Suppose we have a rook and the opponent has a knight next to his king.
Is it possible to win this game theoretically?
E.g., in the following position (white to move), is it possible for white to win?
[FEN "8/8/8/3n4/3k4/8/K7/R7 w - - 0 1"]
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In the specific position that you mention, the answer is a resounding no. The king and the knight will defend each other, and white will not be able to force mate.
However, the knight is a clumsy piece. If the knight is not positioned perfectly, then the rook will frequently be able to drive the knight to a bad square and deprive it of moves. Eventually, the knight may be captured, or the king might be checkmated.
As an example of a position where the rook can win, take the following:
[FEN "8/1n6/8/8/2R2K2/8/5k2/8 w - - 0 1"] [SetUp "1"] [PlyCount "17"] 1. Rc2+ Ke1 2. Ke3 Kd1 (2... Kf1 3. Rf2+ Kg1 4. Rd2 Nc5 5. Kf3 Ne6 6. Kg3 Kf1 7. Rd5 Nc7 8. Re5 Na6 9. Kf3 Kg1 10. Rg5+ Kf1 11. Ra5) 3. Rb2 Nc5 4. Rb6 $22 Na4 5. Rb4 Nc3 (5... Nc5 6. Rd4+ Kc2 7. Rc4+) 6. Kd3 Na2 7. Rb1+ Nc1+ 8. Kc3 Ke2 9. Rxc1 *
Here, white can win starting with
1.Rc2+. The plan is to drive the king to the first rank and then attack the knight while improving the position of the rook.
In your question, you mention specific cases where the king and the knight are close together. In these cases, it is very important to know two key positions:
First, in this position, black can hold the draw:
No matter what white tries, the white king cannot approach the black king due to the knight. Black will be able to squeeze out with
...Kb2. Black's plan is to mark time by playing
...Nb1. White cannot make useful progress.
This position, almost the same, has one major difference - black can no longer mark time with the knight:
If white plays a waiting move, for example
1.Re2, then black must lose immediately. Any move will lose the knight or allow mate in one.
If you're interested in a specific position, you can check out the Nalimov Tablebases online. They have completely solved every position with 6 or fewer pieces on the board.
I believe it's a draw given that the king and knight stick to each other. Someone has already mentioned the "Nalimov Tablebases". Give it a try and explore the variations. However, I just lost a game with the knight (the reason why I was searching for this question XD). I was really low on time and voluntarily moved my king to a square where my knight was pinned. After playing around with the Nalimov Tablebases, here are some tips for defending the draw:
If the knight is close to the king, it is hard to win with a rook. That's because the knight will fend off the enemy pieces that try to approach the king. Most winning positions feature the knight and king far apart so that the opposing king and rook can pick them off one by one (checkmate the king or trap the knight).
An important exception to the above is if the knight is either on a corner square, or is a "knights jump" from the corner square. A knight has only two moves from a corner square, meaning it may not have enough maneuvering room. If the knight can find a safe haven on the open board it can defend its king.
A similar thing is true with a bishop and king against a rook and king.
If it's a normal game, a rook against a knight will always be a draw.
In very rare situation both can win:
A knight wins here in this position
[Title "White to move"] [FEN "8/8/8/8/8/8/r2N4/k1K5 w - - 0 1"]
A rook wins here in this position
[Title "White to move"] [FEN "8/2R5/8/8/7k/3K4/8/4n3 w - - 0 1"]
Out of curiosity I put Stockfish 2.3.1 to work on this and it did find a solution for white to win; however it took 60 more moves. However just glancing at the results it would assume it could be done in less moves.
[FEN "8/8/8/3n4/3k4/8/K7/R7 w - - 0 1"] 1.Rd1+ Kc4 2.Kb2 Ne3 3.Rd8 Nd5 4.Kc2 Ne3+ 5.Kd2 Nd5 6.Kc2 Kd4 7.Rh8 Nc3 8. Kd2 Ne4+ 9.Ke2 Nc3+ 10.Kd2 Nb1+ 11.Kd1 Na3 12.Kd2 Nb1+ 13.Ke2 Nc3+ 14.Kd2 Nb1+ 15.Kd1 Na3 16.Kd2 Nb1+ 17.Kc2 Na3+ 18.Kd1 Nb5 19.Kd2 Kc4 20.Rh4+ Kd5 21.Kd3 Nd6 22.Rh5+ Ke6 23.Ke3 Nf5+ 24.Kf4 Ne7 25.Ra5 Nd5+ 26.Ke4 Nf6+ 27. Kd4 Ng4 28.Rc5 Nf6 29.Rc6+ Kf5 30.Rc5+ Kf4 31.Rc1 Kf5 32.Rc5+ Kf4 33.Rc2 Kf5 34.Rc5+ Kf4 35.Rc1 Kf5 36.Rc5+ Kf4 37.Rc1 Nh5 38.Kd3 Nf6 39.Ra1 Kf5 40.Rb1 Kg4 41.Ra1 Kf5 42.Re1 Nd5 43.Kd2 Nf4 44.Ra1 Ke4 45.Rb1 Nd5 46.Ra1 Nc7 47.Re1+ Kd4 48.Ra1 Nd5 49.Rb1 Nc3 50.Ra1 Nb1+ 51.Rxb1 Ke4 52.Kc3 Ke5 53.Re1+ Kf6 54.Kd3 Kf5 55.Kd4 Kf4 56.Rf1+ Kg5 57.Ke4 Kg6 58.Ke5 Kg7 59.Kf5 Kf7 60.Re1 Kg7 61.Re7+ Kf8 62.Kf6 Kg8 63.Re8+ Kh7 64.Ra8 Kh6 65.Rh8# *