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How to lose a won endgame??

I found this position very instructive http://en.lichess.org/urDu9NPi/white#82 - black has just moved Kd7, giving me (white) the centre and a won game. After thinking for all of 6 seconds I grabbed the e5 pawn "knowing" the rest of the King side pawns were mine for the taking, but it was immediately losing. Seems like there was only one winning move.. not sure if I'd of found it or not, even if I gave it the thinking time it deserved during the game.
Either way I need to stop making reflex moves.
It's a simple but very tricky position. A bit unusal, too. At first I thought h3 could win, but then there's f3 and white loses. Many pawn endings are very hard. You can't trust in intuition in this kind of situation. Your intuition will probably fail because pawn endings tend to be counterintuitive. You should only trust in calculation.
Indeed, I've been trying to wade through John Nunn's excellent endgame books, he shows lots of examples of games analysed and published in books, but modern computer calculations prove them to be wrong for the most part. It really is a tricky part of the game, and it doesn't help that only a small fraction of games make it to a tight endgame at my level.
Black offers draw
White declines draw
Black offers draw
White declines draw
Black offers draw
White declines draw
...
Damn, I haven't seen so many rejections since I was trying to hit on a gril on highschool
Funny, I haven't been burned so bad since I tried to hit a grill.

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