I have witnessed many painful attempts of beginners at the king and rook v. king checkmate. All you need to know is a foolproof method, which works for every situation, and relies on just 2 key patterns and you will never again struggle.
The method which I am suggesting is not the fastest way to checkmate your opponent, but it is certainly the simplest. The first thing you have to decide is which way you are going to push your opponent back, from left to right, up to down, or the reverse of either of these two. The rook acts as a barrier, cutting off the enemy king from all of the files/ranks behind the line of the barrier:
It is also important to understand the idea of the ‘opposition‘. In this endgame, when the black king grabs the opposition, white checks the black king from the side, and this forces the black king to concede ground. This pattern repeats, until the black king is on the back rank, and when white forces black to move back, the black king can`t go back, because there are no more ranks left, and so it is checkmate. Here is an example of this checkmate:
1. Kd3 Kc5 2. Re4 Now we can see that the white rook is cutting off the black king from ranks 1-4.
2…, Kd5 3. Rh4 Ke5 4. Ra4 Kd5 5. Ra5+ Kd6 White forces the black king to concede ground. Now the black king can`t go on ranks 1-5.
6. Kc4 Ke6 7. Kd4 Kf6 8. Ke4 Kg6 9. Kf4 Kh6 10. Kg4 Kg6 Now white can force the black king to retreat, by a repeating pattern. 11. Ra6+
11…, Kf7 12. Kf5 Ke7 13. Rh6 Kd7 14. Ke5 Kc7 15. Kd5 Kb7 16. Kc5 Ka7 17. Kb5 Kb7 18. Rh7+ With this side check, we force the black king onto the back rank… Kc8 19. Kc6 Kd8 20. Ra7 We waste a move, so that the kings are three squares apart and it is our move. If the black king comes back on himself, then the rook check will be mate. Ke8 21. Kd6 Kf8 22. Ke6 Kg8 23. Kf6 Kh8 24. Kg6 Kg8 25. Ra8#
I hope this will help you carry out a rook and king v, king endgame with no problems…One last little tip is to avoid putting your king on the same file/rank as your rook, because it will possibly allow your opponent`s king to pass through the broken barrier of a the rook. This is rather frustrating and time wasting, so just keep it simple, follow the recurring patterns, and you should never again have troubles over this endgame.