View source for User:Bubba73/MySandbox

==chess images==
{| align="right"
| [[Image:Chess kll44.png]]
| [[Image:Chess kld44.png]]
| [[Image:Chess qll44.png]]
| [[Image:Chess qld44.png]]
| [[Image:Chess bll44.png]]
| [[Image:Chess bld44.png]]
| [[Image:Chess nll44.png]]
| [[Image:Chess nld44.png]]
| [[Image:Chess rll44.png]]
| [[Image:Chess rld44.png]]
| [[Image:Chess pll44.png]]
| [[Image:Chess pld44.png]]
| [[Image:Chess l44.png]]
| [[Image:Chess d44.png]]
|}
{| align="right"
| [[Image:Chess kdl44.png]]
| [[Image:Chess kdd44.png]]
| [[Image:Chess qdl44.png]]
| [[Image:Chess qdd44.png]]
| [[Image:Chess bdl44.png]]
| [[Image:Chess bdd44.png]]
| [[Image:Chess ndl44.png]]
| [[Image:Chess ndd44.png]]
| [[Image:Chess rdl44.png]]
| [[Image:Chess rdd44.png]]
| [[Image:Chess pdl44.png]]
| [[Image:Chess pdd44.png]]
|}

{| align="right"
| [[Image:Chess kll45.svg]]
| [[Image:Chess kld45.svg]]
| [[Image:Chess qll45.svg]]
| [[Image:Chess qld45.svg]]
| [[Image:Chess bll45.svg]]
| [[Image:Chess bld45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess nld45.svg]]
| [[Image:Chess rll45.svg]]
| [[Image:Chess rld45.svg]]
| [[Image:Chess pll45.svg]]
| [[Image:Chess pld45.svg]]
| [[Image:Chess l45.svg]]
| [[Image:Chess d45.svg]]
|}

{| align="right"
| [[Image:Chess kdl45.svg]]
| [[Image:Chess kdd45.svg]]
| [[Image:Chess qdl45.svg]]
| [[Image:Chess qdd45.svg]]
| [[Image:Chess bdl45.svg]]
| [[Image:Chess bdd45.svg]]
| [[Image:Chess ndl45.svg]]
| [[Image:Chess ndd45.svg]]
| [[Image:Chess rdl45.svg]]
| [[Image:Chess rdd45.svg]]
| [[Image:Chess pdl45.svg]]
| [[Image:Chess pdd45.svg]]
|}
{| align="right"
| [[Image:Chess x1l45.svg]]
| [[Image:Chess x1d45.svg]]
| [[Image:Chess x2l45.svg]]
| [[Image:Chess x2d45.svg]]
| [[Image:Chess x3l45.svg]]
| [[Image:Chess x3d45.svg]]
| [[Image:Chess x4l45.svg]]
| [[Image:Chess x4d45.svg]]
| [[Image:Chess x5l45.svg]]
| [[Image:Chess x5d45.svg]]
| [[Image:Chess x6l45.svg]]
| [[Image:Chess x6d45.svg]]
| [[Image:Chess x7l45.svg]]
| [[Image:Chess x7d45.svg]]
| [[Image:Chess x8l45.svg]]
| [[Image:Chess x8d45.svg]]
| [[Image:Chess x9l45.svg]]
| [[Image:Chess x9d45.svg]]
|}
{| align="right"
| [[Image:Chess x1l44.png]]
| [[Image:Chess x1d44.png]]
| [[Image:Chess x2l44.png]]
| [[Image:Chess x2d44.png]]
| [[Image:Chess x3l44.png]]
| [[Image:Chess x3d44.png]]
| [[Image:Chess x4l44.png]]
| [[Image:Chess x4d44.png]]
| [[Image:Chess x5l44.png]]
| [[Image:Chess x5d44.png]]
| [[Image:Chess x6l44.png]]
| [[Image:Chess x6d44.png]]
| [[Image:Chess x7l44.png]]
| [[Image:Chess x7d44.png]]
| [[Image:Chess x8l44.png]]
| [[Image:Chess x8d44.png]]
| [[Image:Chess x9l44.png]]
| [[Image:Chess x9d44.png]]
|}

{| align="right"
| [[Image:Chess xol44.png]]
| [[Image:Chess xod44.png]]
| [[File:Chess xwl44.png]]
| [[File:Chess xwd44.png]]
| [[File:Chess xxl44.png]]
| [[File:Chess xxd44.png]]
|}

{{clear}}


{{Chess diagram
|tright
|

|xo|oo|xx|x1|x2|  |  |  
|xo|oo|xx|x1|x2|  |  |  
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|
}}

{{clear}}

==from Wikibooks==

==King and Queen vs. King==
{{Chess diagram
| tright
| 

|  |  |  |  |  |  |  |  
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|  |  |  |kd|  |  |  |  
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|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|kl|ql|  |  |  |  |  |  
| Sample starting position for K+Q vs. K mate
}}
Delivering checkmate with a king and queen against a lone king is quite easy. The basic technique involves driving the king to the edge of the board, which the queen can do by herself.  It's faster if you use your king and queen together, but this increases the probability of a stalemate, so beginners should do it without the king. The technique described below will accomplish the mate in about 10 to 15 moves.

Here's an example (see diagram on right):

'''1.Qb5'''

Cutting the black king off along the fifth rank.

'''1...Ke6'''

1...Kc7 2.Qa6 limits Black's king to the last two ranks.
<br style="clear:both" />
{{Chess diagram
| tright
| 

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |kd|  |  |  
|  |  |ql|  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|kl|  |  |  |  |  |  |  
| Position after 2.Qc5
}}

'''2.Qc5''' (see diagram)

During this phase, notice how White's queen always stays a knight's move away from the black king, and how no checks are necessary (or even desirable).  Moves like 2.Qc6+? only allow Black's king more freedom after 2...Ke5.

'''2...Kf6 3.Qd5 Kg6 4.Qe5 Kf7'''

After 4...Kh6 5.Qg3 White's goal has been achieved: the black king is trapped on the edge. White will then bring his king to f6 to force mate.

'''5.Qd6 Kg7 6.Qe6 Kh7'''

Black's king is forced to the edge of the board no matter what he does, e.g. 6...Kf8 7.Qd7.
<br style="clear:both" />
{{Chess diagram
| tright
| 

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |kd
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |ql|  
|  |  |  |  |  |  |  |  
|  |kl|  |  |  |  |  |  
|  |  |  |  |  |  |  |  
| Position after 8.Kb2
}}

'''7.Qg4 Kh6 8.Kb2''' (see diagram)

Now that Black's king is stuck, the white monarch comes in to finish off his adversary.

'''8...Kh7 9.Kc3'''

9.Qg5 doesn't spoil anything, but it isn't necessary. Unlike the king and rook vs. king mate, here Black's king doesn't have to be trapped in the corner.

'''9...Kh8 10.Kd4'''

10 Qg6=?? [[stalemate]] was what Black was hoping for. Beware of this trap!

'''10...Kh7 11.Ke5 Kh8 12.Kf6 Kh7 13.Qg7#.'''



==fractions==
{{frac|1|5}}
{{frac|3|3|4}}

&frac12;
&frac23;

{{user-full|bubba73}}

==diagram problems==

{{Chess diagram
|tright
|Bruce Moreland

|kl|  |kd|  |  |  |  |  
|  |  |  |  |nd|  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|pl|  |pl|  |  |  |  |  
|  |  |  |  |  |  |  |  
|Black to move checkmates in 17 moves
}}

{{Chess diagram
|tright
|Fischer-Petrosian 1959

|  |  |  |  |  |  |ql|ql
|  |  |  |  |  |  |  |  
|  |kd|pd|qd|  |  |  |  
|  |  |  |  |pd|  |  |  
|  |  |pl|pd|pl|  |  |  
|  |  |nd|pl|  |  |pl|  
|  |  |  |  |  |  |  |  
|qd|  |  |  |  |bl|kl|  
|Position after 37. h8=Q, four queens
}}

{{Chess diagram
|tright
|Fischer-Petrosian 1959

|  |  |  |  |  |  |ql|ql
|  |  |  |  |  |  |  |  
|  |kd|pd|qd|  |  |  |  
|  |  |  |  |pd|  |  |  
|  |  |pl|pd|pl|  |  |  
|  |  |nd|pl|  |  |pl|  
|  |  |  |  |  |  |  |  
|qd|  |  |  |  |bl|kl|  
|Position after 37. h8(Q), four queens
}}


An "=" in the caption messes the diagram up:

::"You need to use {{tl|{{=}}}} when writing = in templates. This is a limitation of all templates."

{{Chess diagram small
|tright
|Fischer-Petrosian 1959

|  |  |  |  |  |  |ql|ql
|  |  |  |  |  |  |  |  
|  |kd|pd|qd|  |  |  |  
|  |  |  |  |pd|  |  |  
|  |  |pl|pd|pl|  |  |  
|  |  |nd|pl|  |  |pl|  
|  |  |  |  |  |  |  |  
|qd|  |  |  |  |bl|kl|  
|Position after 37. h8=Q, four queens
}}

{{Chess diagram
|tright
|

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|
<!-- generated by WikiChessDiagram 2.0--> }}

{{Chess diagram
|tright
|
|=<!-- generated by WikiChessDiagram 2.0-->
|rd|nd|bd|qd|kd|bd|nd|rd
|pd|pd|pd|pd|pd|pd|pd|pd
|  |  |  |  |  |  |  |  
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|  |  |  |  |  |  |  |  
|pl|pl|pl|pl|pl|pl|pl|pl
|rl|nl|bl|ql|kl|bl|nl|rl
|
}}
{{clear}}

==Harv==
* {{harv|Smith|2000|pp=137-39}}
* {{harvcol|Smith|2000|pp=137-39}}

==Two knights==
{| align="right"
| [[Image:Chess kll45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess kdl45.svg]]
|}
The '''two knights endgame''' is a [[chess]] [[Chess endgame|endgame]] with a [[king (chess)|king]] and two [[knight (chess)|knights]] versus a king, possibly with some other [[material (chess)|material]].  The material with the defending king is usually one [[pawn (chess)|pawn]], but some positions studied involve additional pawns or other pieces.  In contrast to a king plus two [[bishop (chess)|bishops]], or a bishop and a knight, a king and two knights cannot force [[checkmate]] against a lone king.  (However, the superior side can force [[stalemate]].)  Although there are checkmate positions, the superior side cannot force them against proper (and easy) defense {{harvcol|Speelman|Tisdall|Wade|1993|p=11}}.  

{{Chess diagram
|tright
|

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
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|  |kl|  |  |  |  |  |  
|  |  |nl|nl|  |  |  |  
|kd|  |  |  |  |  |  |  
|Checkmate position, but it cannot be forced {{harvcol|Seirawan|2003|p=17}}.  The knight on d2 could be on ''c3'' or ''a3'' instead, and the white king could be on ''a3'' instead.
}}
On the other hand, if the lone king has a pawn (and sometimes with more pawns), then checkmate can be forced in some cases.  These positions were studied extensively by [[A. A. Troitzky]].  If the defender's pawn is blocked on or before the "Troitzky line", the stronger side can force checkmate, although it may require up to 115 moves with [[optimal play]].  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 {{harvcol|Müller|Lamprecht|2001|pp=19–20}}. The technique (when it is possible) is to block the pawn with one knight and use the king and other knight to force the opposing king into a corner or near the other knight.  Then when the block on the pawn is removed, the knight can be used to checkmate {{harvcol|Dvoretsky|2006|p=280}}.
{{algebraic notation|pos=tocleft}}

==Two knights cannot force checkmate==
{| align="right"
| [[Image:Chess kll45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess kdl45.svg]]
|}
Although there are checkmate positions with two knights against a king, they cannot be forced.  [[Edmar Mednis]] stated that this inability to force checkmate is "one of the great injustices of chess" {{harvcol|Mednis|1996|p=40}}.

Unlike some other theoretically [[draw (chess)|drawn]] endgames, such as a [[rook and bishop versus rook endgame|rook and bishop versus rook]], the defender has an easy task in all endings with two knights versus a lone king. He simply has to avoid moving into a position in which he can be checkmated on the next move, and he always has another move available in such situations {{harvcol|Speelman|Tisdall|Wade|1993|p=11}}.

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) {{harvcol|Fine|1941|pp=5–6}}.

===In the corner===
{{Chess diagram small
|tright
|Keres

|  |  |  |  |  |  |  |kd
|  |  |  |  |  |  |  |  
|  |  |  |  |nl|  |kl|  
|  |  |  |  |  |nl|  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|Two knights cannot force checkmate
}}
The player with the lone king has to make a [[blunder (chess)|blunder]] to be checkmated.  In this position, 1.Ne7 or 1.Nh6 immediately [[stalemate]]s Black.  White can try instead:

: '''1. Nf8 Kg8'''
: '''2. Nd7 Kh8'''
: '''3. Nd6 Kg8'''
: '''4. Nf6+''' 
now if Black moves 4...Kh8?? then 5.Nf7# is checkmate, but if Black moves
: '''4... Kf8'''
then White has made no progress {{harvcol|Keres|1984|pp=2–3}}.
{{clear}}

{{Chess diagram
|tright
|Berger
|= <!-- Program to generate diagrams: WikiChessDiagram 2.03 by Bubba73 -->
|  |  |  |  |  |  |kd|  
|  |  |  |  |  |  |  |  
|  |  |  |nl|nl|  |kl|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|Draw with either side to move
}}

[[Johann Berger]] 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 {{harvcol|Guliev|2003|p=74}}.
{{clear}}

===On the edge===
{{Chess diagram small
|tright
|

|  |  |kd|  |  |kl|  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |nl|nl|  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|White can also try for a mate on the edge of the board
}}
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).
{{clear}}

===Examples from games===
{{Chess diagram small
|tright
|Benko vs. Bronstein, 1949

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |nl|  
|  |  |kl|  |  |kd|nd|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |nd|  |  
|Position before 106. Nh2+
}}
In this position from a 1949 game<ref>[http://www.chessgames.com/perl/chessgame?gid=1033751 Benko–Bronstein]</ref> between [[Pal Benko]] and [[David Bronstein]], Black had just [[promotion (chess)|underpromoted]] to a knight (104...f2–f1=N+ 105.Kd2–c3 Kg2–f3).  Black did not [[promotion (chess)|promote]] to a queen or any other piece because White could [[fork (chess)|fork]] Black's king and his newly promoted piece (104...f1=Q 105.Ne3+) immediately after the promotion.  White made the humorous move 
:'''106. Nh2+'''
forking Black's king and knight, but [[sacrifice (chess)|sacrificing]] the knight.  Black responded
:'''106... Nxh2''' 
and a [[draw by agreement|draw was agreed]] {{harvcol|Benko|2007|p=133}}.
{{clear}}

Another example is the eighth game from the 1981 [[World Chess Championship 1981|World Chess Championship]] match between [[Anatoly Karpov]] and [[Viktor Korchnoi]].<ref>[http://www.chessgames.com/perl/chessgame?gid=1068236 Karpov vs. Korchnoi]</ref>

==Troitzky line==
{| align="right"
| [[Image:Chess kll45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess nll45.svg]]
| [[Image:Chess kdl45.svg]]
| [[Image:Chess pdl45.svg]]
|}
{{Chess diagram
|tright
|

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |xo|  |  |  |  |xo|  
|  |  |xo|  |  |xo|  |  
|xo|  |  |xo|xo|  |  |xo
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|Troitzky line: two white knights can checkmate if the black pawn is blocked on or before the marked squares
}}
The Troitzky (or Troitsky) line (or Troitzky position) is a key motif in [[chess endgame]] theory in the rare and practically unimportant (but theoretically interesting) ending of two [[knight (chess)|knights]] versus a [[pawn (chess)|pawn]]. The endgame was analyzed by [[Alexey Troitsky|A. A. Troitzky]].

Whilst two knights cannot force [[checkmate]] (with the help of their [[king (chess)|king]]) against a lone king, a decrease in material advantage allowing the defending king to have a pawn can actually cause his demise. This is due to the fact that a common technique in this endgame is that of reducing the defending king to a position that would be a stalemate except for an available pawn move, and allowing the pawn to move can allow the attacking knights to move in for the kill. For the position with White on the attack, Troitsky established that if a black pawn is [[glossary of chess#Blockade|blockaded]] (by one of the white knights) on a square no further forward than the line a4–b6–c5–d4–e4–f5–g6–h4, then White can win the resulting endgame (and similarly in reverse for Black), no matter where the other pieces are placed. However, the checkmate procedure is difficult and long.  In fact, it can require up to 115 moves by White, so in competition often a [[draw (chess)|draw]] by the [[fifty-move rule]] will occur first (but see [http://www.chesscafe.com/text/mueller36.pdf this article] and [[Two knights endgame#Second Troitzky line|Second Troitzky line section]] for the zone where the win can be forced within fifty moves).  Therefore the ending is more of theoretical than practical interest.  If the defending black pawn is past the Troitsky line, there are zones such that if the black king is in one, white still has a theoretical win; otherwise the position is a draw.

[[John Nunn]] analyzed the endgame of two knights versus a pawn with an [[endgame tablebase]] and stated that "the analysis of Troitsky and others is astonishingly accurate" {{harvcol|Nunn|1995|p=265}}.

Even when the position is a theoretical win, it is very complicated and difficult to play correctly.  Even [[grandmaster (chess)|grandmasters]] fail to win it.  [[Andor Lilienthal]] failed to win it twice in a six-year period, see [http://www.chessgames.com/perl/chessgame?gid=1090836 Norman vs. Lilienthal] and [http://www.chessgames.com/perl/chessgame?gid=1090903 Smyslov vs. Lilienthal].  But a fine win is in a game by [[Adolf Seitz|Seitz]], see [http://www.chessgames.com/perl/chessgame?gid=1151993 Znosko-Borovsky vs. Seitz] {{harvcol|Giddins|2012|p=26}}.

Two knights versus pawn is sometimes called the "Halley's Comet" endgame.<ref>[http://www.chessninja.com/dailydirt/2008/01/corus-08-r2-favorites-with-1st-move.htm use of name]</ref>
{{clear}}

===Examples===
{{Chess diagram small
|tright
|

|  |  |  |  |  |  |  |  
|  |  |  |  |  |kl|  |kd
|  |  |  |  |  |  |  |  
|  |  |  |  |  |nl|  |  
|  |  |  |  |  |  |  |  
|  |  |  |pd|  |  |  |  
|  |  |  |nl|  |  |  |  
|  |  |  |  |  |  |  |  
|White to move wins
}}
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.
{{clear}}

{{Chess diagram small
|tright
|

|  |  |  |  |  |  |  |  
|kd|  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |kl|nd|  |  
|  |  |  |  |  |  |  |  
|pl|  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |nd|  |  |  |  |  |  
|Black to move wins in 115 moves
}}
The longest win is this position that requires 115 moves, starting with '''1... Ne7''' (this is not the only possible example of position where 115 moves are required to win).
{{clear}}

===Pawn beyond the Troitsky line===
{{Chess diagram small
|tright
|Chéron, 1955

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |nl|  |  |  |  |  
|  |  |  |  |  |  |pd|  
|kd|  |kl|  |  |  |nl|  
|  |  |  |  |  |  |  |  
|White wins with either side to move
}}
If a pawn is beyond the Troitsky line, the result usually depends on the location of the defending king.  Usually there is a "drawing area" and a "losing area" for the defending king, which was also analyzed by Troitsky.  In this study by [[André Chéron]], White wins even though the pawn is well beyond the Troitsky line {{harvcol|Müller|Lamprecht|2001|p=20}}.
{{clear}}
{{Chess diagram small
|tright
|
|= <!-- Program to generate diagrams: WikiChessDiagram 2.03 by Bubba73 -->
|xx|xx|xx|xx|  |  |  |  
|xx|xx|xx|xx|xx|  |  |  
|xx|xx|xx|nl|xx|xx|kd|  
|  |xx|xx|xx|kl|xx|  |  
|  |  |xx|xx|xx|xx|  |  
|  |  |  |  |  |  |  |pd
|  |  |  |  |  |  |  |nl
|  |  |  |  |  |  |  |  
|Drawing area marked with "X"
}}
In this diagram, if the black king can enter the drawing area and remain there, the game is a draw.  Black cannot be checkmated in the a8 corner because the knight on h2 is too far away – the pawn would advance.  Checkmate can be forced in the a1 and h8 corners {{harvcol|Averbakh|Chekhover|1977|p=119}}.
{{clear}}

===Topalov versus Karpov===
{{Chess diagram small
|tright
|Topalov vs. Karpov, 2000

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |pd|  
|  |kl|  |nl|  |  |  |  
|  |  |  |  |nl|  |  |  
|  |kd|  |  |  |  |  |  
|White won after 74. Ne2, even though the pawn was past the Troitzky line.
}}
[[Anatoly Karpov]] lost an endgame with a pawn versus two knights to [[Veselin Topalov]] <ref>[http://www.chessgames.com/perl/chessgame?gid=1295765 Topalov vs. Karpov]</ref> although he had a theoretical draw with a pawn past the Troitzky 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 {{harvcol|Troitzky|2006}}.)  In this "rapid play" [[time control]], 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.<ref>[http://www.chesscafe.com/text/mueller37.pdf Muller article]</ref>
{{clear}}

===Wang versus Anand===
{{Chess diagram small
|tright
|Wang vs. Anand, 2009
|= <!-- Program to generate diagrams: WikiChessDiagram 2.03 by Bubba73 -->
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |kd|  |  |  |  |  
|kl|  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |pl|  |  |nd|nd|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|Position after 61. Kxa5
}}
This position from a [[blindfold chess|blindfold]] game between [[Yue Wang]] and [[Viswanathan Anand]] is an example of a win with the pawn past the Troitsky line.<ref>[http://www.chessgames.com/perl/chessgame?gid=1541026 Wang vs. Anand]</ref> After
:'''61... Ne4'''
:'''62. c4 Nc5!'''
the pawn is blocked but it is past the Troitsky line – Black has a forced win {{harvcol|Soltis|2010|p=42}}.  However, the actual game was drawn.
{{clear}}

==Second Troitzky line==
{{Chess diagram small
|tright
|

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |xx|  |  |  |  |xx|  
|xo|  |xo|xo|xo|xo|  |xo
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|Second Troitzky line
}}
Since many of the wins when the pawn is blocked on or behind the Troitzky line require more than fifty moves (and thus would be draws under the fifty-move rule) [[Karsten Müller]] asked for the "second Troitzky line", which corresponds to where the knights can win without the fifty-move rule coming into effect. If Black's pawn is blocked by a white knight on or behind one of the dots, White can force a win within fifty moves. If the pawn can be blocked on or behind one of the ''X''s, White can force a win within fifty moves more than 99 percent of the time.<ref>[http://www.chesscafe.com/text/mueller36.pdf The second Troitzky line]</ref>
{{clear}}

==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 (chess)|capture]] all except one.  The knights cannot set up an effective blockade against four [[connected pawns]], so the position generally results in a draw.  Five or more pawns usually win against two knights {{harvcol|Fine|Benko|2003|p=101}}. 
{{Chess diagram
|tleft
|Fine & Benko, diagram 201

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |kd|pd|pd|  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |kl|  |nl|nl|  |  |  
|  |  |  |  |  |  |  |  
|White to move wins
}}
{{Chess diagram
|tleft
|[[Reuben Fine|Fine]], ECE #1778

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |pd|pd|pd|kd|  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |nl|  |kl|nl|  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|White to move wins
}}
{{Chess diagram
|tleft
|[[Henri Rinck|Rinck]], 1935, ECE #1780

|  |  |nd|nd|  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |pl|  |  |  |  
|  |  |  |  |  |  |pl|  
|  |  |  |  |  |pl|  |  
|  |  |  |  |  |  |  |kd
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |kl
|White to move wins
}}
{{clear}}

===Example from game===
{{Chess diagram small
|tright
|Motwana vs. I. Gurevich
|= <!-- Program to generate diagrams: WikiChessDiagram 2.03 by Bubba73 -->
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |kd|  |  |  
|  |  |kl|nd|pl|nd|  |  
|  |  |  |pl|  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|Position after Black's 72nd move
}}
In this 1991 game between Paul Motwana and [[Ilya Gurevich]], Black has blockaded the white pawns.  In ten moves, Black won the pawn on d4.  There were some inaccuracies on both sides, but White [[resign (chess)|resigned]] on move 99 {{harvcol|Speelman|Tisdall|Wade|1993|p=114}}.

{{clear}}

==Position of mutual zugzwang==
{{Chess diagram
|tright
|Troitsky
|= <!-- Program to generate diagrams: WikiChessDiagram 2.03 by Bubba73 -->
|  |  |  |  |  |  |  |kd
|  |  |  |  |  |kl|  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |nl|  |  
|  |  |  |  |  |  |  |  
|  |  |  |pd|  |  |  |  
|  |  |  |nl|  |  |  |  
|  |  |  |  |  |  |  |  
|White to move draws; Black to move loses
}}
There are positions of [[mutual zugzwang]] 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]]:
: '''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 {{harvcol|Averbakh|Chekhover|1977|p=106}}.
{{clear}}

==Checkmate in problems==
The possible checkmate on the edge of the board is the basis of some composed [[chess problem]]s, as well as variations of the checkmate with two knights against a pawn.

===Two knights versus one knight===
{{Chess diagram small
|tright
|

|  |  |  |kd|  |kl|  |  
|  |  |nd|  |  |  |  |  
|  |nl|  |  |  |  |  |  
|  |  |  |nl|  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|White to move… 
}}
In some positions it is possible for two knights to force checkmate against one knight, using the same idea, e.g. the defending knight makes a move available that would avoid stalemate. In position on the right White wins after '''1. Nf4!''' followed by '''2. Ne6#''' . {{citation needed|date=August 2012}}

{{clear}}

=== Angos ===
{{Chess diagram small
|tright
|Angos, 2005

|  |  |  |nl|  |  |kd|  
|  |nd|  |nl|  |  |  |  
|  |  |  |  |  |  |kl|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|White to move and mate in four 
}}
In this problem by Alex Angos, White checkmates in four moves:
:'''1. Ne6! Nd8'''
:'''2. Nf6+ Kh8'''
:'''3. Ng5 N'''–''any'' (Black is in [[zugzwang]] and any knight move must abandon the protection of the f7-square)
:'''4. Nf7#''' {{harvcol|Angos|2005|p=46}}.
{{clear}}
{{Chess diagram small
|tright
|Berger, 1890

|  |  |  |  |nd|  |kd|  
|  |  |  |  |  |  |  |  
|  |  |  |nl|nl|  |kl|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|White to move and mate in four 
}}
A similar problem was composed by [[Johann Berger]] in 1890.  The solution is:
:'''1. Nf7! Nd6'''
:'''2. Nh6+ Kh8'''
:'''3. Ng5'''
followed by 
:'''4. Ngf7#''' {{harvcol|Matanović|1993|pp=492–93}}.
{{clear}}

===de Musset===
{{Chess diagram small
|tright
|Alfred de Musset, 1849

|  |nd|  |  |kd|  |kl|  
|  |  |  |  |  |  |  |rl
|  |  |  |  |  |  |  |  
|  |  |  |  |nl|  |  |  
|  |  |  |  |  |  |nl|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|White to move and mate in three
}}
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#''' {{harvcol|Hooper|Whyld|1992}}.
{{clear}}

===Sobolevsky===
{{Chess diagram
|tright
|P. Sobolevsky, [[Shakhmaty v SSSR]], 1951

|  |  |  |  |  |  |  |  
|  |  |  |  |  |kd|  |nl
|  |  |  |  |  |bl|nl|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |kl
|  |  |  |  |  |  |nd|bd
|  |  |  |  |  |  |  |  
|White to move and win
}}
In this problem 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#''' {{harvcol|Nunn|1981|p=6}}.
{{clear}}

===Nadanian===
{{Chess diagram
|tright
|Ashot Nadanian, [[ChessBase]], 2009<ref>{{cite web|url=http://www.chessbase.com/puzzle/christmas2009/chr09-05.htm|title=ChessBase Christmas Puzzles: A tale of seven knights|date=2009-12-29|publisher=ChessBase|accessdate=6 February 2010}}</ref>

|  |  |  |  |  |  |  |rl
|  |  |  |nd|  |  |rd|  
|  |  |  |nl|  |  |nd|pd
|  |  |  |  |  |  |kd|pl
|  |  |  |nl|  |pd|  |  
|  |  |  |  |  |  |  |kl
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|White to move and win
}}
In this problem composed by [[Ashot Nadanian]], 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 [[zugzwang]]; two white knights deliver four different checkmates:<ref>{{cite web|url=http://www.chessbase.com/puzzle/christmas2009/chr09-sol1.htm|title=Solutions to 2009 Christmas Puzzles|date=2010-02-02|publisher=ChessBase|accessdate=6 February 2010}}</ref> 
* '''3... R'''–''any'' '''4. Ng7#'''
* '''3... Nd'''–''any'' '''4. Nf6#'''
* '''3... Ng'''–''any'' '''4. Nf4#'''
* '''3... f3''' '''4. Ng3#'''
{{clear}}

==History==
The fact that two knights can sometimes win against one or more pawns was known at least as early as 1780, when Chapsis did a partial analysis of three positions with the pawn on f4 or h4 {{harvcol|Troitzky|2006|p=200}}.  In 1851 [[Bernhard Horwitz|Horwitz]] and [[Josef Kling|Kling]] published three positions where the knights win against one pawn and two positions where they win against two pawns {{harvcol|Horwitz|Kling|1986|pp=64–68}}.  The analysis by Chapsis was revised by Guretsky-Cornitz and others, and included by [[Johann Berger]] in ''Theory and Practice of the Endgame'', first published in 1891.  However, the analysis by Guretsky-Cornitz was incorrect and the original analysis by Chapsis was in principle correct {{harvcol|Troitzky|2006|p=200}}.  Troitsky started studying the endgame in the early 20th century and published his extensive analysis in 1937 {{harvcol|Mednis|1996|p=43}}.  Modern computer analysis found it to be very accurate {{harvcol|Nunn|1995|p=265}}.

Master games with this ending are rare – Troitzky 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 [[draw (chess)|draw]] from an opponent who did not know how to win.  The first master game with a win was in 1931 when [[Adolf Seitz]] beat [[Eugene Znosko-Borovsky]] {{harvcol|Troitzky|2006|pp=197–99}}.<ref>[http://www.chessgames.com/perl/chessgame?gid=1151993 Znosko-Borovsky vs. Seitz]</ref>

{{Chess diagram
|tleft
|Horwitz & Kling, 1851

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |pd|  |  |  |  
|  |kl|  |nl|  |  |  |  
|  |  |  |  |  |  |  |  
|  |kd|  |  |  |  |nl|  
|White to move checkmates in six moves
}}
{{Chess diagram
|tleft
|Horwitz & Kling, 1851

|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |nl|  
|  |  |  |  |  |nl|  |  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |  |  |pd
|  |  |  |  |kl|  |kd|pd
|  |  |  |  |  |  |  |  
|White to move wins
}}
{{Chess diagram
|tleft
|[[William Pollock (chess player)|Pollock]] vs. [[Jackson Whipps Showalter|Showalter]], c. 1890

|  |  |  |  |  |  |  |  
|pd|  |  |  |  |  |  |  
|  |  |  |  |  |  |  |  
|nl|  |  |  |  |  |  |  
|  |  |nl|  |  |  |kd|  
|  |  |  |  |  |  |  |  
|  |  |  |  |  |kl|  |  
|  |  |  |  |  |  |  |  
|White to move.  Pollock refused to continue and [[draw by agreement|agreed to a draw]] six moves later, but White has a winning position {{harvcol|Troitzky|2006|p=197}}.
}}
{{clear}}

==See also==
* [[Checkmate]]
* [[Chess endgame]]

==Notes==
{{reflist}}

==References==
* {{Citation
 |last=Angos|first=Alex|authorlink=
 |title=You Move ... I Win!: A Lesson in Zugzwang
 |year=2005
 |publisher=Thinkers' Press, Inc.
 |isbn= 978-1-888710-18-2 
}}
* {{Citation
 |last=Averbakh|first=Yuri|authorlink=Yuri Averbakh
 |last2=Chekhover|first2=Vitaly
 |title=Knight Endings
 |year=1977
 |publisher=Batsford
 |isbn= 0-7134-0552-X
}}
* {{Citation
 | last = Dvoretsky| first = Mark| authorlink = Mark Dvoretsky
 | year = 2006
 | title = Dvoretsky's Endgame Manual
 | edition = 2nd
 | publisher = Russell Enterprises
 | isbn= 1-888690-28-3
}}
*{{Citation
|last=Fine|first=Reuben|authorlink=Reuben Fine 
|year=1941
|title=[[Basic Chess Endings]]
|publisher=McKay
|isbn= 0-679-14002-6
}}
*{{Citation
|last=Fine|first=Reuben <!-- |authorlink=Reuben Fine -->
|last2=Benko|first2=Pal|authorlink2=Pal Benko
|year=2003
|edition=2nd
|title=[[Basic Chess Endings]] (1941)
|publisher=McKay
|isbn= 0-8129-3493-8
}}
*{{citation
|last=Giddins|first=Steve|authorlink=Stephen Giddins
 | year=2012
 | title=The Greatest Ever Chess Endgames
 | publisher=[[Everyman Chess]]
 | isbn=978-1-85744-694-4}}
* {{citation
|last= Guliev |first=Sarhan| authorlink=Sarhan Guliev
| title=The Manual of Chess Endings
|publisher=Russian Chess House
| year=2003
| isbn=5-94693-020-6}}
* {{Citation
 |last=Hooper|first=David|authorlink=David Vincent Hooper
 |last2=Whyld|first2=Kenneth|authorlink2=Kenneth Whyld
 |title=[[The Oxford Companion to Chess]]
 |year=1992
 |edition=s2nd
 |publisher=Oxford University Press 
 |isbn= 0-19-866164-9
}} Reprint: (1996) {{ISBN|0-19-280049-3}}
* {{citation
 |last=Horwitz|first=Bernhard|authorlink=Bernhard Horwitz
 |last2=Kling|first2=Josef|authorlink2=Josef Kling
 |title=Chess Studies and End-Games (1851, 1884)
 |year=1986
 |publisher=Olms
 |isbn= 3-283-00172-3
}}
* {{Citation
 |last=Keres|first=Paul|authorlink=Paul Keres
 |title=Practical Chess Endings
 |year=1984
 |publisher=Batsford
 |isbn= 0-7134-4210-7
}}
* {{Citation
 |last=Matanović|first=Aleksandar|authorlink=Aleksandar Matanović
 |title=Encyclopedia of Chess Endings (minor pieces)
 |volume=5
 |year=1993
 |publisher=[[Chess Informant]]
 |isbn=
}}
*{{citation
 | last = Mednis| first = Edmar | authorlink = Edmar Mednis
 | title = Advanced Endgame Strategies
 | publisher = Chess Enterprises
 | year = 1996
 | isbn = 0-945470-59-2
 }}
*{{Citation
|last=Müller|first=Karsten|authorlink=Karsten Müller
|last2=Lamprecht|first2=Frank|authorlink2=Frank Lamprecht
|year=2001
|title=Fundamental Chess Endings
|publisher=Gambit Publications
|isbn= 1-901983-53-6
}}
*{{Citation
|last=Nunn|first=John|authorlink=John Nunn
|year=1981
|title=Tactical Chess Endings
|publisher=Batsford
|isbn= 0-7134-5937-9
}}
*{{Citation
|last=Nunn|first=John <!-- |authorlink=John Nunn -->
|year=1995
|title=Secrets of Minor-Piece Endings
|publisher=Batsford
|isbn= 0-8050-4228-8
}}
*{{Citation
 |last=Seirawan|first=Yasser|authorlink=Yasser Seirawan
 |title=Winning Chess Endings
 |year=2003
 |publisher=Everyman Chess
 |isbn= 1-85744-348-9
}}
*{{citation
 | last = Soltis |first = Andy |authorlink=Andy Soltis
 | year = 2010
 | month = January
 | title = Chess to Enjoy: EGTN
 | journal = [[Chess Life]] 
 | volume = 2010
 | issue = 1
 | pages = 42–43
 | doi =
 | id = 
 | url = 
 | format = 
 | accessdate = 
 }}
*{{Citation
 |last=Speelman|first=Jon|authorlink=Jon Speelman
 |last2=Tisdall|first2=Jon|authorlink2=Jonathan Tisdall
 |last3=Wade|first3=Bob|authorlink3=Robert Wade (chess player)
 |title=Batsford Chess Endings
 |year=1993
 |publisher=B. T. Batsford
 |isbn= 0-7134-4420-7
}} <!-- This is the currently-available version. isbn= 0-8050-2947-8 the 1993 edition. -->
*{{citation
|last=Troitzky|first=Alexey|authorlink=Alexey Troitsky
|year=2006
|title=Collection of Chess Studies (1937)
|publisher=Ishi Press
|isbn= 0-923891-10-2}} The last part (pages 197–257) is a supplement containing Troitzky's analysis of two knights versus pawns.

==External links==

[[International Grandmaster|Grandmaster]] and endgame specialist [[Karsten Müller]] wrote a helpful two-part article on this endgame called ''The Damned Pawn'' (in [[Portable Document Format|PDFs]]):
#[http://www.chesscafe.com/text/mueller35.pdf Part 1] about the Troitzky line and the technique
#[http://www.chesscafe.com/text/mueller36.pdf Part 2: the second Troitzky line solved] the winning line taking into account the 50-move rule, and more winning techniques and drawing zones.
*[http://www.chessvideos.tv/endgame-training/two-2-knights-vs-pawn-12.php Two Knights vs King and Pawn Trainer]
*[http://www.chessgames.com/perl/chessgame?gid=1090903 Smyslov vs. Lilienthal]
*[http://www.chessgames.com/perl/chessgame?gid=1090836 Norman vs. Lilienthal]

{{Chess}}

{{DEFAULTSORT:Two Knights Endgame}}
[[:Category:Chess endgames]]
[[:Category:Chess problems]]
[[:Category:Chess theory]]

== inverse function==
If ''x'', ''y'', and ''z'' are positive integers and ''z'' is a function of ''x'' and ''y'', ''z(''x'',''y'')'' = (''y''^2-3''y'')/2+''x''+1, given ''z'', what are ''x'' and ''y''? [[User:Bubba73|Bubba73]] <sup>[[User talk:Bubba73|You talkin' to me?]]</sup> 02:32, 7 November 2021 (UTC)