White to move. Can white win this? How should white proceed?
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White to move. Can white win this? How should white proceed?
1. e5 Kd7
2. Ke2 Ke7
3. Kd3 Kd7
4. Kd4 Kc6
5. Ke4 Kc5
6. Kf3 Kc4
7. Kg4 Kd5
8. Kf4 Kc4
9. Kg5 Kb4
10. Kf6 Kc3
11. Kxe6 Kc4
12. Kf7 Kd3
13. e6 Kd2
14. e7 Kc1
15. e8=Q
So why is 1. e5 the only move to win?
1. Kd2 e5 2. Kc3 Kd7 3. Kb4 Kc6 4. Kc4 Kd6 5. Kb5 wins.
1. Kd2 Kd7 2. Kc3 Kc6 3. Kd4 Kd6 4. e5+ Kc6 and now 5. Ke4 Kc5 6. Kf4 Kc6 7. Kg5 Kc5 8. Kf6 Kd5 9. e4+ and White wins so
6. … Kd5 7. e4+ Kc6 8. Kg5 Kd7 9. Kf6 and White wins
But
1. Kd2 Kd7 2. Kc3 Kd6 3. Kd4 e5+ 4. Kc4 Kc6 draws
Also
1. Kd2 Kd7 2. e5 Kc6 forcing the advance of the next WP 3. e4 Kd7 and draws
So 1. e5 must win because the WK can go towards the opposite side of the pawns to the BK, but isn’t that also true of 1. Ke2 ?
1. Ke2 Kd7 2. Kf3 Ke7 3. Kf4 Kf7 4. Kg5 Kg7 5. e5 Kf7 6. Kh6 Kf8 7. Kg6 Ke7 8. Kg7 and White wins
3. Kf4 Kf6 4. e5+ Kg6 5. Ke4 Kf7 6. Kd4 Ke7 7. Kc5 and White wins
Also 1. Ke2 Ke7 2. e5 Ke8 3. Kf3 Kf7 4. Kf4 wins
But
1. Ke2 e5 2. Kf3 Kf7 3. Kg4 Kg6 draws,
So
1. e5 Kd7 2. Kf2/Ke2 wins and 1. e5 Kf7 2. Kd2/Ke2 wins because the BK can’t force the advance of the second pawn if the WK can defend the e5 pawn from the opposite side.
I have not looked at every piece of your comment to verify it is correct, but your conclusion is likely pretty solid- by playing 1.e5 white puts the decision on black to go to the f-file or the d-file, leaving white the time to take to the opposite side of the pawns. There is likely an additional wrinkle in the variations after 1.e5 Ke7 where white has more flexibility to simply because of the hole left at e4 in some lines- in other words, white can win, too, by directly opposing the black king because the white king can get through to the other side via the e4 square when needed (i.e. some of the lines are drawn if white is forced to play the backward pawn to e4 at some point).
An important line that I missed was
1. e5 Kd7 2. Kf2 Kc6 3. Kf3 Kc5! 4. Kg4! Kc4 5. Kg5 Kc5 6. Kf6 Kd5 7. e4+ and yes, the BK has forced the advance of the second pawn but it’s too late for it to matter.
1. Kd2 Kd7 2. Kc3 Kc6 3. Kd4 Kd6 4. e5+ Kc6 and now 5. Ke4 Kc5 6. Kf4?? Kc6?? might not have been very good play, I was thinking that 6. Kf4 Kd5 could be met by 7. e4+ because I thought the d4 square was covered by the pawn at e3 which had just moved.
6. Kf3 was the right move.