1.
[
{F:B}{B:8/8/8/7K/1k6/3P1Rp1/8/6B1 BLACK}
{V:In longer helpmates, it is even more important to start by finding the mating position. The astronomical number of possibilities means that although a trial and error approach might work for a computer, the human has to proceed by using a combination of logic, intuition and experience. Black to move. Helpmate in 4. Try your solution on the board.;b4c3;That's the right one!;Incorrect.;It isn't hard to see that White cannot mate the black king on the queenside, the rook and bishop simply aren't up to the job by themselves. In other words, the black king has to head for the kingside, where White's own king can lend a hand.}]

2.
[
{F:B}{B:8/8/8/7K/1k6/3P1Rp1/8/6B1}
This problem is relatively easy to solve, because there is only one plausible mating position, with Black's king on f4, the white bishop on d4 and his rook on the f-file. It takes Black's king four moves to reach f4, and since c4 and c5 are covered, the first move must be ...]

3.
[
{S:b4c3}
1...Kc3
Black's king must reach f4 via either e3 or e5. But if Black's last move is ...Ke3-f4, then White's bishop cannot have been on d4 the move before, in other words White's last move must be Bd4, giving mate by discovery. The bishop cannot have come from f2 (because Black's king was on e3), so it must have come from f6, but there isn't enough time for White to reach f6, for example 2 Rf8 Kd2 3 Bd4 Ke3?? 4 Bf6 Kf4 5 Bd4 would be a solution, except for the illegal move in the middle. Therefore Black's king must go via d4 and e5, so White's second move must be Re3 or Rf2. Now it doesn't take long to find the solution.]

4.
[
{S:f3f2}
2  Rf2
First the rook blocks the bishop ...]

5.
[
{S:c3d4}
2...Kd4]

6.
[
{S:f2f1}
3 Rf1+]

7.
[
{S:d4e5}
3...Ke5]

8.
[
{S:g1f2}
4 Bf2
... and then the bishop blocks the rook.]

9.
[
{S:e5f4}
4...Kf4]

10.
[
{S:f2d4}
5 Bd4 mate.]