Answers.Body: <p>@SmallChess is right. This position is absolutely hopeless for White, and there is a very human strategy for Black, which consists here in not exchanging pieces, but <strong>hunting the white king</strong>. This strategy will force White to exchange pieces unfavorably or White will lose material to avoid being mate.</p>
  
  <p>The principle behind this strategy is simple <strong><em>"with three or more pieces in the attack the enemy king will not be able to escape."</em></strong></p>
  
  <p>Here is an example where nothing is forced but it's very difficult to suggest good ideas for White. White is completely squeezed.</p>
  
  <pre><code>[White "White"]
  [Black "Black to move"]
  [FEN "8/rp4p1/1N1nkb2/3n4/6R1/1R6/8/2K5 b - - 1 38"]
  
  1...Nc3! {Black chases the king and cuts off the two rooks. They can't easily coordinate.} 2.Nc4 Nde4! {same strategy: coordinating the pieces to go against the White king} 3.Kc2 (3.Ra3 {is probably better but now Black is happy to exchange and push the g pawn} 3...Rxa3 4. Nxa3 Kf5 5.Rg2 g5-+) b5 4.Ne3 (4.Rxb5 Nxb5 5.Rxe4 Kf5-+ {does not help White}) Rc7 {here with get our principle of three or more pieces in the attack and the White king feels unsafe} 5.Rg2 (5.Kd3 Nf2-+) Na2! 6.Kb1? (6. Kd1 {is better but the King does not escape} Rc1+ 7. Ke2 Nac3+ 8. Kf3 Rh1-+) Rc1 7.Kxa2 Ra1#
  </code></pre>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-04-26T12:19:26.707Z
Answers.Text: I am asking for a sure strategy, not one just based on general principles, but thank you for your 'general principle'. I actually believe that it would be easier to prove a winning strategy by exchanging the knight. If you do not think so, why, and can you prove your strategy correct? If you look at the links I just added to my question, you will see that sometimes human analysis is not only possible but also can be superior to computer analysis. That is why I am not interested in how StockFish or other engines would play from this position.
Answers.UserDisplayName: user21820
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-04-27T04:17:35.810Z
Answers.Text: Clearly, you do not understand the meaning of "perfect play" and "optimal strategy", which are basic concepts in combinatorial game theory, which includes the study of chess. If you have basic mathematical knowledge, including induction and recursive definition, I can briefly explain it to you. Otherwise, you may want to find an introductory book on combinatorial game theory and read it up on your own.
Answers.UserDisplayName: user21820
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-04-25T18:36:57.003Z
Answers.LastActivityDate: 2020-04-26T09:04:55.940Z
Answers.LastEditDate: 2020-04-26T09:04:55.940Z
Answers.UserDisplayName: Kortchnoi
Answers.Body: <p>The position looks clearly winning for Black, and I don't think it makes much of a difference whether you exchange knights or not. It might make your task simpler, but Black's knight could be a bit stronger than White's.</p>
  
  <p>To win all you really have to do is push one of your pawns, with the support of the king. Your other pieces can be used to block checks, and the minor pieces can also chase any blockading rooks out of the pawn's way. Eventually things will get to the point where you win a White rook for the pawn and a minor piece (in the worst case scenario), which will result in an easily won position.</p>
  
  <p>I'm not sure how to give a certain proof that guarantees a win, as there's no forced mate yet. It's just a position where Black is up 4 points, so a winning edge should be taken for granted at this point.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-04-26T12:21:26.360Z
Answers.Text: I agree with your second paragraph, and in fact that is precisely what motivated my question, because it feels like there may be a general principle along these lines that can be proven. I am not sure what you mean by "the b6-knight is kind of cut off" since it can escape if Black does not trade now.
Answers.UserDisplayName: user21820
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Answers.CreationDate: 2020-04-26T12:31:46.127Z
Answers.Text: To give more detail, we already know that a KRK endgame is won, so if we eliminate all the White pieces without sacrificing our rook, then we are done. Let "A→B" mean "A protects B", and label the pawns P1,P2 and knights N1,N2. Then we can have R→N1→P1 and K→N2→P2 and use B to block perpetual checks. This is just a bit short, because we need to prove that we can handle a double rook attack as well as push the pawn P2. Once we can prove that, we are done, because it forces White to sacrifice a R for N2+P2, leaving us with KRBNP against KR, where it should be easy to prove a win.
Answers.UserDisplayName: user21820
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Answers.CreationDate: 2020-04-26T12:32:51.273Z
Answers.Text: Okay I saw your edit, but I'm still not convinced there is a clear sense in which Black's knight is stronger than White's.
Answers.UserDisplayName: user21820
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-04-26T12:37:28.077Z
Answers.Text: @user21820 I'm not really sure how a formal proof like the one you refer to would work in chess. At some point doing it seems like overkill, since the position is clearly winning. A practical "proof" would be letting yourself play Stockfish and seeing that you're able to consistently beat it. Also I edited my answer because you're right the knight can escape. Black's knight still might be slightly better though because if Black meets Nc4 with ...Nde4, the knight will be temporarily undefended on c4. Meanwhile, Black's d5-knight could go to the strong c3-square.
Answers.UserDisplayName: Inertial Ignorance
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Answers.Text: See the links in my (edited) question for examples of what I would consider as complete human analysis of optimal strategy. The motivation for looking for such a proof is that anyone who comes up with a human-readable proof would have to distill the strategy to a handful of provable principles, which would be relevant not only to this position but infinitely many others. In other words, the core ingredients of such proofs would have very wide applicability.
Answers.UserDisplayName: user21820
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Answers.LastEditDate: 2020-04-26T12:31:44.143Z
Answers.UserDisplayName: Inertial Ignorance
Body: <p><a href="https://chess.stackexchange.com/posts/comments/46636">This line of play</a> leads to this position:</p>
  
  <pre><code>[White "White"]
  [Black "Black to move"]
  [FEN "8/rp4p1/1N1nkb2/3n4/6R1/1R6/8/2K5 b - - 1 38"]
  </code></pre>
  
  <p>StockFish 10+ assures me that Black has a big advantage, but I am interested to know whether it is possible to humanly analyze it and be completely certain of a winning strategy. Although SF suggests keeping my pieces, my instinct is to exchange as many pieces as possible except the pawns, while pushing the pawns to queen, which includes taking the knight immediately. However, is there any hope of <strong>proving</strong> a winning strategy here, rather than relying on intuition? I think this is somewhat plausible because Black has enough minor pieces for the king to hide behind (even after exchanging 2 of them), and so White cannot make sustained checks. The question is whether there is a systematic way for Black to push the pawns that is guaranteed to work.</p>
  
  <p>I do not see any convincing reason that there cannot be a proof of a winning strategy, because there are numerous examples of endgames with human analysis of the optimal strategy (e.g. <a href="https://chess.stackexchange.com/q/9003">a</a>,<a href="https://chess.stackexchange.com/q/28225">b</a>,<a href="https://chess.stackexchange.com/q/12220">c</a>,<a href="https://chess.stackexchange.com/q/18601">d</a>,<a href="https://chess.stackexchange.com/a/1725">e</a>). If the position was not so imbalanced, I know it would be much harder to prove a win. But here, Black has sufficiently many pieces that I feel there should be a proof that we can force some trades to get to a certain winning position. That is also why I would choose to take the knight immediately to hopefully make a full analysis humanly tractable.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-04-25T10:00:59.410Z
Comments.Text: @bof: Obviously, we are not asking about proving a strategy for Black very early in the game, otherwise it would surely be infeasible. My mathematical intuition tells me that this game might have a feasible human analysis. As for the board, yes Black's side is at the bottom as the default viewer setting makes it (if the FEN says "black to move"), so Black's pawns are at their starting squares.
Comments.UserDisplayName: user21820
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-04-25T11:24:13.803Z
Comments.Text: (-1) I don't understand... This is a total overwhelming position for Black. White should resign here. Absolutely hopeless. No hope at all. As long as Black doesn't blunder a pawn/piece, simply push whenever there is an opportunity, move the pieces forward etc is 100% win.
Comments.UserDisplayName: SmallChess
Tags: <analysis><strategy><endgame><pawn-promotion><rook-and-minor-endgame>
Title: Human analysis for a KRBNNPP against KRRN ending?
ThreadText: # Human analysis for a KRBNNPP against KRRN ending?
  
  (Asked by: username_0 on 2020-04-25)
  
  [This line of play](https://chess.stackexchange.com/posts/comments/46636) leads to this position:
  
  ```
  [White "White"]
  [Black "Black to move"]
  [FEN "8/rp4p1/1N1nkb2/3n4/6R1/1R6/8/2K5 b - - 1 38"]
  
  ```
  
  StockFish 10+ assures me that Black has a big advantage, but I am interested to know whether it is possible to humanly analyze it and be completely certain of a winning strategy. Although SF suggests keeping my pieces, my instinct is to exchange as many pieces as possible except the pawns, while pushing the pawns to queen, which includes taking the knight immediately. However, is there any hope of **proving** a winning strategy here, rather than relying on intuition? I think this is somewhat plausible because Black has enough minor pieces for the king to hide behind (even after exchanging 2 of them), and so White cannot make sustained checks. The question is whether there is a systematic way for Black to push the pawns that is guaranteed to work.
  
  I do not see any convincing reason that there cannot be a proof of a winning strategy, because there are numerous examples of endgames with human analysis of the optimal strategy (e.g. [a](https://chess.stackexchange.com/q/9003),[b](https://chess.stackexchange.com/q/28225),[c](https://chess.stackexchange.com/q/12220),[d](https://chess.stackexchange.com/q/18601),[e](https://chess.stackexchange.com/a/1725)). If the position was not so imbalanced, I know it would be much harder to prove a win. But here, Black has sufficiently many pieces that I feel there should be a proof that we can force some trades to get to a certain winning position. That is also why I would choose to take the knight immediately to hopefully make a full analysis humanly tractable.
  
  **Tags:** analysis, strategy, endgame, pawn-promotion, rook-and-minor-endgame
  
  > **Comments:**
  > **username_0**: @bof: Obviously, we are not asking about proving a strategy for Black very early in the game, otherwise it would surely be infeasible. My mathematical intuition tells me that this game might have a feasible human analysis. As for the board, yes Black's side is at the bottom as the default viewer setting makes it (if the FEN says "black to move"), so Black's pawns are at their starting squares.  
  > **username_1**: (-1) I don't understand... This is a total overwhelming position for Black. White should resign here. Absolutely hopeless. No hope at all. As long as Black doesn't blunder a pawn/piece, simply push whenever there is an opportunity, move the pieces forward etc is 100% win.  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_2 on 2020-04-25. Score: 3
  
  @username_1 is right. This position is absolutely hopeless for White, and there is a very human strategy for Black, which consists here in not exchanging pieces, but **hunting the white king**. This strategy will force White to exchange pieces unfavorably or White will lose material to avoid being mate.
  
  The principle behind this strategy is simple ***"with three or more pieces in the attack the enemy king will not be able to escape."***
  
  Here is an example where nothing is forced but it's very difficult to suggest good ideas for White. White is completely squeezed.
  
  ```
  [White "White"]
  [Black "Black to move"]
  [FEN "8/rp4p1/1N1nkb2/3n4/6R1/1R6/8/2K5 b - - 1 38"]
  
  1...Nc3! {Black chases the king and cuts off the two rooks. They can't easily coordinate.} 2.Nc4 Nde4! {same strategy: coordinating the pieces to go against the White king} 3.Kc2 (3.Ra3 {is probably better but now Black is happy to exchange and push the g pawn} 3...Rxa3 4. Nxa3 Kf5 5.Rg2 g5-+) b5 4.Ne3 (4.Rxb5 Nxb5 5.Rxe4 Kf5-+ {does not help White}) Rc7 {here with get our principle of three or more pieces in the attack and the White king feels unsafe} 5.Rg2 (5.Kd3 Nf2-+) Na2! 6.Kb1? (6. Kd1 {is better but the King does not escape} Rc1+ 7. Ke2 Nac3+ 8. Kf3 Rh1-+) Rc1 7.Kxa2 Ra1#
  
  ```
  
  > **Comments:**
  > **username_0**: I am asking for a sure strategy, not one just based on general principles, but thank you for your 'general principle'. I actually believe that it would be easier to prove a winning strategy by exchanging the knight. If you do not think so, why, and can you prove your strategy correct? If you look at the links I just added to my question, you will see that sometimes human analysis is not only possible but also can be superior to computer analysis. That is why I am not interested in how StockFish or other engines would play from this position.  
  > **username_0**: Clearly, you do not understand the meaning of "perfect play" and "optimal strategy", which are basic concepts in combinatorial game theory, which includes the study of chess. If you have basic mathematical knowledge, including induction and recursive definition, I can briefly explain it to you. Otherwise, you may want to find an introductory book on combinatorial game theory and read it up on your own.  
  
  ---
  
  ### Answer by username_3 on 2020-04-26. Score: 2
  
  The position looks clearly winning for Black, and I don't think it makes much of a difference whether you exchange knights or not. It might make your task simpler, but Black's knight could be a bit stronger than White's.
  
  To win all you really have to do is push one of your pawns, with the support of the king. Your other pieces can be used to block checks, and the minor pieces can also chase any blockading rooks out of the pawn's way. Eventually things will get to the point where you win a White rook for the pawn and a minor piece (in the worst case scenario), which will result in an easily won position.
  
  I'm not sure how to give a certain proof that guarantees a win, as there's no forced mate yet. It's just a position where Black is up 4 points, so a winning edge should be taken for granted at this point.
  
  > **Comments:**
  > **username_0**: I agree with your second paragraph, and in fact that is precisely what motivated my question, because it feels like there may be a general principle along these lines that can be proven. I am not sure what you mean by "the b6-knight is kind of cut off" since it can escape if Black does not trade now.  
  > **username_0**: To give more detail, we already know that a KRK endgame is won, so if we eliminate all the White pieces without sacrificing our rook, then we are done. Let "A→B" mean "A protects B", and label the pawns P1,P2 and knights N1,N2. Then we can have R→N1→P1 and K→N2→P2 and use B to block perpetual checks. This is just a bit short, because we need to prove that we can handle a double rook attack as well as push the pawn P2. Once we can prove that, we are done, because it forces White to sacrifice a R for N2+P2, leaving us with KRBNP against KR, where it should be easy to prove a win.  
  > **username_0**: Okay I saw your edit, but I'm still not convinced there is a clear sense in which Black's knight is stronger than White's.  
  > **username_3**: @username_0 I'm not really sure how a formal proof like the one you refer to would work in chess. At some point doing it seems like overkill, since the position is clearly winning. A practical "proof" would be letting yourself play Stockfish and seeing that you're able to consistently beat it. Also I edited my answer because you're right the knight can escape. Black's knight still might be slightly better though because if Black meets Nc4 with ...Nde4, the knight will be temporarily undefended on c4. Meanwhile, Black's d5-knight could go to the strong c3-square.  
  > **username_0**: See the links in my (edited) question for examples of what I would consider as complete human analysis of optimal strategy. The motivation for looking for such a proof is that anyone who comes up with a human-readable proof would have to distill the strategy to a handful of provable principles, which would be relevant not only to this position but infinitely many others. In other words, the core ingredients of such proofs would have very wide applicability.  
  
  ---
  
  