Is there an easy way to detect a draw in a King + Pawn vs King endgame? I know the basics of opposition, but I am struggling to evaluate positions for drawishness in King + Pawn VS King endgame. A lot of times it is useful to know whether you should make an exchange. For example, in this position:

[Title "Black to move"]
[FEN "k7/8/8/8/1PR4r/2K5/8/8 w - - 0 1"]


Black can simply exchange the rook and then move to b8 knowing that he has a draw. But in order to do so, he should know that position after exchange is drawish. 

The problem is, that looking at positions like this

[Title "White to move"]
[FEN "8/4k3/8/8/8/2P5/2K5/8 w - - 0 1"]


this

[Title "Black to move"]
[FEN "k7/8/8/8/1PR4r/2K5/8/8 w - - 0 1"]


or this

[Title "White to move"]
[FEN "4k3/8/8/8/8/2P5/8/6K1 w - - 0 1"]


I can not quickly determine whether it is drawish or not. I end up thinking for 5 minutes and in the end I can be wrong (so to me it looks like I am simply guessing).

So is there an easy way to determine win/draw in King + Pawn vs King endgame (I am not asking how to secure a draw/win. I hope that I will be able to find it, knowing the evaluation in advance).
There is a very easy way to detect whether King and Pawn endgames are drawn or not.

This method I use is a very easy to understand one from Karsten Mueller and Frank Lamprecht's excellent book Secrets of Pawn Endings

It concerns key squares and opposition 

The rule states that if pawn has not reached or crossed the central line (5th rank for White and 4th rank for Black) then there are 3 key squares 2 squares in front of it.

For example this starting position, the key squares are b5, c5 and d5. If white's king can occupy any of these key squares then the game will be a win for white no matter what.

  [Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "White to Move"]
[Black ""]
[Result "1/2-1/2"]
[SetUp "1"]
[FEN "4k3/8/8/8/8/2P5/8/6K1 w - - 0 1"]
[PlyCount "27"]

1. Kf2 {The Key squares in this position are b5, c5 and d5. If white's king
can occupy any of these key squares then the game will be a win for white no
matter what.} Kd7 2. Ke3 Kc6 3. Kd3 (3. Kd4 Kd6 $11 {Opposition}) 3... Kd5 {
Unfortunately black has occupied the key squares mentioned, there is no way
for white's king to enter the key squares, therefore, the game is a draw.
Analysis will be continued to prove this} 4. c4+ Kc5 5. Kc3 Kc6 6. Kd4 Kd6 7.
c5+ Kc6 8. Kc4 Kc7 9. Kd5 Kd7 10. c6+ Kc7 11. Kc5 Kc8 12. Kb6 Kb8 13. c7+ Kc8
14. Kc6 $11 (14. Kb5 Kxc7) 1/2-1/2


It turns out white could not occupy the key squares in time, therefore the game is a draw.

The second rule is that if a pawn is on the 5th rank (or 4th for black) or beyond, there are 3 key squares in front of it.

In this position the key squares for white's king are f6, g6 and h6

White is able to occupy one of these squares with his king and the win is secured. The variation shown shows if white had chosen a different route, black would then be able to secure the opposition and the key squares and the game would be a draw.

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round ""]
[White "White to Move"]
[Black ""]
[Result "1-0"]
[SetUp "1"]
[FEN "4k3/8/8/6P1/6K1/8/8/8 w - - 0 1"]
[PlyCount "9"]

1. Kh5 (1. Kf5 Kf7 2. g6+ Kg7 3. Kg5 Kg8 4. Kf6 Kf8 5. g7+ Kg8 $11) 1... Kf7 2.
Kh6 Kf8 3. g6 Kg8 4. g7 Kf7 5. Kh7 1-0


One last key squares example, with a position you gave.

This is a win because White can occupy the key squares of the pawn b5, c5 and d5 (3 squares since the pawn is on the 3rd rank) without any problems

However the variations I gave are to show if white had chosen the wrong route to get to the squares, 1.Kd3 would be a heartbreaking failure as black can secure distant opposition with 1...Kd7 and can therefore control the key squares. 

The second variation 2.Kc4 would also be met by brutal opposition 2...Kc6 and control of the key squares for Black.

The text moves show white occupying the key squares, sidestepping any tricks black has (and there are plenty as shown) and securing the win.

[Event "?"]
[Site "?"]
[Date "????.??.??"]
[Round "?"]
[White "White to Move"]
[Black ""]
[Result ""]
[SetUp "1"]
[FEN "8/4k3/8/8/8/2P5/2K5/8 w - - 0 1"]
[PlyCount "5"]

1. Kb3 (1. Kd3 Kd7 2. Kc4 Kc6) 1... Kd7 2. Kb4 (2. Kc4 Kc6) 2... Kd6 3. Kb5 1-0


One final position, I hope you know this one! 

OK, Black has the opposition, isn't he safely drawing the game then? Well, there are exceptions to every rule, and this happens to be one. Opposition in King and Pawn endgames always works, except when the pawn is on the 5th rank and your king is also in front of it. This is an important pattern to know!

This position is a good one to emphasize the fact that King and Pawn endgames are very concrete and are based on calculation, judgement can be used with opposition and key squares but in the end stone cold analysis is the ultimate weapon; although it is best equipped with the knowledge of opposition and key squares! :)

[Event "?"]
[Site "?"]
[Date ""]
[Round "?"]
[White "White to Move"]
[Black ""]
[Result "1-0"]
[SetUp "1"]
[FEN "6k1/8/6K1/6P1/8/8/8/8 w - - 0 1"]
[PlyCount "7"]

1. Kf6 Kf8 2. g6 Kg8 3. g7 Kh7 4. Kf7 1-0