The two knights endgame is a

chess endgame In chess and other similar games, the endgame (or end game or ending) is the stage of the game when few pieces are left on the board. The line between middlegame and endgame is often not clear, and may occur gradually or with the quick exchange o ...

with a

king King is the title given to a male monarch in a variety of contexts. The female equivalent is queen, which title is also given to the consort of a king. *In the context of prehistory, antiquity and contemporary indigenous peoples, the tit ...

and two

knights A knight is a person granted an honorary title of knighthood by a head of state (including the Pope) or representative for service to the monarch, the Christian denomination, church or the country, especially in a military capacity. Knighthood ...

versus a king. In contrast to a king and two

bishops A bishop is an ordained clergy member who is entrusted with a position of authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance of dioceses. The role or office of bishop is ca ...

(on opposite-colored squares), or a bishop and a knight, a king and two knights cannot

checkmate Checkmate (often shortened to mate) is any game position in chess and other chess-like games in which a player's king is in check (threatened with ) and there is no possible escape. Checkmating the opponent wins the game. In chess, the king is ...

against a lone king (however, the superior side can force

stalemate Stalemate is a situation in the game of chess where the player whose turn it is to move is not in check and has no legal move. Stalemate results in a draw. During the endgame, stalemate is a resource that can enable the player with the inferior ...

). Although there are checkmate positions, a king and two knights cannot them against proper, relatively easy defense. Paradoxically, although the king and two knights cannot checkmate of the lone king, there are positions in which the king and two knights can force checkmate against a king and some additional material. The extra material of the defending side provides moves that prevent the defending king from being stalemated. The winning chances with two knights are insignificant except against a few pawns. These positions were studied extensively by

A. A. Troitsky Alexey Alexeyevich Troitsky (russian: Алексе́й Алексе́евич Тро́ицкий; March 14, 1866 – August 1942; also ''Alexei'', ''Troitzky'', ''Troitzki'') was a Russian chess theoretician. He is widely considered to have ...

, who discovered the Troitsky line, a line on or behind which the defending side's pawn must be securely blockaded for the attacking side to win. If the side with the knights carelessly captures the other side's extra material, the game devolves to the basic two knights endgame, and the opportunity to force checkmate may be lost. When the defender has a single pawn, the technique (when it is possible) is to block the pawn with one knight, and use the king and the other knight to force the opposing king into a corner or nearby the blocking knight. Then, when the block on the pawn is removed, the knight that was used to block the pawn can be used to checkmate.

Checkmate possibilities

In general, two knights cannot force checkmate, but they can force stalemate. Three knights can force checkmate, even if the defending king also has a knight or a bishop.

Edmar Mednis Edmar John Mednis ( lv, Edmārs Džons Mednis; March 22, 1937 – February 13, 2002) was a Latvian-American chess player and writer of Latvian origin. He was awarded the title of Grandmaster by FIDE in 1980. Biography Mednis' family were ...

stated that this inability to force checkmate is "one of the great injustices of chess." Unlike some other theoretically drawn endgames, such as a rook and bishop versus rook, the defender has an easy task in all endings with two knights versus a lone king. Players simply have to avoid moving into a position in which the king can be checkmated on the next move, and there is always another move available in such situations.

Two knights

In the corner

The player with the lone king has to make a blunder to be checkmated. In this position, 1.Ne7 or 1.Nh6 immediately

s Black. White can try instead: : 1. Nf8 Kg8 : 2. Nd7 Kh8 : 3. Nd6 Kg8 : 4. Nf6+ and now if Black moves 4...Kh8?? then 5.Nf7# is checkmate, but if Black moves : 4... Kf8! then White has made no progress.

Johann Berger Johann Nepomuk Berger (11 April 1845, Graz – 17 October 1933) was an Austrian chess master, theorist, endgame study composer, author and editor. In September 1870, he won the first tournament in the Austro-Hungarian Empire at Graz. In 1875, ...

gave this position, a draw with either side to move. With White to move: : 1. Nf5 Kh8 : 2. Ng5 Kg8 : 3. Ne7+ Kf8! (Black just avoids 3...Kh8? which leads to a checkmate on the next move with 4.Nf7#) : 4. Kf6 Ke8 and White has made no progress. With Black to move: : 1... Kh8 : 2. Nf7+ Kg8 : 3. Nh6+ Kh8 : 4. Ng5 gives stalemate.

On the edge

There are also checkmate positions with the inferior side's king on the edge of the board (instead of the corner), but again they cannot be forced ( + ). In the position at right, White can try 1. Nb6+, hoping for 1...Kd8?? 2.Ne6#. Black can easily avoid this with, for example, 1... Kc7. This possible checkmate is the basis of some problems (see below).

Examples from games

In this position from a 1949 game between

Pal Benko Pál C. Benkő ( hu, Benkő Pál; July 15, 1928 – August 26, 2019) was a Hungarian-American chess player, author, and composer of endgame studies and chess problems. Early life Benko was born on July 15, 1928 in Amiens, France, where his ...

and

David Bronstein David Ionovich Bronstein (russian: Дави́д Ио́нович Бронште́йн; February 19, 1924 – December 5, 2006) was a Soviet and Ukrainian chess player. Awarded the title of International Grandmaster by FIDE in 1950, he narr ...

, Black underpromoted to a knight. Black did not promote to a queen or any other piece because White could

fork In cutlery or kitchenware, a fork (from la, furca 'pitchfork') is a utensil, now usually made of metal, whose long handle terminates in a head that branches into several narrow and often slightly curved tines with which one can spear foods ei ...

Black's king and his newly promoted piece (e.g. 104...f1=Q 105.Ne3+) immediately after the promotion. :104...f1=N+ :105. Kc3 Kf3. White made the humorous move :106. Nh2+ forking Black's king and knight, but

sacrificing Sacrifice is the offering of material possessions or the lives of animals or humans to a deity as an act of propitiation or worship. Evidence of ritual animal sacrifice has been seen at least since ancient Hebrews and Greeks, and possibly ex ...

the knight. Black responded :106... Nxh2 and a draw was agreed. (A draw by

threefold repetition In chess, the threefold repetition rule states that a player may claim a draw if the same position occurs three times during the game. The rule is also known as repetition of position and, in the USCF rules, as triple occurrence of position.Articl ...

could have been claimed on move 78 and at other times.) Another example is the eighth game of the 1981

World Chess Championship The World Chess Championship is played to determine the world champion in chess. The current world champion is Magnus Carlsen of Norway, who has held the title since 2013. The first event recognized as a world championship was the World Chess ...

match between

Anatoly Karpov Anatoly Yevgenyevich Karpov ( rus, links=no, Анато́лий Евге́ньевич Ка́рпов, p=ɐnɐˈtolʲɪj jɪvˈɡʲenʲjɪvʲɪtɕ ˈkarpəf; born May 23, 1951) is a Russian and former Soviet chess grandmaster, former World Ches ...

and

Viktor Korchnoi Viktor Lvovich Korchnoi ( rus, Ви́ктор Льво́вич Корчно́й, p=vʲiktər lʲvovʲɪtɕ kɐrtɕˈnoj; 23 March 1931 – 6 June 2016) was a Soviet (before 1976) and Swiss (after 1980) chess grandmaster (GM) and chess writer. He ...

. Black forces a draw by * 80... Nf7! * 81. h7 Ng5! * 82. Ne7+ Kb7 * 83. Nxg6 Nxh7 * 84. Nxh7 draw

Three knights

Three knights and a king can force checkmate against a lone king within twenty moves (unless the defending king can win one of the knights). Also, a complete computational

retrograde analysis In chess problems, retrograde analysis is a technique employed to determine which moves were played leading up to a given position. While this technique is rarely needed for solving ordinary chess problems, there is a whole subgenre of chess pr ...

revealed that they can force checkmate only on the edge of the board.

Two knights versus a pawn

In some positions with two knights versus a pawn, the knights can force checkmate by gaining a

tempo In musical terminology, tempo (Italian, 'time'; plural ''tempos'', or ''tempi'' from the Italian plural) is the speed or pace of a given piece. In classical music, tempo is typically indicated with an instruction at the start of a piece (often ...

when the pawn has to move.

Troitsky line

Even though two knights cannot force

(with the help of their

) against a lone king (with the exception of positions where White wins in one move), decreasing the material advantage and allowing the defending king to have a pawn can actually allow for a forced checkmate. The reason that checkmate can be forced is that the pawn gives the defender a piece to move and deprives him of a stalemate defense. Another reason is that the pawn can block its own king's path without necessarily moving (e.g. Kling & Horwitz position right). The Troitsky line (or Troitsky position) is a key motif in

theory in the rare but theoretically interesting ending of two

versus a

pawn Pawn most often refers to: * Pawn (chess), the weakest and most numerous piece in the game * Pawnbroker or pawnshop, a business that provides loans by taking personal property as collateral Pawn may also refer to: Places * Pawn, Oregon, an his ...

. The line, assuming White has the two knights and Black the pawn, is shown left. The Russian theoretician Troitsky made a detailed study of this endgame and discovered the following rule: An example of the application of this rule is given in the diagram Müller and Lamprecht right; "... the position would be lost no matter where the kings are." However, the checkmate procedure is difficult and long. In fact, it can require up to 115 moves by White (assuming perfect play), so in competition often a

draw Draw, drawing, draws, or drawn may refer to: Common uses * Draw (terrain), a terrain feature formed by two parallel ridges or spurs with low ground in between them * Drawing (manufacturing), a process where metal, glass, or plastic or anything ...

by the

fifty-move rule The fifty-move rule in chess states that a player can claim a draw if no has been made and no pawn has been moved in the last fifty moves (for this purpose a "move" consists of a player completing a turn followed by the opponent completing a tur ...

will occur first. Troitsky showed that "on any placement of the black king, White undoubtedly wins only against black pawns standing on

he Troitsky line He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana へ * He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...

and above".

John Nunn John Denis Martin Nunn (born 25 April 1955) is an English chess grandmaster, a three-time world champion in chess problem solving, a chess writer and publisher, and a mathematician. He is one of England's strongest chess players and was former ...

analyzed the endgame of two knights versus a pawn with an

endgame tablebase An endgame tablebase is a computerized database that contains precalculated exhaustive analysis of chess endgame positions. It is typically used by a computer chess engine during play, or by a human or computer that is retrospectively analysin ...

and stated that "the analysis of Troitsky and others is astonishingly accurate". He undertook this checking after the very ending occurred in a critical variation of his post mortem analysis of a game he lost to Korchnoi in the 1980 Phillips and Drew Tournament in London. Neither player knew whether the position was a win for the player with the knights (Korchnoi). Even when the position is a theoretical win, it is very complicated and difficult to play correctly. Even grandmasters fail to win it.

Andor Lilienthal Andor (André, Andre, Andrei) Arnoldovich LilienthalReuben Fine, ''The World's Great Chess Games'', Dover Publications, 1983, p. 216. . (5 May 1911 – 8 May 2010) was a Hungarian and Soviet chess player. In his long career, he played against ten ...

failed to win it twice in a six-year period, se
Norman vs. Lilienthal
an
Smyslov vs. Lilienthal
But a fine win is in a game by Seitz, se
Znosko-Borovsky vs. Seitz

Examples

This diagram shows an example of how having the pawn makes things worse for Black (here Black's pawn is past the Troitsky line), by making Black have a move available instead of being stalemated. :1. Ne4 d2 :2. Nf6+ Kh8 :3. Ne7 (if Black did not have the pawn at this point, the game would be a draw because of stalemate) :3... d1=Q :4. Ng6# If Black did not have the pawn move available, White could not force checkmate. The longest wins require 115 moves; this is one example starting with 1... Ne7. This position is winnable, but White pawn can be allowed to move only after 84 moves, making the win almost impossible in practical match due to fifty-move rule.

Pawn beyond the Troitsky line

In this study by

André Chéron André Chéron (September 25, 1895 – September 12, 1980) was a French chess player, endgame theorist, and a composer of endgame studies. He was named a FIDE International Master of Chess Composition in 1959, the first year the title was awar ...

, White wins even though the pawn is well beyond the Troitsky line. Black to move is quicker. With White to move, he must maneuver to give the move to Black, as follows. 1.Kc3 Kb1 2.Kd2 Ka1 3.Kc1 Ka2 4.Kc2 (White then maneuvers to get the same position with vertical instead of horizontal opposition) 4...Ka1 5.Kb3 Kb1 6.Nb2 Kc1 7.Kc3 Kb1 8.Nd3 Ka1 9.Kc4 Ka2 10.Kb4 Ka1 11.Ka3 Kb1 12.Kb3 (Now White has enough time to bring the blockading N in to generate a mating net in time) 12...Ka1 13.Ne3 g2 14.Nc2+ Kb1 15.Na3+ Ka1 16.Nb4 g1=Q 17.Nbc2# In the situation with Black's rook pawn blockaded on h3, if the black king can enter and remain in the area marked with crosses in the adjacent diagram, the game is a draw. Otherwise, White can force the black king into one of the corners not located in the drawing zone and deliver checkmate. Black cannot be checkmated in the a8-corner because the knight on h2 is too far away to help deliver mate: Black draws by pushing the pawn as soon as White moves the knight on h2. White to play in the diagram can try to prevent Black to enter the drawing zone with 1.Ke6, but Black then plays 1...Kg5 aiming to attack the knight on h2. White is compelled to stop this with 2.Ke5 which allows Black to return to the initial position with 2...Kg6, and White has made no progress.

Topalov versus Karpov

lost an endgame with a pawn versus two knights to

Veselin Topalov Veselin Aleksandrov Topalov (pronounced ; bg, Весели́н Александров Топа́лов; born 15 March 1975) is a Bulgarian chess International Grandmaster, grandmaster and former FIDE World Chess Championship, World Chess Champ ...

although he had a theoretical draw with a pawn past the Troitsky line; because of its rarity, Karpov seemed not to know the theory of drawing and headed for the wrong corner. (Depending on the position of the pawn, checkmate can be forced only in certain corners.) In this "rapid play"

time control A time control is a mechanism in the tournament play of almost all two-player board games so that each round of the match can finish in a timely way and the tournament can proceed. Time controls are typically enforced by means of a game clock, ...

, the position in the game was initially a draw, but Karpov made a bad move which resulted in a lost position. Topalov later made a bad move, making the position a draw, but Karpov made another bad move, resulting in a lost position again.

Wang versus Anand

This position from a

blindfold A blindfold (from Middle English ') is a garment, usually of cloth, tied to one's head to cover the human eye, eyes to disable the wearer's visual perception, sight. While a properly fitted blindfold prevents sight even if the eyes are open, a ...

game between Wang Yue and

Viswanathan Anand Viswanathan "Vishy" Anand (born 11 December 1969) is an Indian chess grandmaster and a former five-time World Chess Champion. He became the first grandmaster from India in 1988, and is one of the few players to have surpassed an Elo rating of ...

leads to an example with a forced win even though the pawn is past the Troitsky line. The game continued :61... Kc5, blocking the pawn with the wrong piece. Black should have played 61...Ne4 62. c4 Nc5!, blocking the pawn on the Troitsky line with a knight, with a forced win. The game continued: :62. c4 Ne4 :63. Ka4 Nd4 :64. Ka5. Black still has a theoretical forced win in this position, even after letting the pawn advance past the Troitsky line: :64... Nc6+ :65. Ka6 Kd6!! :66. c5+ Kc7 and Black has a forced checkmate in 58 more moves. However, the actual game was drawn.

More pawns

Two knights can win in some cases when the defender has more than one pawn. First the knights should blockade the pawns and then capture all except one. The knights cannot set up an effective blockade against four

connected pawns In chess, connected pawns are two or more pawns of the same color on adjacent , as distinct from isolated pawns. These pawns are instrumental in creating pawn structure because, when diagonally adjacent, like the two rightmost white pawns, they f ...

, so the position generally results in a draw. Five or more pawns usually win against two knights.

Example from game

In this 1991 game between Paul Motwani and

Ilya Gurevich Ilya Mark Gurevich (born February 8, 1972) is a Soviet-born American chess player. Born in Kyiv, he emigrated to the U.S. in January 1980. He was a student at Yeshiva Academy in Worcester, Massachusetts. In 1983, Gurevich won the U.S. National ...

, Black has blockaded the white pawns. In ten moves, Black won the pawn on d4. There were some inaccuracies on both sides, but White

resigned Resignation is the formal act of leaving or quitting one's office or position. A resignation can occur when a person holding a position gained by election or appointment steps down, but leaving a position upon the expiration of a term, or choos ...

on move 99.

Position of mutual zugzwang

There are positions of

mutual zugzwang Zugzwang (German for "compulsion to move", ) is a situation found in chess and other turn-based games wherein one player is put at a disadvantage because of their obligation to make a move; a player is said to be "in zugzwang" when any legal move ...

in the endgame with two knights versus one pawn. In this position, White to move draws but Black to move loses. With Black to move: : 1... Kh7 : 2. Ne4 d2 : 3. Nf6+ Kh8 : 4. Ne7 (or 4.Nh4) d1=Q : 5. Ng6# With White to move, Black draws with correct play. White cannot put Black in

zugzwang Zugzwang (German for "compulsion to move", ) is a situation found in chess and other turn-based games wherein one player is put at a disadvantage because of their obligation to make a move; a player is said to be "in zugzwang" when any legal move ...

: : 1. Kf6 Kh7 : 2. Kf7 Kh8 : 3. Kg6 Kg8 : 4. Ng7 Kf8 : 5. Kf6 Kg8 : 6. Ne6 Kh7! (but not 6...Kh8? because White wins after 7.Kg6!, which puts Black to move) : 7. Kg5 Kg8 : 8. Kg6 Kh8 and White has no way to force a win.

Checkmate in problems

The possible checkmate on the edge of the board is the basis of some composed

chess problem A chess problem, also called a chess composition, is a puzzle set by the composer using chess pieces on a chess board, which presents the solver with a particular task. For instance, a position may be given with the instruction that White is to ...

s, as well as variations of the checkmate with two knights against a pawn. ;Angos, 2005 In this problem by Alex Angos, White checkmates in four moves: :1. Ne6! Nd8 :2. Nf6+ Kh8 :3. Ng5 N–''any'' (Black is in

and any knight move must abandon the protection of the f7-square) :4. Nf7# ;;Berger, 1890 A similar problem was composed by

in 1890. The solution is: :1. Nf7! Nd6 :2. Nh6+ Kh8 :3. Ng5 followed by :4. Ngf7#. ;de Musset, 1849 In this composition by Alfred de Musset, White checkmates on the edge of the board in three moves with: :1. Rd7 Nxd7 :2. Nc6 N–''any'' :3. Nf6#. ;Sobolevsky, 1951 In this

study Study or studies may refer to: General * Education **Higher education * Clinical trial * Experiment * Observational study * Research * Study skills, abilities and approaches applied to learning Other * Study (art), a drawing or series of drawi ...

composed by Sobolevsky, White wins by checkmating with two knights: : 1. Nh8+ Kg8 : 2. Kxg2 Bf4 : 3. Ng6 Bh6! : 4. Ng5 Bg7! : 5. Ne7+ Kh8 : 6. Nf7+ Kh7 : 7. Bh4! Bf6! : 8. Ng5+ Kh6 : 9. Ng8+ Kh5 : 10. Nxf6+! Kxh4 : 11. Nf3# ;Nadanian, 2009 In this study composed by

Ashot Nadanian Ashot Nadanian (sometimes transliterated as Nadanyan; hy, Աշոտ Նադանյան; born 19 September 1972) is an Armenian chess International Master (1997), chess theoretician and chess coach. His highest achievements have been in openin ...

, White wins by checkmating with two knights: : 1. Rg8!! Rxg8 If 1...Re7, then 2.N6f5! Re1 3.Rxg6+ Kxh5 4.Rxh6+ Kg5 5.Nf3+ and White wins. : 2. Ne4+ Kxh5 : 3. Ne6 and checkmate on the next move, due to

; two white knights deliver four different checkmates: * 3... R–''any'' 4. Ng7# * 3... Nd–''any'' 4. Nf6# * 3... Ng–''any'' 4. Nf4# * 3... f3 4. Ng3#

History

The first known composition where two knights win against one pawn is, according to Lafora, by

Gioachino Greco Gioachino Greco (c. 1600 – c. 1634) ( ελληνικά/greek: Τζοακίνο Γκρέκο), surnamed Cusentino and more frequently ''il Calabrese'', was an Italian chess player and writer. He recorded some of the earliest chess games kno ...

in 1620. In 1780, Chapais did a partial analysis of three positions with the pawn on f4 or h4. In 1851

Horwitz Horwitz is a surname, current among Ashkenazi Jews. It is derived from the Yiddish pronunciation of the name of the town of Hořovice in Bohemia. For detailed historical background see the Horowitz page . Notable people with the surname include: ...

and Kling published three positions where the knights win against one pawn and two positions where they win against two pawns. The analysis by Chapais was revised by Guretsky-Cornitz and others, and it was included by

in ''Theory and Practice of the Endgame'', first published in 1891. However, the analysis by Guretsky-Cornitz was incorrect, and the original analysis by Chapais was, in principle, correct. Troitsky started studying the endgame in the early 20th century and published his extensive analysis in 1937. Modern computer analysis found it to be very accurate. Master games with this ending are rare — Troitsky knew of only six when he published his analysis in 1937. In the first four (from c. 1890 to 1913), the weaker side brought about the ending to obtain a

from an opponent who did not know how to win. The first master game with a win was in 1931 when

Adolf Seitz Jakob Adolf Seitz (February 14, 1898, Meitingen, Germany – April 6, 1970, Switzerland) was a German–Argentine chess master and journalist. Career In 1920, he tied for 2-4th in Canterbury, took 10th in Berlin, and tied for 4-5th in Kulmbach. I ...

beat

Eugene Znosko-Borovsky Eugene Znosko-Borovsky (russian: Евге́ний Алекса́ндрович Зноско-Боро́вский, Yevgeny Alexandrovich Znosko-Borovsky; 16 August 1884 – 31 December 1954) was a Russian chess player, music critic, music and drama ...

.Znosko-Borovsky vs. Seitz
/ref>

References

Bibliography * * * * * * * * * * * Reprint: (1996) * * * * * * * * * * * * The last part (pages 197–257) is a supplement containing Troitsky's analysis of two knights versus pawns.

External links

* Grandmaster and endgame specialist

Karsten Müller Karsten Müller (born November 23, 1970 in Hamburg, West Germany) is a German chess Grandmaster and author. He earned the Grandmaster title in 1998 and a PhD in mathematics in 2002 at the University of Hamburg. He had placed third in the 1996 Ger ...

wrote a helpful two-part article on this endgame called ''The Damned Pawn'' (in PDFs):
Part 1
about the Troitsky line and the technique
Part 2: the second Troitsky line solved
the winning line taking into account the 50-move rule, and more winning techniques and drawing zones.
Two Knights vs King and Pawn TrainerSmyslov vs. LilienthalNorman vs. Lilienthal
{{DEFAULTSORT:Two Knights Endgame Chess endgames Chess problems Chess theory