Rule 1: King is on its 6th rank in front of the Pawn.
If the king of the side with the extra pawn has reached the 6th (3rd for Black) rank then all positions are winning regardless of who plays. The pawn must be advanced to the 7th rank without giving check.
1. Ke6! (1. e6?? Kf8 2. e7+ Ke8 3. Ke6 with stalemate!)
1... Kd8 (1... Kf8 2. Kd7! +-)
2. Kf7 Kd7 3. e6+ Kd6 4. e7 and the pawn will be promoted.
White wins even if it is Black to play
1... Kf8 2. e6 Ke8 3. e7 (This time there is no stalemate.)
3... Kd7 4. Kf7+- and White will promote the pawn in the next move.
The Knight-pawn
The win with the Knight-pawn (b and g file) is kind of trickier, but if you know the winning plan, then it is also elementary. White wins only if he goes with his king to the h-file.
1. Kh6! {1. Kf6 is less accurate because of 1... Kh7!? 2. Kf7 Kh8 and now White must play 3. Kg6! (3. g6??stalemate!)}
1... Kh8 2. g6 Kg8 3. g7 Kf7 4. Kh7+-.
The Rook-pawn
The h-pawn is an exception in many different endgames. If Black king has managed to enter into the corner, White cannot promote the pawn as there is always stalemate when the pawn reaches the 7th rank.
1. h6 Kh8 2. h7 stalemate.
Even if it was Black's move the game is drawn.
Remember
If the King is on its 6th rank in front of the Pawn, all pawn endgames are winning regardless of who plays. The only exception is when the king of the side with the extra pawn has an a or h-pawn. With the b and g-pawn, you must bring the king to the a or h-file, otherwise there is a stalemate idea.
