Knight and King versus Rook and King is a theoretical draw. Yet, you might lose if you don't know the method. How do you hold a draw?
The weaker side needs to keep Knight close to his King in order to achieve draw.
There are some special cases where the stronger side wins even in those situations, like when Knight is cornered or pinned in such a way that puts weaker side in zugzwang.
If the Knight is far away from the King then the result of the game depends whether or not the defending side is able to connect his pieces.
Here are some demonstrations:
[Title "Rook vs Knight, typical drawing plan"] [fen "3nk3/2R5/8/4K3/8/8/8/8 w - - 0 1"] 1.Kd6 Nf7+! ( 1...Kf8?? 2.Rc8 Ke8 3.Ra8+- ) 2.Ke6 Nd8+ 3.Kf6 Kf8 4.Rd7 Ke8 ( 4...Nc6?? 5.Rd6+- ) 5.Re7+ Kf8 6.Re1 Nb7 7.Ke6 Ke8 8.Rb1 Nd8+ 9.Kd6 Nf7+ 10.Ke6 Nd8+ 1/2-1/2
It is useful to mention that in these position the defender should not allow vertical opposition(
e6-e8 in the above example ), but should strive for the diagonal one(
Even in the following position, Black is able to maintain a draw:
[Title "Drawing plan in a difficult position"] [fen "5nk1/4R3/8/6K1/8/8/8/8 w - - 0 1"] 1.Kf6 Nh7+! 2.Kg6 Nf8+ 3.Kh6 Kh8 4.Rf7 Kg8 5.Rg7+ Kh8 6.Rg1 Nd7! ( 6...Nh7?? 7.Kg6! Kg8 8.Rg2 Nf8+ 9.Kf6+ Kh8 10.Kf7+- ) ( 6...Ne6 7.Kg6! Nf8 ( 7...Kg8 8.Kf6+! ) 8.Kf7 Nh7 9.Rg8# ) 7.Kg6 Kg8 8.Rd1 Nf8+ 9.Kf6 Nh7+ 10.Kg6 Nf8+ 1/2-1/2
Winning motifs for the stronger side:
We already saw one above-the decisive pinning of the knight that leads to zugzwang:
[Title "Decisive zugzwang"] [fen "R2nk3/8/3K4/8/8/8/8/8 w - - 0 1 "] 1.Rc8 Kf7 2.Rxd8+-
These motifs happen due to bad positioning of the King:
[Title "Defenders pieces are badly positioned : Cornered King"] [fen "6nk/5R2/8/7K/8/8/8/8 w - - 0 1 "] 1.Kg6+- ( 1.Kg5+- )
or badly posted Knight:
[Title "Defenders pieces are badly positioned : Knight is away from his King"] [fen "5k2/3R4/2n2K2/8/8/8/8/8 w - - 0 1"] 1.Rd6 Ne7 2.Rd8# 1-0
Another important motif is cornering the Knight :
[Title "Cornered Knight"] [fen "6k1/5n2/6K1/8/8/3R4/8/8 w - - 0 1 "] 1.Rd5! Nh8+ 2.Kf6 Nf7 3.Rd7 Nh6 ( 3...Nh8 4.Ra7+- ) 4.Kg6+-
Sometimes defenders units are not badly placed but just separated. In that case stronger side can win if it can push Knight away from the king since then it will be able to trap it. Here is an example:
[Title "Knight is too far from his King"] [fen "8/8/8/2k1K3/8/3R4/4n3/8 w - - 0 1 "] 1.Re3! Ng1 2.Kf5! ( 2.Kf4? Kd4! 3.Re1 Nh3+ 4.Kg3 Ng5 5.Kf4 Nh3+= ) 2...Kd4 3.Kf4 Kc4 4.Kg3 Kd4 5.Re1+-
The defender must keep his pieces together and avoid being cornered. Maintaining diagonal opposition with kings is desired.
The stronger side has several winning motifs:
Corner the opposing King and then execute double attack with the rook which threatens mate and attacks the Knight.
Corner the Knight and create zugzwang.
Push away the Knight as far away as possible from his King and then trap him.
In some special cases, exploit bad piece coordination to create decisive zugzwang.
Examples and instructions are taken from the book:
Y.Averbakh - Comprehensive Chess Endings Volume 2.