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Suppose we have just rook and our opponent has just one knight which is near to his king is it possible to win this game theoretically?

For example, 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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3 Answers

up vote 11 down vote accepted

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:
Knight vs Rook Endgame where black holds 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 ...Na3 and ...Nb1. White cannot make useful progress.

This position, almost the same, has one major difference - black can no longer mark time with the knight:
Knight vs Rook Endgame where white wins
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.

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good answer and great link, thanks – Saeed Amiri Jun 11 '12 at 14:44
I just edited the answer as well. I added two key positions that you should know in order to win these positions as the stronger side as well as to hold the draw as the weaker side. – Andrew♦ Jun 11 '12 at 14:51
1  
Interestingly, the key position could arise after black's underpromotion, something to keep in mind for K + R vs. K + P. Here is an example: chessgames.com/perl/chessgame?gid=1400994 – Akavall Jun 11 '12 at 15:05
@Akavall Great mind think alike! ;-) I just posted this answer with the same underpromotion. I was reminded of it from the current question. – Andrew♦ Jun 11 '12 at 15:09
@Andrew, Wow! That answer fits underpromotion question perfectly. – Akavall Jun 11 '12 at 15:16
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up vote 1 down vote

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.

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up vote 1 down vote

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# *
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1  
At least 50's move for black is totally wrong, also If after 50 move both rook and knight remain in the game, match is draw. – Saeed Amiri Mar 29 at 19:21
2  
Stockfish is choosing some weird moves there. The original position is indeed drawn, as can be verified with tablebases. 49...Nc3? throws away the draw; 49...Ke5, among other moves, keeps the draw. White gave the draw back with 50.Ra1?, and then I don't even know what to say about 50...Nb1+??? 50...Nd5, for example, would have been a draw again. Perhaps Stockfish thought it would be able to claim a 50-move draw after 50...Nb1+ so it didn't matter what it played. – dfan Mar 29 at 19:23

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