1.
[{B:3K1B2/1p6/pp6/rk2N3/b1p5/1pP5/1P2P3/8}
{V:Very long problems may appear intimidating, but in fact they are often quite easy to solve. Usually Black is completely tied up and White can plan long-range maneuvers without worrying too much about enemy counter-play. This is a case in point, White cannot afford to let Black's rook and bishop out of the box, so he must restrict Black to king moves throughout the solution. White to play and mate in 20.;d8e7;Correct;Incorrect.;Long problems can often be solved by making a wish and then seeing how the wish can be made to come true. In this position White could mate in two by 1 Nf3 Kc6 2 Nd4 mate, if only the d5-square were covered. So White's wish is to have his e-pawn on e4. It takes 18 moves for the wish to become reality, but all Black's moves are forced, which makes life much easier for White.}]

2.
[{B:3K1B2/1p6/pp6/rk2N3/b1p5/1pP5/1P2P3/8}
Such long problems have a peculiar logic all their own. White's ultimate aim is to mate by Nf3 and Nd4 mate, but at the moment 1 Nf3 Kc6 2 Nd4+ fails to 2...Kd5, so first of all White must advance his e-pawn to e4 with gain of tempo. This large problem can be broken down into smaller sub-problems. White would like to maneuver with his bishop, for example by 1 Bh6 Kc5 2 Be3+, but once again Black's king escapes via d5. So the first step is to put a temporary guard on d5 to free the bishop.]

3.
[{S:d8e7}
1 Ke7]

4.
[{S:b5c5}
1...Kc5]

5.
[{S:e7e6}
2 Ke6+]

6.
[{S:c5b5}
2...Kb5
Although d5 is now guarded, White cannot yet mate by 3 Nf3 Kc6 4 Nd4+ because he has given up control of c7. d5 has to be covered by the e-pawn if this mating idea is to work, but that is still a long way off.]

7.
[{S:f8h6}
3 Bh6]

8.
[{S:b5c5}
3...Kc5
Black's moves are forced throughout the solution.]

9.
[{S:h6e3}
4 Be3+]

10.
[{S:c5b5}
4...Kb5
White is now free to maneuver with his bishop, but it isn't obvious how this helps to advance the e-pawn. The key idea is to transfer the bishop to f2 with gain of tempo, then e2-e3 will lift the stalemate, forcing ...Kc5, and after e3-e4+ White's wish has been granted.]

11.
[{S:e3g5}
5 Bg5
The bishop couldn't transfer from f8 to f2 directly, because the f8-h6 diagonal wasn't long enough. Once again a tricky problem can be broken down into smaller problems, first transfer the bishop to the d8-h4 diagonal, and then use the h4 square to reach f2.]

12.
[{S:b5c5}
5...Kc5]

13.
[
{S:g5e7}
6 Be7+]

14.
[{S:c5b5}
6...Kb5]

15.
[
{S:e7h4}
7 Bh4]

16.
[
{S:b5c5}
7...Kc5]

17.
[
{S:h4f2}
8 Bf2+]

18.
[
{S:c5b5}
8...Kb5
At long last the e-pawn can make its first step forward.]

19.
[
{S:e2e3}
9 e3]

20.
[{S:b5c5}
9...Kc5]

21.
[{S:e3e4}
10 e4+]

22.
[
{S:c5b5}
10...Kb5
The e-pawn has reached its final destination, but White's king and bishop are on the wrong squares for the final mating idea. The next step is to execute all the previous maneuvers in reverse. First of all the bishop has to return to f8.]

23.
[
{S:f2h4}
11 Bh4]

24.
[
{S:b5c5}
11...Kc5]

25.
[
{S:h4e7}
12 Be7+]

26.
[
{S:c5b5}
12...Kb5]

27.
[
{S:e7g5}
13 Bg5]

28.
[
{S:b5c5}
13...Kc5]

29.
[
{S:g5e3}
14 Be3+]

30.
[
{S:c5b5}
14...Kb5]

31.
[
{S:e3h6}
{S:b5c5}
15 Bh6  Kc5]

32.
[
{S:h6f8}
16 Bf8+]

33.
[
{S:c5b5}
16...Kb5
Now White's initial king maneuver can be reversed.]

34.
[
{S:e6e7}
17 Ke7]

35.
[
{S:b5c5}
17...Kc5]

36.
[
{S:e7d8}
18 Kd8+]

37.
[
{S:c5b5}
18..Kb5
The king controls c7 and the pawn controls d5, so the stage is set for the final act.]

38.
[
{S:e5f3}
19 Nf3]

39.
[
{S:b5c6}
19...Kc6]

40.
[
{S:f3d4}
20 Nd4 mate.]