1.
[{B:7k/8/5N1P/8/2p5/2N5/8/3K3R}
{V:This is a gentle introduction to the longer problems. White only needs to give check in order to mate, but at the same time he must avoid stalemating black. White to play and mate in four.;c3b1;Correct;Incorrect.;All the obvious moves either lead to stalemate, or allow Black to promote with check and thereby delay the mate beyond the fourth move. One example is 1 Nb5 c3 2 Ke2 c2 3 Rg1 c1N+ and White has to waste a tempo moving his king again. But if White could force Black to promote his pawn on a different square, he would no longer be able to promote with check and White would save a tempo. How can this be achieved?}]

2.
[{B:7k/8/5N1P/8/2p5/2N5/8/3K3R}
{S:c3b1}
The answer is Nb1. After 1 Ncd5, for example, Black plays 1...c3 2 Rg1 c2+ and White has the choice between stalemating Black and allowing a promotion with check (for example, 3 Ke2 c1N+) when mate in four is impossible. The same problem arises after all the other moves of the c3-knight, except for one. White's plan is to force Black to promote on b1 instead of c1, when he will not be in a position to check White's king.]

3.
[{B:7k/8/5N1P/8/2p5/8/8/1N1K3R}
{S:c4c3}
1...c3.]

4.
[{B:7k/8/5N1P/16/2p5/8/1N1K3R}
{S:d1e2}
2 Ke2]

5.
[{B:7k/8/5N1P/16/2p5/4K3/1N5R}
{S:c3c2}
2...c2]

6.
[{B:7k/8/5N1P/24/2p1K3/1N5R}
{S:h1c1}
3  Rc1!
The only way to prevent a promotion on c1 is to block the square with the rook. Thanks to White's first move, Black is not stalemated.]

7.
[{B:7k/8/5N1P/24/2p1K3/1NR5}
{S:c2b1}{P:BQb1}
3...cxb1Q.]

8.
[{B:7k/8/5N1P/24/4K3/1qR5}
{S:c1c8}
4  Rc8 mate.]