Hello,
in the puzzle position
4k3/1p3p2/1p2pP2/3p1p2/2pP1K2/2P5/PP3P2/8 w - - 0 1
the system thinks 1.a4 is only move to win. But 1.Ke5 is also completely winning.
This is a real simple example showing that something is wrong with the puzzle generation.
Nothing's wrong with the puzzle, king to e5 is a positional mistake because the enemy king can go to d7 blocking the white king's entry into the enemy pawn structure. after that the engine recommends to move the king back to f4 in order to execute the original idea of a4 and then travel to the g7 or f8 square to attack the base of black's pawn structure.
I do understand your confusion, When you play a4 first, the top two and almost equal moves is to go to f8, protecting the pawn structure, or d7, preventing the king from entering trough e5. depending on the move you either travel to attack the f7 pawn or play e5.
This puzzle is basacally a zugzwang, playing a4 forces black to make a bad move, e5 doesn't. SF-13 gives it a +6.8 for a4 and +3 for e5, that's enough difference to say one is clearly better. how high is the rating of this puzzle?
lichess.org/training/V6ACRKe5 is a bit strange, but it is winning. Seems like a faulty puzzle indeed. I think it might have something to do with the engine not considering lines that lead to a repetition of moves as winning (as white will have to go Kf4 later if Ke5 is played initially).
"This is a real simple example showing that something is wrong with the puzzle generation."
I guess... it's an algorithm, it's not going to be perfect. If you can make a perfect puzzle generation algorithm, I'm sure lichess will be very interested.
That being said, I agree that the puzzle does have two solutions.
" If you can make a perfect puzzle generation algorithm, I'm sure lichess will be very interested."
Of course. Why I mentioned it: People developing an algorithm are interested in as simple as possible examples where it goes wrong.
This puzzle is pretty amazing. a4 doesn't look at all obvious at first, but it really restricts black from pushing the b-pawn. (If he does, axb5 or a5 are both good). Then, the White king will come around via e5-d6-e7 to tie the Black king to the defense of the f7 pawn, and then f4 will be played to take away Black's only remaining temporizing pawn moves, when Black must either abandon the f7 pawn or make a bad pawn move... ...b5 can be answered with a5 and something is queening soon. Black can only delay the inevitable by throwing away pawns.
The key here is calculating to make sure that it will be BLACK who has the move once the kings are in zugzwang and all the good pawn moves are gone.
Very instructional.