How do you find the move?

Below is a relatively simple position.

What is the thought process of a good player here?

White must maintain the opposition and push black away from the queening square, before letting black snap the backward pawn and marching the remaining pawn down the board. 

The thought process of a good player in this position is likely- I know this is a win(becuase I know my endgames well).

If a good player doesn't know the position(I am not a good player-but have forgetten how to win so I will give u my poor thought process)

If I push the pawn and exchange pawns then black is always ready to take opposition(and draw the resultant down pawn endgame). 

So I need to get to the key square d6 or e6.  So need to get a form of opposition.  So k-d3 is a form of opposition(light squared box of the kings if u know what I mean).  Next thing I notice is the king can't come to e5(and blacks king can't use e6-IMPORTANT).  So we have to make sure we don't lose the opposition because of that e5 square fact and or use the e6 square to our benefit.

I also notice I have to be careful of blacks king coming to snatch the backward pawn if my king goes to far away so I do have to be accurate. 

So 1.k-d3! and the easy part is if black goes to either e8, or e7 it is easy...

1...K-e8 we do have to be  careful (because of that e5 square) if 2. ke4 then k-e7 and black takes the opposition.  So instead 2. k-c4 maintaing opposition and I can get to the key d6 or c6 square...

1...ke7 and simply k-e4 taking the opposition and  now I am realizing how much I really need to get back to working on my endgame( as it still isn't simple after gaining the opposition)....

If k-g7 I am seeing it as a draw.  Am I even right?  Or have I forgot this much about endgames?

So that is how a rusty decent player thinks about this position otb.  That probably didn't help lol

 This position is far more complicated than u seem to think.  Remember that black may be able to come around and grab that backward pawn(I am having trouble calculating that part without a board myself.All right so I am having trouble calculating a lot of it lol)

OK, thanks for the input.

Actually, I played around in Nalimov and came up with this, I thoought it was interesting so I posted here.

There is only one winning move, Kd4, everything else draws.

How is a patzer like me to KNOW in a game it's d4, and not e4 or erikido's d3?

Is there a method?

AHHH, k-d4 it is so simple once u see it.  K-d4 and u can get to the c6 square...another ke square.

Hhow do u see it?  First u have to know the key squares.  THen u have to calculate a bit(like making sure the backward pawn isn't taken...I was thinking the backward pawn could be taken with k-d4-which I miscalculated.   you can get the trebuchet position by touching the black pawn first(from e7 and once the black king protects u move to e6 with the trebuchet position-whoever is to move loses).  So basically to win in endgames u have to know some key motifs(eg the trebuchet) recognize key squares(e6,c6 and d6) and calculate well(which I didn't do well.  I thought white got the pawn before black got to e7 before but now looking at it again they are a move short).  This is why endgame study is so important and(helps your middlegame so much)

That was supposed to be an edit.  I just switched black and whhite in the 2nd to last sentence

I basically went through the same things as erikido23:

1. White wins if he can capture the black pawn on f6 without losing his own pawn on f5.  This is because f6 is one of the six key squares (e6,f6,g6,e7,f7,g7) of the f5-pawn in a K+P vs K ending.
2. How can I capture the black pawn on f6?  Clear the board and put down only the black pawn on f6 and the white pawn on f5.  The key squares for capturing the black pawn on f6 are c6,d6,e6,g6,h6,i6 (if there is an i-th file).  So if I can put the White king on any of the key squares while the Black king stays on the 7th or 8th ranks, I know I will eventually capture the black pawn on f6 (and win by #1).
3. Now I start looking at the three adjacent key squares c6,d6,e6.  Opposition is basically a method to gain control of one of three adjacent squares, so you can think of this part in terms of opposition.  I happened to recall a maneuver in an endgame study where the king heads towards the key square which is furthest away.  White threatens to go Ke3-Kd4-Kc5, then stepping on the key square c6.  The only square the Black king can reach in two moves to prevent this is Kd7.  But now White plays Kd5 and he has the opposition and eventually completes #2 and #1.
4. So it is hopeless for Black to defend the c6,d6,e6 key squares.  How if he tries to counterattack on the other side with 1. Kd4 Kg7?  This part reminds me of the Trebuchet, but we actually have a more favorable version of it as the black king must go through g5!  So 1.Kd4 Kg7 2.Kd5 Kh6 3.Ke6 Kg5 4.Ke7 Kxg4 5.Kxf6 and we complete #1.  In a Trebuchet, 3.Ke6 only loses/draws so we have to play 3.Kd6 followed by 4.Ke7 instead.

From #4, we can deduce that White still wins even without the white pawn on g4.

"The Final Countdown" by Herman Claudius Van Riemsdijk and Willem Diederik Hajenius is a good resource if you want to learn to think in terms of key squares in pawn endings.  "Secrets of Pawn Endings" by Karsten Muller and Frank Lamprecht is also very good for this.  I was really confused by all the talk about distant opposition until I read those two books.

The diagonal move Kd4 as opposed to Kd3 and Ke4 suggests itself because I have seen the following endgame study:

The only way to fight for the key squares a6,b6,c6 is by moving as quickly as possible towards the furthest key square.

I guess another reason I thought about Kd4 first is because we use the Reti diagonal moves all the time in pawn endings and K+R vs. K+P.

I know this is not quite a method.

In this position, the outflanking technique should be used by moving Kd4,  and we will reach the f7 square.and the f6 pawn will fall, after which simple.

Just calculate. After Kd4 it’s all over, because you have to think of your opp’s moves.  What can he do? he has to do something, and that means the white K can infiltrate the position. 

Also, all other moves draw “with best play”. If the guy has no clue what he’s doing, any move may win here.  :D

The candidate moves here are easy to figure out because there's only 4 moves possible. 1.g5 looks like the worst of the lot so that leaves 3 possible K moves, and from there I just use a brute force search since  95% of my games are turn-based and time is not critical - although it would be nice to know these positions automatically in case they come up in a blitz game, not to mention saving time in my CC games to work on other less obvious positions.

Of course it helps a whole lot to know about the principle of opposition here, and another basic K+P (vs K) endgame rule: If you can get the pawn to the 7th rank WITHOUT checking the enemy King the pawn queens, otherwise it's a stalemate. 

See Fine's Basic Chess Endings and Euwe & Hooper's A Guide to Chess Endings if you want to be able to play endings like this one correctly and instantaneously (or at least very quickly)

The big mistake is to say this is a relatively simple position. I would say that to ninety nine percent of chess players it is a very difficult position. I'm an average player and I'm constantly misjudging positions in every department of the game. I can also say from reading about other much stronger players they are also doing the same thing.

Thats what makes chess so fascinating its a very difficult game at all stages.

I think most good players don't have much of a thought process in this particular ending. They know from study/practice this is a win and they simply proceed to win with their " technique ". I know this ending and others like it and if you study/understand the elements of pawn endings ( outflanking, opposition, key squares...etc ) . Its easy to calculate that if black defends " actively " by going after the g pawn white will take the f pawn when black takes the g pawn with an easily won ending... the f pawn queens in a few moves. This leaves only passive defense for black which means he must keep the white king out and its easy to see that black cannot keep the white king out . 

The key to this kind of position is 'direct opposition' starting with 1.Kd4 Ke7 2. Kc5 (diagonal opposition) Kd7 3.Kd5 Ke7 4.Kc6 Kf7 5.Kd7 and white wins.

@GlennBk: Of course it's a relatively simple position.  Try mating with K+B+N vs K or winning with  K+Q+QNP vs K+Q so you can figure out the difference between relatively simple endgames and impossible endgames, such as K + 2 Kt vs K+P which is an exception to the 50-move rule (it can take up to 66 moves in the most unfavorable winning position, but TS FIDE abolished all the exceptions in 1992)

The fact that 90% or 99% of chess players don't find this position simple doesn't make it difficult relative to other endings such as K + 3 Kt vs K which I've never been able to learn BTW! Also the percentages here are just someone's opinion not necessarily mine...

1.Ke4 Ke8 and Black has a distant opposition in place.  Now it is impossible for White to force his way to any of the key squares c6,d6,e6.  There is no hope to penetrate from the other side as we do not have the i-th file, so after 1.Ke4 the game is drawn with best play.

In the position given by rooperi, my aim as white is to capture the pawn f6 with the king.

(I do not consider trying to exchange pawns with g4-g5 because I know that the pawn ending with the black king in front of the remaining pawn is a draw.)

So white needs to try to penetrate on the queenside with the king.

A) can black try to defend against that by blocking that path with his king?

1.Kd4 Ke7 2.Kc5! (fighting to gain the opposition with a known technique - many people will know this as the so-called diagonal opposition. Note that white would not have this option after 1.Ke4? Ke7 2.Kd5 Kd7, and black has the opposition and holds a draw) 2...Kd7 3.Kd5! white has won the opposition and is winning easily (3...Ke7 (3...Kc7 4.Ke6 followed by 5.Kxf6) 4.Kc6 and the black king cannot stop the white king from continuing to penetrate all the way through to f6)

B) can black try to be faster with his king on the kingside by trying a king race?

1.Kd4 Kg7 2.Kd5 Kh6 3.Kd6 ( a usual way of fighting for tempos in such situations (even winning equivalent positions without the pawn on g4). However, the easier and more direct approach also works here: 3.Ke6 Kg5 4.Kf7 Kxg4 5.Kxf6 also wins for white)

3...Kg5 4.Ke6 Kxg4 5.Kxf6 wins for white

After these considerations and these calculations, I can say that white is winning in this position.

I admit that I actually did not start calculating lines with 1.Ke4 because this move, while also trying to penetrate on the queenside, simply gives white less options while not providing any speed advantage in a king race.