Just for the record, the longest win in this endgame is 7 moves:
[FEN "8/8/8/8/p7/8/N7/k1K5 w - - 0 1"]
1. Nb4 a3 2. Nc2+ Ka2 3. Nd4 Ka1 4. Kc2 Ka2 5. Ne2 Ka1 6. Nc1 a2 7. Nb3#
The idea in this position (and other similar positions) is to stalemate the king in the corner; that forces a pawn move and when the pawn reaches a2, you'll need to have ...
In the general case K+N vs K+P is of course a draw - or a win for the pawn if it can promote unhindered.
There is however a famous construction were the knight can force a mate against a king stuck in front of its own well-advanced rook pawn:
[fen "8/3N4/8/8/8/p7/k7/2K5 w - - 0 1"]
1.Nc5 Ka1 2.Kc2 Ka2 3.Nd3 Ka1 4.Nc1 a2 5.Nb3#
There are many ...
As you rightly point out Kc4 is forced for white and then whichever king has to move first loses.
It is worth quickly checking if white can abandon the d5 pawn and go for the a5 with the intention of queening the a4 pawn. You do this by counting to see who queens first.
For white it goes Kb5, Kxa5, Kb4 (to threaten the c pawn), a5, a6, a7, a8=Q - 7 moves.
I think the term 'intersect' is not really intuitive here.
From c2, the white king can go to d2 and c3 in one move.
That means the black king must find a square from which it can reach f3 and e3 in one move. That square is f4; this would be true for e2 and f2 as well, but then the black king is outside the pawn's square and it can run to promotion.
King and pawn endgames are much easier than ones where each side also has a rook. The key principles are fairly simple. You want to try and queen a pawn. With pawns on both sides of the board and an imbalance as here there are two key points:
Getting your king to the middle of the board is very important.
The only pawns you are likely to queen are ones on ...
In the starting position you give it is easy for white to create a passed pawn on the kingside and more difficult for black to do the same on the queenside. To guarantee that black cannot create a queenside passed pawn you need to aim for this position:
[fen "4k3/1pp2ppp/p1p5/8/P1P1P3/1P6/5PPP/4K3 w - - 0 1"]
From here the win is still not easy or ...
This is answered by Wikipedia's article on the Ruy Lopez, Exchange variation:
If White can exchange all pieces, the pawn structure is a big advantage in the endgame. Max Euwe gave the pure pawn ending (without pieces) resulting after the exchange of White's d-pawn for Black's e-pawn as a win for White. The winning procedure is detailed in Secrets of Pawn ...
In this position you basically pawn up. Even though it is not easy to win, There are two things you must remember
1- Centralize your King. This is almost important in every endgame position.
2- Do not allow your opponent to fix his queen-side pawn structure such as exchanging the pawns.
After do that, You must use 4 to 3 pawn structure with forcing the king-...
and in the second, what to do to win strategically?
OK, I've just seen your second edit with another position. Strictly speaking it would be better as a new question, but I'll add another answer looking specifically at this.
The first two things that strike me looking at the final position are that black has a much better king position and white has 2 "...
Without looking at those gamescores, I'm guessing that you're not taking charge in the center when it's there for the taking.
You know Sicilian players are supposed to win more of the endings, right? White wins the shorter, sharper games that get into magazines, and Black wins the longer ones after weathering the storm.
After weathering that storm, Black is ...
I have made a search on pawn endgames in my database, and here is the definite answer! 21 full moves, it really cracked me up to see it hahahaha.
By the way your above example is not 16 half-moves (that would be 8 full moves). It's 34 full moves.