Answers.Body: <p>This was a great game! <a href="https://en.wikipedia.org/wiki/Hou_Yifan" rel="noreferrer">Hou Yifan</a> brilliantly outplayed her opponent  <a href="https://en.wikipedia.org/wiki/Fabiano_Caruana" rel="noreferrer">Fabiano Caruana</a> in a very positional middle game, and mind you using the <a href="https://en.wikipedia.org/wiki/Petrov&#39;s_Defence" rel="noreferrer">Petroff's defence</a> in the opening, which is Caruana's specialty by any stretch of the word! The endgame was very tricky, and Caruana proved his resilience and held the game to a draw in a very resourceful way, despite there having been missed chances for black.</p>
  
  <p>The short answer is: <strong>yes</strong> there was at least one key moment where black had a forced win in the <code>N+4p vs B+3p.</code> Having traded into a better endgame, pawn up and knight vs bishop, one of the critical moments of the game was after white's <code>64.Bb7</code> move, harassing black's pawn backbone which all lie on light-squares, at this moment black had in fact the possibility of achieving a <strong>pawn breakthrough</strong>, and create a passer!</p>
  
  <pre><code> [title "Position after white's inaccurate 64.Bb7"]
   [fen "8/1B6/p7/1p1p4/2p2n2/P1P1k3/1KP5/8 b - - 13 64"]
   [startflipped "0"]
  </code></pre>
  
  <p>To see the pawn breakthrough, the idea to spot is: noting that white's bishop is locked out of play by black's pawns on light-squares, once the black king gets to <code>d2</code>, the two kings on <code>b2</code> and <code>d2</code> are in opposition, and had there been no white blockaders on <code>c2-c3</code>, black would just advance their own <code>c</code> pawn and promote (diagram):</p>
  
  <p><img src="https://i.sstatic.net/yQuTW.jpg" width="250" height="250"></p>
  
  <p>So the question is: how could we create this desired pawn breakthrough, or in other words, </p>
  
  <h2>Can white's <code>c2-c3</code> pawns be removed after Kd2?</h2>
  
  <p>First key move is <code>Nd3+!!</code> The knight sacrifice removes the <code>c2</code> blockader, and if white were to accept the sacrifice, then <code>d4</code>
  removes the 2nd blockader (<code>c3</code>) and black's <code>c</code> pawn advance with check and promotes since the bishop is too slow in reaching the <code>a4-d1</code> diagonal to eye <code>c2.</code> Summary in diagrams:</p>
  
  <p><img src="https://i.sstatic.net/YCDcE.jpg" width="220" height="220">
  <img src="https://i.sstatic.net/7AaoA.jpg" width="220" height="220">
  <img src="https://i.sstatic.net/g6Eiw.jpg" width="220" height="220"></p>
  
  <h2>But does white have to take the knight on d3?</h2>
  
  <p>Well, let's consider the only reasonable king move: <code>Kb1</code>, as anything else just leaves both <code>c</code> pawns hanging to black's king. Then, the same pawn breakthrough on the <code>c</code>-file can be achieved, with the difference now that we first <em>win</em> the <code>c2</code> pawn with <code>Ne1-Nxc2</code> and then play <code>d4</code>. Let's see in diagrams:</p>
  
  <p><img src="https://i.sstatic.net/7Z3xC.jpg" width="220" height="220">
  <img src="https://i.sstatic.net/0ss1B.jpg" width="220" height="220">
  <img src="https://i.sstatic.net/FuzEk.jpg" width="220" height="220"></p>
  
  <p>and black again promotes a pawn on the <code>c</code>-file, as in the rightmost diagram: </p>
  
  <ul>
  <li><p>if white takes on <code>c4</code> then <code>dxc3+ Ka2</code> and <code>Nd4</code> wins by preventing <code>Bb3</code> and the <code>c</code> pawn runs.</p></li>
  <li><p>instead, if white plays <code>Ba4</code> immediately to cover <code>c2</code>, then <code>dxc3+ Ka2</code> and <code>Nd4</code> wins again since now black has two <code>c</code> pawns and white's bishop can only be sacrificed for one of them.</p></li>
  </ul>
  
  <p>And here's a summary of all the discussed variations:</p>
  
  <pre><code> [title "Pawn breakthrough for black"]
   [fen "8/1B6/p7/1p1p4/2p2n2/P1P1k3/1KP5/8 b - - 1 1"]
   [startflipped "0"]
  
   1... Kd2 {key starting move of the winning line, attacking both white pawns and covering the path for a pawn to promote along the c file.} 2. Bxa6 (2. Bc8 {an attempt to immediately re-route towards the defense of c2, but it is too slow and irrelevant as Nd3 still works} Nd3+ 3. Kb1 {cxd3 loses similarly to d4} Ne1 4. Bf5 Nxc2 5. Bxc2 Kxc3 {and black's connected passers win easily.}) Nd3+ 3. cxd3 (3. Kb1 Ne1 4. Bxb5 Nxc2 {Nxa3+ fork threatened} 5. Kb2 d4 6. Bxc4 (6. cxd4 c3+ 7. Ka2 Nb4+ 8. axb4 c2) (6. Ba4 dxc3+ 7. Ka2 Nd4 {and one of the c pawns promotes}) dxc3+ 7. Ka2 Nd4 {preventing Bb3 and c-pawn runs}) d4 4. cxd4 c3+ 5. Ka2 c2 {and black promotes}
  </code></pre>
  
  <p>So it's safe to say, <code>Kd2</code> followed by <code>Nd3+</code> was winning by force. What a beautiful pattern!</p>
  
  <p>Note that, at the stage of the game that we are talking about here (around move <code>64</code>), the players were playing by rote as they were relatively low on time, so Hou Yifan played the very natural <code>a5</code> in reply to <code>Bb7</code> (which in hindsight now we know to have been a blunder) and the winning pawn breakthrough was missed. She might have considered it but maybe couldn't work out all the details, e.g. the lines where white doesn't accept the sacrifice. Therefore, they played pragmatically considering their energy and remaining time on the clocks. Admittedly, even with more time, these strange pawn breakthroughs are never easy to spot, and tend to require a more relaxed mindset (like when one solves studies, and thus <em>knows</em> there's something to be found), and are often times missed/dismissed in practical play.</p>
  
  <p>Reminds me of Kramnik vs Anand (world championship 2008), where Anand found a beautiful breakthrough in this position: </p>
  
  <pre><code> [title "black to play and win"]
   [fen "8/1R3p1p/4pk2/8/PP3pn1/8/5PPP/2r2BK1 b - - 0 1"]
   [startflipped "0"]
  </code></pre>
  
  <p>The analysis is left to the reader ;)</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T14:11:38.723Z
Answers.Text: We were, clearly, both working diligently at the same time on our answers. :)
Answers.UserDisplayName: PhishMaster
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T12:32:45.003Z
Answers.LastActivityDate: 2020-02-02T12:32:45.003Z
Answers.UserDisplayName: Ellie
Answers.Body: <p>You left out THE single biggest factor in the position, at least initially: The knight has zero mobility. That said, she did, indeed miss two opportunities to win involving a queening scenario, but they were very tricky for a human.</p>
  
  <p>I remember this endgame when they played it, and now, the winning idea comes back to me.</p>
  
  <pre><code> [Event "Grenke Chess Classic 5th"]
   [Site "Karlsruhe/Baden-Baden"]
   [Date "2018.04.06"]
   [Round "6"]
   [White "Caruana, Fabiano"]
   [Black "Hou, Yifan"]
   [Result "1/2-1/2"]
   [ECO "C42"]
   [WhiteElo "2784"]
   [BlackElo "2654"]
   [PlyCount "196"]
   [EventDate "2018.03.31"]
   [EventType "tourn"]
   [EventRounds "9"]
   [EventCountry "GER"]
   [EventCategory "20"]
   [FEN ""]
  
   1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. Nc3 Nxc3 6. dxc3 Be7 7. Be3 O-O 8. Qd2 Nd7 9. O-O-O Nf6 10. h3 c5 11. Bf4 Be6 12. a3 d5 13. Ng5 Bd7 14. g4 Bc6 15. Bg2 Re8 16. Qd3 Bd6 17. Qg3 Bxf4+ 18. Qxf4 h6 19. Nf3 Qb8 20. Qxb8 Raxb8 21. Nd2 Re2 22. Rhf1 b6 23. Rde1 Rbe8 24. Rxe2 Rxe2 25. Kd1 Re6 26. f4 Kf8 27. Re1 Ke7 28. Rxe6+ Kxe6 29. Ke2 g5 30. fxg5 hxg5 31. Nf3 Ne4 32. Ke3 f6 33. Bf1 Nd6 34. Nd2 f5 35. gxf5+ Nxf5+ 36. Kf2 Ke5 37. Nf3+ Kf4 38. Ne1 c4 39. Ng2+ Ke5 40. Be2 Ba4 41. Bd1 Be8 42. Bg4 Bg6 43. Be2 Nd6 44. Ne3 Ne4+ 45. Kf3 Nf6 46. Kf2 Bf7 47. Ng2 Be6 48. h4 Ne4+ 49. Kf3 Bh3 50. hxg5 Nxg5+ 51. Kg3 Bxg2 52. Kxg2 Ne6 53. Kf2 Nc5 54. Bh5 Na4 55. Be8 Nxb2 56. Ke2 a6 57. Kd2 b5 58. Bh5 Na4 59. Bf7 Nc5 60. Kc1 Ne6 61. Kb2 Nf4 62. Be8 Ke4 63. Bc6 Ke3 64. Bb7 a5 (64...Kd2 $1 65. Bxa6 Nd3+ 66. Kb1 (66. cxd3 d4 {Queens.}) Ne1 67. Bxb5 Kxc3 68. Ba4 Kd2 69. Kb2 d4 70. Kb1 Nxc2 71. Bxc2 d3 72. Ba4 c3 $19) 65. Kc1 Ne2+ 66. Kb2 Kd2 (66... Nf4 $1 67. Bc6 Kd2 68. Bxb5 Ne2 69. Bd7 Nxc3 70. Bc6 d4 71. Be8 Nd1+ 72. Kb1 Ne3 73. Bg6 c3 74. a4 Nxc2 75. Bxc2 d3 76. Bb3 c2+ $19) 67. Bxd5 Nxc3 68. Bc6 Nd1+ 69. Kb1 Ne3 70. Be4 c3 71. Bh7 Nc4 72. Ka2 Ne5 73. Be4 Ke3 74. Bh7 Nc6 75. Bg6 Kd2 76. Be4 Nd4 77. Kb1 Kd1 78. Bd5 Nf5 79. Be4 Ne3 80. Bg6 Kd2 81. Bd3 Nc4 82. Ka2 Nb6 83. Kb1 Nc4 84. Ka2 a4 85. Bg6 Kc1 86. Bd3 Nd6 87. Ka1 Nb7 88. Ka2 Nc5 89. Bf5 Na6 90. Bg6 Kd2 91. Bh7 Nb8 92. Bd3 Nc6 93. Bxb5 Nd4 94. Bxa4 Nxc2 95. Kb3 Nd4+ 96. Kc4 Nc2 97. Kb3 Ne3 98. Kb4 c2 1/2-1/2
  </code></pre>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T19:54:20.000Z
Answers.Text: and that idea was ?  or are we supposed to play through all the moves until the lightbulb shines over our head.
Answers.UserDisplayName: edwina oliver
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T20:02:47.457Z
Answers.Text: There are only two variations, and one small subvariation, to play over. If you do not understand them, ask a question.
Answers.UserDisplayName: PhishMaster
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-05T19:11:07.580Z
Answers.Text: @user929304 They are basically the same, and we were working on them simultaneously, posting our respective answers only three minutes apart, with hers taking a bit longer to make. It does not bother me either way, and the public liked the diagrams more (they usually do). Feel free to accept hers. They are truly pretty equal, and if they were not, I would tell you why I believed that, and explain it logically. That is not the case here.
Answers.UserDisplayName: PhishMaster
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-05T21:43:35.763Z
Answers.Text: @PhishMaster cool, thank you for your candour!
Answers.UserDisplayName: user929304
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T12:27:32.187Z
Answers.LastActivityDate: 2020-02-02T12:32:58.467Z
Answers.LastEditDate: 2020-02-02T12:32:58.467Z
Answers.UserDisplayName: PhishMaster
Answers.Body: <p>It is a draw.
  Black can protect the horsie.  And then his pawn chain but then:</p>
  
  <p>There is no viable way to force a pawn through where it wont be captured leaving K+N vs K.  Which is a draw. </p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T20:14:33.670Z
Answers.Text: It is not a draw. When Stockfish 11 registers an evaluation over +10, that means it is a clear win. https://en.chessbase.com/post/grenke-chess-classic-2018-round-6
Answers.UserDisplayName: PhishMaster
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T20:27:22.947Z
Answers.Text: Here is a video of Peter Leko showing the win. I hope you believe a former World Championship challenger. https://www.youtube.com/watch?v=NhgPCy492dA
Answers.UserDisplayName: PhishMaster
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-02T19:58:41.340Z
Answers.LastActivityDate: 2020-02-02T19:58:41.340Z
Answers.UserDisplayName: edwina oliver
Body: <p>I am currently studying the classical game of <a href="https://www.chessgames.com/perl/chessgame?gid=1917432" rel="noreferrer">Fabiano Caruana vs Hou Yifan played in 2018</a>. The game ended in a fighting draw, but for the most part of the ending, starting from move <code>55</code> onward (see diagram), black was a pawn up, in a knight vs bishop and 4 vs 3 pawn structure. </p>
  
  <pre><code> [title "Fabiano Caruana vs Hou Yifan 2018"]
   [fen ""]
   [startply "110"]
  
   1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. Nc3 Nxc3 6. dxc3 Be7 7. Be3 O-O 8. Qd2 Nd7 9. O-O-O Nf6 10. h3 c5 11. Bf4 Be6 12. a3 d5 13. Ng5 Bd7 14. g4 Bc6 15. Bg2 Re8 16. Qd3 Bd6 17. Qg3 Bxf4+ 18. Qxf4 h6 19. Nf3 Qb8 20. Qxb8 Raxb8 21. Nd2 Re2 22. Rhf1 b6 23. Rde1 Rbe8 24. Rxe2 Rxe2 25. Kd1 Re6 26. f4 Kf8 27. Re1 Ke7 28. Rxe6+ Kxe6 29. Ke2 g5 30. fxg5 hxg5 31. Nf3 Ne4 32. Ke3 f6 33. Bf1 Nd6 34. Nd2 f5 35. gxf5+ Nxf5+ 36. Kf2 Ke5 37. Nf3+ Kf4 38. Ne1 c4 39. Ng2+ Ke5 40. Be2 Ba4 41. Bd1 Be8 42. Bg4 Bg6 43. Be2 Nd6 44. Ne3 Ne4+ 45. Kf3 Nf6 46. Kf2 Bf7 47. Ng2 Be6 48. h4 Ne4+ 49. Kf3 Bh3 50. hxg5 Nxg5+ 51. Kg3 Bxg2 52. Kxg2 Ne6 53. Kf2 Nc5 54. Bh5 Na4 55. Be8 Nxb2 56. Ke2 a6 57. Kd2 b5 58. Bh5 Na4 59. Bf7 Nc5 60. Kc1 Ne6 61. Kb2 Nf4 62. Be8 Ke4 63. Bc6 Ke3 64. Bb7 a5 65. Kc1 Ne2+ 66. Kb2 Kd2 67. Bxd5 Nxc3 68. Bc6 Nd1+ 69. Kb1 Ne3 70. Be4 c3 71. Bh7 Nc4 72. Ka2 Ne5 73. Be4 Ke3 74. Bh7 Nc6 75. Bg6 Kd2 76. Be4 Nd4 77. Kb1 Kd1 78. Bd5 Nf5 79. Be4 Ne3 80. Bg6 Kd2 81. Bd3 Nc4 82. Ka2 Nb6 83. Kb1 Nc4 84. Ka2 a4 85. Bg6 Kc1 86. Bd3 Nd6 87. Ka1 Nb7 88. Ka2 Nc5 89. Bf5 Na6 90. Bg6 Kd2 91. Bh7 Nb8 92. Bd3 Nc6 93. Bxb5 Nd4 94. Bxa4 Nxc2 95. Kb3 Nd4+ 96. Kc4 Nc2 97. Kb3 Ne3 98. Kb4 c2 1/2-1/2
  </code></pre>
  
  <p>However, at a first glance, I can see that other than being a pawn up, black is also positionally better, since:</p>
  
  <ul>
  <li><p>The white king stands defensively throughout the game, the black king is well active and centralised. </p></li>
  <li><p>the white pawns are two isolated islands, and doubled on the c-file.</p></li>
  <li><p>there are only pawns on one side of the board (queenside), which makes the bishop's long-ranged scope less effective.</p></li>
  <li><p>Black has a healthy pawn chain, with remaining flexibility in how they want to fix the pawn structure.</p></li>
  <li><p>And once the pawn structure fixed, the knight can be a threat to pawns on both colours, while the bishop's limited to light-squares.</p></li>
  </ul>
  
  <p>With all these in mind, my question is:</p>
  
  <ul>
  <li>Was this endgame really a drawn one, or did Hou Yifan miss a win?</li>
  </ul>
  
Tags: <endgame><planning><bishop-endgame><knight-endgame>
Title: Knight vs bishop endgame in Fabiano Caruana vs Hou Yifan 2018
ThreadText: # Knight vs bishop endgame in Fabiano Caruana vs Hou Yifan 2018
  
  (Asked by: username_0 on 2020-02-02)
  
  I am currently studying the classical game of Fabiano Caruana vs Hou Yifan played in 2018. The game ended in a fighting draw, but for the most part of the ending, starting from move 55 onward (see diagram), black was a pawn up, in a knight vs bishop and 4 vs 3 pawn structure. 
   [title "Fabiano Caruana vs Hou Yifan 2018"]
   [fen ""]
   [startply "110"]
  
   1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. Nc3 Nxc3 6. dxc3 Be7 7. Be3 O-O 8. Qd2 Nd7 9. O-O-O Nf6 10. h3 c5 11. Bf4 Be6 12. a3 d5 13. Ng5 Bd7 14. g4 Bc6 15. Bg2 Re8 16. Qd3 Bd6 17. Qg3 Bxf4+ 18. Qxf4 h6 19. Nf3 Qb8 20. Qxb8 Raxb8 21. Nd2 Re2 22. Rhf1 b6 23. Rde1 Rbe8 24. Rxe2 Rxe2 25. Kd1 Re6 26. f4 Kf8 27. Re1 Ke7 28. Rxe6+ Kxe6 29. Ke2 g5 30. fxg5 hxg5 31. Nf3 Ne4 32. Ke3 f6 33. Bf1 Nd6 34. Nd2 f5 35. gxf5+ Nxf5+ 36. Kf2 Ke5 37. Nf3+ Kf4 38. Ne1 c4 39. Ng2+ Ke5 40. Be2 Ba4 41. Bd1 Be8 42. Bg4 Bg6 43. Be2 Nd6 44. Ne3 Ne4+ 45. Kf3 Nf6 46. Kf2 Bf7 47. Ng2 Be6 48. h4 Ne4+ 49. Kf3 Bh3 50. hxg5 Nxg5+ 51. Kg3 Bxg2 52. Kxg2 Ne6 53. Kf2 Nc5 54. Bh5 Na4 55. Be8 Nxb2 56. Ke2 a6 57. Kd2 b5 58. Bh5 Na4 59. Bf7 Nc5 60. Kc1 Ne6 61. Kb2 Nf4 62. Be8 Ke4 63. Bc6 Ke3 64. Bb7 a5 65. Kc1 Ne2+ 66. Kb2 Kd2 67. Bxd5 Nxc3 68. Bc6 Nd1+ 69. Kb1 Ne3 70. Be4 c3 71. Bh7 Nc4 72. Ka2 Ne5 73. Be4 Ke3 74. Bh7 Nc6 75. Bg6 Kd2 76. Be4 Nd4 77. Kb1 Kd1 78. Bd5 Nf5 79. Be4 Ne3 80. Bg6 Kd2 81. Bd3 Nc4 82. Ka2 Nb6 83. Kb1 Nc4 84. Ka2 a4 85. Bg6 Kc1 86. Bd3 Nd6 87. Ka1 Nb7 88. Ka2 Nc5 89. Bf5 Na6 90. Bg6 Kd2 91. Bh7 Nb8 92. Bd3 Nc6 93. Bxb5 Nd4 94. Bxa4 Nxc2 95. Kb3 Nd4+ 96. Kc4 Nc2 97. Kb3 Ne3 98. Kb4 c2 1/2-1/2
  
  However, at a first glance, I can see that other than being a pawn up, black is also positionally better, since:
  
  The white king stands defensively throughout the game, the black king is well active and centralised. 
  the white pawns are two isolated islands, and doubled on the c-file.
  there are only pawns on one side of the board (queenside), which makes the bishop's long-ranged scope less effective.
  Black has a healthy pawn chain, with remaining flexibility in how they want to fix the pawn structure.
  And once the pawn structure fixed, the knight can be a threat to pawns on both colours, while the bishop's limited to light-squares.
  
  With all these in mind, my question is:
  
  Was this endgame really a drawn one, or did Hou Yifan miss a win?
  
  **Tags:** endgame, planning, bishop-endgame, knight-endgame
  
  
  ---
  
  ## 3 answers
  
  ### Answer by username_1 on 2020-02-02. Score: 16
  
  This was a great game! Hou Yifan brilliantly outplayed her opponent  Fabiano Caruana in a very positional middle game, and mind you using the Petroff's defence in the opening, which is Caruana's specialty by any stretch of the word! The endgame was very tricky, and Caruana proved his resilience and held the game to a draw in a very resourceful way, despite there having been missed chances for black.
  The short answer is: yes there was at least one key moment where black had a forced win in the N+4p vs B+3p. Having traded into a better endgame, pawn up and knight vs bishop, one of the critical moments of the game was after white's 64.Bb7 move, harassing black's pawn backbone which all lie on light-squares, at this moment black had in fact the possibility of achieving a pawn breakthrough, and create a passer!
   [title "Position after white's inaccurate 64.Bb7"]
   [fen "8/1B6/p7/1p1p4/2p2n2/P1P1k3/1KP5/8 b - - 13 64"]
   [startflipped "0"]
  
  To see the pawn breakthrough, the idea to spot is: noting that white's bishop is locked out of play by black's pawns on light-squares, once the black king gets to d2, the two kings on b2 and d2 are in opposition, and had there been no white blockaders on c2-c3, black would just advance their own c pawn and promote (diagram):
  
  So the question is: how could we create this desired pawn breakthrough, or in other words, 
  Can white's c2-c3 pawns be removed after Kd2?
  First key move is Nd3+!! The knight sacrifice removes the c2 blockader, and if white were to accept the sacrifice, then d4
  removes the 2nd blockader (c3) and black's c pawn advance with check and promotes since the bishop is too slow in reaching the a4-d1 diagonal to eye c2. Summary in diagrams:
  
  
  
  But does white have to take the knight on d3?
  Well, let's consider the only reasonable king move: Kb1, as anything else just leaves both c pawns hanging to black's king. Then, the same pawn breakthrough on the c-file can be achieved, with the difference now that we first win the c2 pawn with Ne1-Nxc2 and then play d4. Let's see in diagrams:
  
  
  
  and black again promotes a pawn on the c-file, as in the rightmost diagram: 
  
  if white takes on c4 then dxc3+ Ka2 and Nd4 wins by preventing Bb3 and the c pawn runs.
  instead, if white plays Ba4 immediately to cover c2, then dxc3+ Ka2 and Nd4 wins again since now black has two c pawns and white's bishop can only be sacrificed for one of them.
  
  And here's a summary of all the discussed variations:
   [title "Pawn breakthrough for black"]
   [fen "8/1B6/p7/1p1p4/2p2n2/P1P1k3/1KP5/8 b - - 1 1"]
   [startflipped "0"]
  
   1... Kd2 {key starting move of the winning line, attacking both white pawns and covering the path for a pawn to promote along the c file.} 2. Bxa6 (2. Bc8 {an attempt to immediately re-route towards the defense of c2, but it is too slow and irrelevant as Nd3 still works} Nd3+ 3. Kb1 {cxd3 loses similarly to d4} Ne1 4. Bf5 Nxc2 5. Bxc2 Kxc3 {and black's connected passers win easily.}) Nd3+ 3. cxd3 (3. Kb1 Ne1 4. Bxb5 Nxc2 {Nxa3+ fork threatened} 5. Kb2 d4 6. Bxc4 (6. cxd4 c3+ 7. Ka2 Nb4+ 8. axb4 c2) (6. Ba4 dxc3+ 7. Ka2 Nd4 {and one of the c pawns promotes}) dxc3+ 7. Ka2 Nd4 {preventing Bb3 and c-pawn runs}) d4 4. cxd4 c3+ 5. Ka2 c2 {and black promotes}
  
  So it's safe to say, Kd2 followed by Nd3+ was winning by force. What a beautiful pattern!
  Note that, at the stage of the game that we are talking about here (around move 64), the players were playing by rote as they were relatively low on time, so Hou Yifan played the very natural a5 in reply to Bb7 (which in hindsight now we know to have been a blunder) and the winning pawn breakthrough was missed. She might have considered it but maybe couldn't work out all the details, e.g. the lines where white doesn't accept the sacrifice. Therefore, they played pragmatically considering their energy and remaining time on the clocks. Admittedly, even with more time, these strange pawn breakthroughs are never easy to spot, and tend to require a more relaxed mindset (like when one solves studies, and thus knows there's something to be found), and are often times missed/dismissed in practical play.
  Reminds me of Kramnik vs Anand (world championship 2008), where Anand found a beautiful breakthrough in this position: 
   [title "black to play and win"]
   [fen "8/1R3p1p/4pk2/8/PP3pn1/8/5PPP/2r2BK1 b - - 0 1"]
   [startflipped "0"]
  
  The analysis is left to the reader ;)
  
  > **Comments:**
  > **username_2**: We were, clearly, both working diligently at the same time on our answers. :)  
  
  ---
  
  ### Answer by username_2 on 2020-02-02. Score: 8
  
  You left out THE single biggest factor in the position, at least initially: The knight has zero mobility. That said, she did, indeed miss two opportunities to win involving a queening scenario, but they were very tricky for a human.
  I remember this endgame when they played it, and now, the winning idea comes back to me.
   [Event "Grenke Chess Classic 5th"]
   [Site "Karlsruhe/Baden-Baden"]
   [Date "2018.04.06"]
   [Round "6"]
   [White "Caruana, Fabiano"]
   [Black "Hou, Yifan"]
   [Result "1/2-1/2"]
   [ECO "C42"]
   [WhiteElo "2784"]
   [BlackElo "2654"]
   [PlyCount "196"]
   [EventDate "2018.03.31"]
   [EventType "tourn"]
   [EventRounds "9"]
   [EventCountry "GER"]
   [EventCategory "20"]
   [FEN ""]
  
   1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. Nc3 Nxc3 6. dxc3 Be7 7. Be3 O-O 8. Qd2 Nd7 9. O-O-O Nf6 10. h3 c5 11. Bf4 Be6 12. a3 d5 13. Ng5 Bd7 14. g4 Bc6 15. Bg2 Re8 16. Qd3 Bd6 17. Qg3 Bxf4+ 18. Qxf4 h6 19. Nf3 Qb8 20. Qxb8 Raxb8 21. Nd2 Re2 22. Rhf1 b6 23. Rde1 Rbe8 24. Rxe2 Rxe2 25. Kd1 Re6 26. f4 Kf8 27. Re1 Ke7 28. Rxe6+ Kxe6 29. Ke2 g5 30. fxg5 hxg5 31. Nf3 Ne4 32. Ke3 f6 33. Bf1 Nd6 34. Nd2 f5 35. gxf5+ Nxf5+ 36. Kf2 Ke5 37. Nf3+ Kf4 38. Ne1 c4 39. Ng2+ Ke5 40. Be2 Ba4 41. Bd1 Be8 42. Bg4 Bg6 43. Be2 Nd6 44. Ne3 Ne4+ 45. Kf3 Nf6 46. Kf2 Bf7 47. Ng2 Be6 48. h4 Ne4+ 49. Kf3 Bh3 50. hxg5 Nxg5+ 51. Kg3 Bxg2 52. Kxg2 Ne6 53. Kf2 Nc5 54. Bh5 Na4 55. Be8 Nxb2 56. Ke2 a6 57. Kd2 b5 58. Bh5 Na4 59. Bf7 Nc5 60. Kc1 Ne6 61. Kb2 Nf4 62. Be8 Ke4 63. Bc6 Ke3 64. Bb7 a5 (64...Kd2 $1 65. Bxa6 Nd3+ 66. Kb1 (66. cxd3 d4 {Queens.}) Ne1 67. Bxb5 Kxc3 68. Ba4 Kd2 69. Kb2 d4 70. Kb1 Nxc2 71. Bxc2 d3 72. Ba4 c3 $19) 65. Kc1 Ne2+ 66. Kb2 Kd2 (66... Nf4 $1 67. Bc6 Kd2 68. Bxb5 Ne2 69. Bd7 Nxc3 70. Bc6 d4 71. Be8 Nd1+ 72. Kb1 Ne3 73. Bg6 c3 74. a4 Nxc2 75. Bxc2 d3 76. Bb3 c2+ $19) 67. Bxd5 Nxc3 68. Bc6 Nd1+ 69. Kb1 Ne3 70. Be4 c3 71. Bh7 Nc4 72. Ka2 Ne5 73. Be4 Ke3 74. Bh7 Nc6 75. Bg6 Kd2 76. Be4 Nd4 77. Kb1 Kd1 78. Bd5 Nf5 79. Be4 Ne3 80. Bg6 Kd2 81. Bd3 Nc4 82. Ka2 Nb6 83. Kb1 Nc4 84. Ka2 a4 85. Bg6 Kc1 86. Bd3 Nd6 87. Ka1 Nb7 88. Ka2 Nc5 89. Bf5 Na6 90. Bg6 Kd2 91. Bh7 Nb8 92. Bd3 Nc6 93. Bxb5 Nd4 94. Bxa4 Nxc2 95. Kb3 Nd4+ 96. Kc4 Nc2 97. Kb3 Ne3 98. Kb4 c2 1/2-1/2
  
  > **Comments:**
  > **username_3**: and that idea was ?  or are we supposed to play through all the moves until the lightbulb shines over our head.  
  > **username_2**: There are only two variations, and one small subvariation, to play over. If you do not understand them, ask a question.  
  > **username_2**: @username_0 They are basically the same, and we were working on them simultaneously, posting our respective answers only three minutes apart, with hers taking a bit longer to make. It does not bother me either way, and the public liked the diagrams more (they usually do). Feel free to accept hers. They are truly pretty equal, and if they were not, I would tell you why I believed that, and explain it logically. That is not the case here.  
  > **username_0**: @username_2 cool, thank you for your candour!  
  
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  ### Answer by username_3 on 2020-02-02. Score: -5
  
  It is a draw.
  Black can protect the horsie. And then his pawn chain but then:
  
  There is no viable way to force a pawn through where it wont be captured leaving K+N vs K. Which is a draw.
  
  > **Comments:**
  > **username_2**: It is not a draw. When Stockfish 11 registers an evaluation over +10, that means it is a clear win. https://en.chessbase.com/post/grenke-chess-classic-2018-round-6  
  > **username_2**: Here is a video of Peter Leko showing the win. I hope you believe a former World Championship challenger. https://www.youtube.com/watch?v=NhgPCy492dA  
  
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