The Troitzky line
An important variation in the analysis of the game Magnus Carlsen-Wang Yue ended in the following position with two knights against two pawns, which is won although two knights alone can't checkmate. Due to several requests, I would like to consider this problem once more in detail:
65.Nf4!
It is of vital importance to safely block the f-pawn on the Troitzky line a4-b6-c5-d4-e4-f5-g6-h4. This always wins, however, possibly not with regard to the 50-moves rule. Taking this into account, the new line is a5, c5, d5, e5, f5, h5 , always with positions existing on the b- and g-file which can't be won within 50 moves. But those are very rare. The additional black d-pawn plays no part here since it can easily be raked in which, after blockading the f-pawn, is White's next goal now:
65...Kb4
66.Kc2
Kc5
67.Nf7
Kc4
68.Ne5+
Kb5
69.Kd3
Kc5
70.Nc4
Kb4
71.Kxd4
Now the black king must be forced into a corner in which White can checkmate it. In this very favourable case, White principally can checkmate in all four corners since the blockade knight on f4 is placed quite centrally. However, there are examples where this is different. There is one rule in particular: if the pawn has passed the Troitzky line, then there are drawing and losing zones for the defending king. The drawing zones can be safe edges or corners, but not necessarily. In the present case, of course there is no salvation. The defending king will try to finally escape in the direction of the north-eastern corner in order to survive as long as possible. Most favourable for him would be squares like g5, g4 or f3, yet in this case he can't get there if White doesn't let him.
71...Ka4
72.Ne5
Kb5
73.Ned3
Kc6
[If the defender tries to escape towards the south, play might continue as follows: 73...Ka4
74.Kc4
Ka3?!
75.Nc5
Kb2
76.Kd3
Kc1
77.Na4
Kd1
Now follows a typical technique, known as 'Henry's side check': 78.Nb2+
presenting Black the choice in which corner he wants to get checkmated: 78...Ke1
(78...Kc1
79.Kc3
Kb1
80.Nbd3
Ka2
81.Kb4
Kb1
82.Kb3
Ka1
83.Nd5
f4
84.Nc3
f3
85.Nb4
f2
86.Nc2#
) 79.Ke3
Kf1
80.Nbd3
Kg1
81.Ne1
Kf1
82.Neg2
Kg1
83.Ke2
Kh2
84.Kf3
Kg1
85.Ne3
Kh2
86.Kf2
Kh1
87.Kg3
Kg1
88.Ne2+
Kh1
89.Nd1
f4+
90.Kh3
f3
91.Nf2#
]
74.Kc4
Kd6
75.Kb5
Ke7
76.Ne5
Kd6
77.Nf3
Since the knights are forming a barrier, now the king can advance. This knights' placement is often very effective at least against central or bishop pawns.
77...Kd7
78.Kc5
Kc7
79.Nd4
Kc8
80.Kc6
Kd8
[80...Kb8?!
runs into the direct 81.Nb5
Kc8
82.Ne6
f4
83.Nd6+
Kb8
84.Kb6
f3
85.Nc7
f2
86.Na6+
Ka8
87.Nb5
f1Q
88.Nbc7#
]
81.Nde6+
Ke7
82.Kc7
Kf7
83.Kd7
Since the knights are not able to lose tempi, it is very important to choose the right squares with the king. Triangle manoeuvres often play a vital part in this endgame too.
83...Kf6
84.Kd6
Kf7
85.Nd4
Ke8
[After 85...Kf6
86.Nf3
Kf7
87.Kd7
Kf6
the white king bypasses his counterpart with 88.Ke8
Kg7
89.Ke7
Kg8
90.Nd4
Kg7
91.Nde6+
Kh7
92.Kf7
Kh6
93.Kf6
Kh7
94.Kg5
Kg8
95.Kg6
Kh8
96.Nd5
f4
97.Nf6
f3
98.Ng5
f2
99.Nf7#
]
86.Nc6
Kf7
87.Ke5
Kf8
88.Kf6
Ke8
89.Ke6
Kf8
90.Ne5
Ke8
91.Nf7
Kf8
92.Nd6
Kg7
93.Ke7
Kh7
94.Kf7
Kh6
95.Kf6
Kh7
96.Kg5
Kg7
and now finally the moment has come where the blockade knight initiates the final attack:
97.Ne6+
Kh7
98.Ne8
f4
99.Nf6+
Kh8
100.Kg6
f3
101.Ng5
f2
102.Nf7#
For those who want to delve even deeper into this fascinating territory, I would like to indicate some sources in the following:
A. Troitzky, Collection of Chess Studies, 1937
Richard Forster, Late Knight column 24, Archive of www.chesscafe.com,
April 2000, Late Knights' Tango: The Troitzky Endgame
Karsten Müller, Endgame Corner column 35 and 36, Archive of www.chesscafe.com, November and December 2003, The damned pawn
Balashov, Prandstetter, Basic Endgames, Prague Chess Agency 1992, S.55-65
Cheron, Text and handbook of endgames, Volume 2, 2nd edition,
Verlag Das Schach Archiv 1964, p. 220–250
Speelman, Tisdall, Wade, Batsford Chess Endings, Batsford 1993, p. 112–115 1-0