Suppose White has only a rook, knight, and king, and Black has only a king. This ending is always won. If the kings are separated and can be kept that way, then the rook and knight can deliver mate by forcing the Black king to an edge of the board. There are two basic mating patterns. The Black king can be mated in the corner like this:
kR6/8/2N5/8/8/8/8/K7
Black is Mated
Or the Black king can be mated in the middle of an edge like this:
4k3/4R3/4N3/8/8/8/8/K7
Black is Mated
Let's look at an example of how to deliver mate when the kings are separated. Consider:
8/8/8/4k3/5N2/8/3R4/K7 w
White to Move and Win
1. Rd4 Kf5 2. Nh3 Ke5 3. Kb1 Kf5 4. Re4 Kf6 5. Re5 Kg6 6. Rf5 Kg7 7. Rf6 Kh7 8. Rg6 Kh8 9. Nf4 Kh7 10. Nd5 Kh8 11. Kc1 Kh7 12. Nf6+ Kh8 13. Rg8#
Here is an example of mating in the middle of an edge:
5k2/6R1/8/8/3N4/8/8/2K5 w
White to Move and Win
1. Ne6+ Ke8 2. Re7#
This type of mate with the rook and knight happens most often when other pieces are still on the board.
Now let's suppose the kings are connected. Then White will use a setup to separate the kings. A setup that works is: king on b1, rook on b2, knight on b3 (or Kg1, Rg2, Ng3, or Kb8, Rb7, Nb6, or Kg8, Rg7, Ng6).
The setup described in "King and Two Queens vs. King" doesn't work. For example, consider:
8/8/8/8/8/8/Nk6/KR6 b
Black to Move
Here Black should play 1. ... Kc2 (not 1. ... Ka3 2. Rb2 etc.). Then if 2. Rb2+ Black reconnects kings with Kb1.
Now suppose White's king is on b1, his rook on b2, and his knight on b3. Then, if the kings are connected, the Black king will be on a1, a2, c1, or c2, and in each case it may be White's turn or Black's turn. We will show how to separate the kings in all eight cases. It is not necessary to strictly memorize all eight cases. Just memorize the setup and after you play through all the cases now, you'll get the idea, and it will be easy to come up with the winning method in practice.
Case 1:
8/8/8/8/8/1N6/1R6/kK6 w
White to Move and Win
1. Ka2 Kb1 2. Na1 Kc1 3. Ka3 (separated)
Case 2:
8/8/8/8/8/1N6/1R6/kK6 b
Black to Move and White to Win
1. ... Ka2 2. Na1 Ka3 3. Kc1 (separated)
Case 3:
8/8/8/8/8/1N6/kR6/1K6 w
White to Move and Win
1. Na1 Ka3 2. Kc1 (separated)
Case 4:
8/8/8/8/8/1N6/kR6/1K6 b
Black to Move and White to Win
1. ... Ka3 (1. ... Ka1 is back to case 1) 2. Kc1 (separated)
Case 5:
8/8/8/8/8/1N6/1R6/1Kk5 w
White to Move and Win
1. Ka2 Kb1 2. Na1 Kc1 3. Ka3 (separated)
Case 6:
8/8/8/8/8/1N6/1R6/1Kk5 b
Black to Move and White to Win
1. ... Kd1 (1. ... Kc2 leads to case 7) 2. Ka1 (separated)
Case 7:
8/8/8/8/8/1N6/1Rk5/1K6 w
White to Move and Win
1. Nc1 Kd1 (1. ... Kc3 2. Ka1 (separated)) 2. Rc2 (separated)
Case 8:
8/8/8/8/8/1N6/1Rk5/1K6 b
Black to Move and White to Win
1. ... Kd1 (1. ... Kc1 leads to case 5, 1. ... Kd3 2. Rc2, 1. ... Kc3 2. Ka1) 2. Ka1 (separated)
So, you can see it's very easy from the given setup to separate the kings; it never takes more than 3 moves.
Now let's put it all together. Consider:
8/8/8/8/3N4/4k3/3RK3/8 w
White to Move and Win
1. Kd1 Ke2 2. Kc1+ Kd1 3. Rb2 Kd2 4. Kb1 Kc2 5. Nb3 Kc1 6. Ka2 Kb1 7. Na1 Kc1 8. Ka3 Kd1 9. Nb3 Ke1 10. Rd2 Kf1 11. Re2 Kg1 12. Rf2 Kh1 13. Nd2 Kg1 14. Nf3+ Kh1 15. Rh2#