Corresponding Square Problem
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Easy right?
This study appears rather involved, but can actually be solved with just the basics. When I first did this a few weeks ago, I saw that the initial position was a zugzwang and that white cannot win immediately. It took some time to work out the correspondences for all three of the black king's squares and triangulate the win. A second look revealed that it is really a basic opposition problem in disguise. The g4 square is unavailable so white maneuvers through g2 to gain the opposition.
White to move to draw.
Easy Draw
Even though I appreciate concrete variations, like the ones Wolf183 gives, I still think the OP's corresponding square problem deserves more of an explanation in words in order for more inexperienced players to fully understand what's going on in the variations above:
If you already know that it's a corresponding square problem you should go about explaining it in the following sequence:
1. Determine the key squares
- The first key square is obviously f5 because if White captures the pawn then the 2-1 pawn majority will win easily.
- The second key square is g5 because if White's king can occupy that square the f5-pawn is also doomed (Black can't have his king on e5 to protect his f5-pawn since he has to remain within the square of White's e-pawn).
2. Determine the corresponding squares right next to the key squares
This is determined by looking for the squares the kings have to use in order to threaten to move to the key squares:
- f4/h5 and f6 correspond since from f4 White's king threatens both Kxf4 and Kg5 (reaching the key squares) and Black must counter this with ...Kf6. Black can't meet Kf4 with ...Kg6 because even though this immediately counters White's king's threats Black still loses due to: 1.Kf4 Kg6 2.e7! Kf7 3.Kxf5 Kxe7 4.Kg6! - here g6 is a critical square of the d6-pawn which is now lost after a few more moves and with it the game since after capturing the black pawn White's king will be on a key square of his own d5-pawn. Note that with Kh5 White also puts Black into zugzwang if his king is on f6 since he can't continue guarding g5.
- h4 and g6 also correspond. This is because White's king is threatening to penetrate to the key square g5 from h4. This threat can only be met with ...Kg6 since ...Kh6 leaves the square of the white passed pawn, thus allowing its advance, while ...Kf6 does guard g5 against immediate penetration but can be met with a further king advance: 1.Kh4 Kf6 2.Kh5! - Black is in zugzwang (se above that h5 and f6 correspond) and can't continue guarding g5 - thus the f-pawn is lost.
3. Determine the rest of the corresponding square system
This is done by examining how the kings can threaten to occupy not the key squares themselves this time but instead the corresponding squares already found above (or some which we will find below):
- g3 and g7 correspond. This is because White's king by playing 1.Kg3 is threatening to go to both of the corresponding squares f4 and h4. Black must counter this move by going to a square which allows him to respond to both of these threats - in other words a square which gives him the option of going to either f6 or g6 (both options must be available), and this is only possible by countering 1.Kg3 with 1...Kg7. Black's options are restricted to this since 1...Kg5 leaves the square of White's passed pawn, allowing 2.e7! with a win, and also the f7 and f5 squares are unavailable because of the pawns.
- f3 and g6 also correspond. By playing 1.Kf3 White is threatening both 2.Kf4 and 2.Kg3. Black has to be able to meet this with a move which allows both ...Kf6 and ...Kg7 (thus arriving at the corresponding squares to f4 and g3 respectively). This is only possible by countering 1.Kf3 with 1...Kg6 since f7 is unavailable due to the white e-pawn.
- f2/g2 and f6 correspond. This is the final and decisive part of the puzzle. From either f2 or g2 White's king threatens both 1.Kf3 and 1.Kg3. Black has to meet either 1.Kf2 or 1.Kg2 with a move which allows his king to counter White's threats by moving to the corresponding squares of f3 and g3 (the squares g6 and g7). He has four squares for this: f6, f7, h6 and h7. But f7 is again unavailable due to the e-pawn while the last two squares lie outside the square of the white e-pawn so only f6 remains.
This last piece of information is what's really needed since White can maintain his threats by moving between f2 and g2 while Black can't remain on f6 since he has no other piece which can transfer the move to White (moving the f-pawn causes its downfall) - thus Black is in zugzwang.
4. Show the solution by variation
The solution is thus:
1.Kf3 (1.Kg3 also works, for example: 1...Kg7 2.Kg2 (or the other way by 2.Kf2) 2...Kf6 3.Kf2! Kg6 4.Kf3 Kg7 5.Kg3 Kg6 6.Kh4! Kf6 7.Kh5 and White is winning)
1...Kg6 (Black counters thus far) 2.Kg2 (2.Kf2 works the same way) 2...Kf6 (2...Kg7 3.Kg3! transposes to the variation above) 3.Kf2! (Black can't stay on f6 which corresponds to f2 and is thus in zugzwang) 3...Kg7 (3...Kg6 4.Kf3! Kg7 5.Kg3! transposes to the main line) 4.Kg3! Kg6 (4...Kf6 5.Kf4! and Black is in zugzwang) 5.Kh4! Kf6 6.Kh5! (Black loses control of g5) 6...Ke7 7.Kg5 and White wins the pawn and the game.
A very interesting position in my opinion even though it is a relatively simple corresponding square problem. Thanks for uploading EndgameStudy!
White to move and win