Very hard mate in 1

The second one I posted, you need to prove that en passant is legal. In other words, you need to prove that (1) it's now black to move, and (2) white's last move must have been g2-g4.

This can be done by contradiction. Suppose on the contrary that it's white to move, then black just moved; this last move must have been ...exf4, but this is impossible because ____?

Once you prove that you can see that it's black to move, and white's last move must give black a different last move before that, or else we'd have the same impossibility. And this white move would be ___ because then black has the move ___ before that. And of course you can then show that the position is legal by making a proof game leading to the position.

You're going to struggle making a proof game where you get a pawn on e2 without capturing any pieces.

This is an impossible position. The pawns cannot be doubled, when no pieces are taken.

Also if the board happens to be in the wrong way (if we are looking from Black's perspective), then Bxd4 is mate.

@Illion on the first puzzle, can't the solution also not be a draw? It is possible that black's last move was exf4, then white played Qf6 and it's stalemate. fxg3# is only possible if you mention that the current position isn't a stalemate already

I found it pretty quickly because its the only way to attack h8 square, since queen f6 loses the knight

@CrushKing
No, like I said in my previous post, ...exf4 cannot have been the previous move. One of the steps in the solution is for you to work out why not (in some sense, retro problems are closer to logic problems.)

Notice that white is missing 3 units (1 pawn, 2 knights) while black is missing 13 units (everything but the 2 pawns and king.) Also, white has made at least 13 pawn captures to reach the pawn structure (count them); therefore, exactly 13 pawn captures. (Question: which white pawns have not made captures?)

Therefore the black pawn on h3 has already made at least 1 capture (do you see why?)
Also the black pawn that started on a7 must have been captured by the white pawns, so 1 white unit was lost to allow this (do you see why that's the case even if it was promoted on a1?)
Finally, the black pawn that started on h7 must also have been captured by white pawns, so it too captured at least 1 white unit (do you see why it can't have gone straight down the h-file?)

That's the 3 missing white units. So no, black can't have played ...exf4. (Fill in the gaps in my logic, I already practically gave the solution.)

So. White made the last move. Now can you say with certainty what the last move was, and why? Once you have that, then there's only one thing left to do. (And then you realise the position practically spoke for itself and didn't need any instructions or conditions attached.)

@grapejam
I'm not talking about the OP's problem, I'm talking about the second one I posted in post #22.

@Illion ok, now I see it's actually a lot more difficult than i thought first. can you post a sample game (pgn) how the initial position can be reached? I tried and couldn't find it

If you followed my logic from the start and filled in the gaps, then finding the proof game should be easy compared to that (since you know what needs to happen and roughly in what order.) As I said, the point of the problem isn't in the moves but in the logical reasoning - perhaps a different type of chess problem than most are used to.

Of course, the proof game I show is not unique (believe it or not, there are problems where the goal is to find a unique proof game, and those can be similarly hard, but I digress.) However, do note that it's not sufficient to demonstrate one proof game to justify black to move and en passant; rather, you need to prove that -all- possible proof games lead to black to move and legal en passant, which is what the outline of the reasoning I posted is for.

I want to emphasise that a proof game only proves the position is legal. It does not answer the problem; the expected solution is a sound logical argument. (But typically in retros, if you're capable of constructing a proof game, then you've probably already reasoned out the logic since it's nigh impossible to find any proof game without the logic to help you construct it.)

PGN in text format because the point really isn't the moves (and I'm lazy), rather it's to complete the logical reasoning I posted before. In any case one can easily view it on lichess.

1. Na3 c5 2. Nc4 b5 3. Ne5 c4 4. Ng4 h5
5. h4 hxg4 6. Nh3 gxh3 7. h5 c3 8. bxc3 d5
9. Ba3 d4 10. cxd4 e5 11. dxe5 Nf6 12. exf6 Qd3
13. Bd6 Bf5 14. Bh2 b4 15. cxd3 b3 16. Bg1 bxa2
17. Rc1 a1=Q 18. fxg7 Be4 19. dxe4 Bc5 20. Rc2 Be3
21. dxe3 Qf6 22. Qd4 Qf3 23. exf3 Ke7 24. Bc4 Rh6
25. Ke2 Rg6 26. Bd5 Rg4 27. fxg4 a5 28. Kf3 a4
29. Kg3 a3 30. Kh2 a2 31. Re2 a1=Q 32. Qc4 Kf6
33. Qd4+ Kg5 34. Qd3 Qf6 35. Qc3 Kh4 36. Qb2 Qf5
37. exf5 Nd7 38. e4 Ne5 39. Qc3 Ng6 40. fxg6 Ra5
41. Bc6 f5 42. Qd4 f4 43. g5 Rf5 44. exf5 Kg4
45. Qf6 Kh4 46. Be4 Kg4 47. f3+ Kh4 48. g4

*also, small erratum in the reasoning I posted earlier - the black pawn that started on h7 could have been captured, or is the one now on h3. Either way, another white unit has to be captured. The h7-pawn can't be the one now on f4, though - that's again impossible because insufficient captures and such.

Quite impressive