The Troitsky line square is d6. If the pawn is on or behind d5 and a knight can firmly occupy d6 in front of it, the game is theoretically won for the side with two knights—provided the blockade is maintained while the king and the other knight drive the opposing king to a corner.
• Rook-pawn case (a- or h-pawn): The line is the 4th rank. For instance, for a pawn heading toward a8, a knight blockade on a4 (with the pawn on a3 or a2) falls within the winning zone; beyond a4 (e.g., pawn on a5 without a blockade in front), defense is typically easier and often drawn.

Practical visualization tip: draw a “hill” across the board—4th rank on the edges, rising to the 6th rank in the center: a4–b4–c5–d6–e6–f5–g4–h4.

How the win is executed (outline)

The winning method follows a consistent pattern:

1. Fix the pawn with a knight on the Troitsky-line square; the other knight and your king exert a net.
2. Shepherd the defending king toward a corner while keeping the pawn frozen (or under tight control).
3. At the final stage, release the pawn at a precise moment to give the defender a legal move (preventing stalemate) and deliver a prearranged mate with the knights and king.

Typical mating nets feature the defending king trapped in a corner, one knight delivering check while the other covers escape squares and the pawn’s last move breaks stalemate.

Important caveats

• If the pawn is beyond the Troitsky line, the endgame is usually drawn, although there are exceptional positions where the win is still possible (e.g., if the defending king is already badly boxed in).
• The 50-move rule matters: some theoretical wins can require long maneuvers. In practice, the stronger side often engineers the pawn’s advance or a capture to reset the 50-move count at key moments.
• You must establish and keep the blockade. If the pawn breaks free early, defensive chances increase drastically.

Historical notes and anecdotes

Alexey Troitsky (1866–1942) is widely regarded as the father of the modern endgame study. He introduced the Troitsky line in analysis published in the early 1900s. Decades later, computer tablebases confirmed the essence of his findings, a remarkable validation of pre-computer endgame theory. Troitsky himself died during the Siege of Leningrad, but his analytical legacy remains central to endgame literature.

Related concepts

• Two knights vs pawn endgame
• Blockade
• Zugzwang and stalemate-avoidance technique
• Tablebase-confirmed endgame theory

Quick reference

• Troitsky line squares (pawn advancing upward): a4, b4, c5, d6, e6, f5, g4, h4
• Mirror these ranks for a pawn advancing downward.
• Blockade on or in front of the line to aim for a win with KNN vs KP.