What a delightful puzzle! The answer to the main question is that
White can capture en passant
and to see why this is necessarily the case involves a beautiful
parity argument.
For the rest of the explanantion I will not use spoiler blocks, so proceed with caution. :)
Let's start with a bit of inventory. White is missing the queen, a knight and the light squared bishop, while Black is short their queen and the dark squared bishop. Where did these captures take place? Well, since the white king never moved, the white queen never left her home square. The same is clearly true for both missing bishops. Hence, the missing white knight is the only piece that could offer itself up on b6, while the black queen must have fallen victim to White's f-pawn. Moreover, since the black queen must have escaped through c7, her capture must have taken place after c7xNb6.
As we can also see from the diagram, this crucial pawn capture cxb6 also sealed in the white knight in the corner since its only other escape square would have delivered an illegal check. As a consequence, White is really pressed for time following cxb6 - only four pawn moves are available from that point onward since every other piece is immobile (or already dead).
Within these four moves, Black now somehow has to find a way to offer their queen up on either of g3, g4 and g5. I claim that out of these, only g5 is possible and, moreover, that the queen necessarily had to pass through f5 on the way. To see why this is the case takes a bit of effort:
It's easy to see that g3 is not possible - White has only one move available without the f-pawn, which means that the queen would have to reach g3 in only one move. Similarly, g4 is out - the queen needs at least three moves to get there, at which point the f-pawn is already too far up. So the only option is indeed g5. In order to reach that square in time, the queen can take one of three paths, namely
(1) d8-c7-c5-g5,
(2) d8-c7-e5-g5 or
(3) d8-c7-g3-g5
Note that the third option uses g3 which is aligned with the white king. So this option is only possible if the line of sight is blocked at that point. The only piece that could possibly achieve this would be the black knight (on h3 in the diagram). BUT, blocking this check either uses up another black move or requires that the white f-pawn has to start its walk one step up. Either way, White is left one tempo short, so that this third path is not available. The other two options then do both pass through f5, which shows the claim.
As a consequence, we see that the black f-pawn must reach f5 only after the black queen is captured, which must therefore have been the last move of the game so far. Since it cannot have been a capture, it was either f6-f5 or f7-f5. Only in the latter case is White allowed to capture en passant. So, which is it? Well, as the above analysis shows, there is no room for variation following cxb6 beyond the move order of the white pawns and the two different paths of the black queen. In particular, all of these variations start at the same critical position, which must be either variant A (where White can capture e.p.):
Or variant B (where e.p. is impossible):
Why are not both of these possible? They don't look so different at first glance! But, giving it a closer look, we can observe that up until this critical position, only the knights, Black's a8-rook and possibly Black's f-pawn could have made any moves. Crucially, these pieces are all bound by a parity constraint (meaning that they can only revisit a square after an even number of moves)! If we tally them up, we can see that the players have both spent an even number of moves with their knights (since both players have one on each colour). In contrast, the a8-rook has spent an odd number of turns, while the f-pawn has made either no move (A) or exactly one move (B). Summing them up, we see that in variant A, an odd number of moves have taken place, while variant B requires an even number of moves. Since the next move from this position must be from black, we conclude that we must be in variant A, so that the last move of the game must indeed have been f7-f5 which allows White to follow up with en passant!
For completeness, here is a proof game leading to the diagram position
- Nf3 Nf6 2. Ng5 Ne4 3. Ne6 Nc3 4. Nxf8 Nxd1 5. Ne6 Ne3 6. Nc5 Nxf1 7. Na4 Ng3 8. Nb6 Nh5 9. Nc3 Nc6 10. Ncd5 Rb8 11. Na8 Ne5 12. Ndb6 Ng6 13. Nd5 Nf8 14. Ndb6 Nf4 15. Nd5 Nh3 16. Ndb6 cxb6 17. a3 Qc7 18. f3 Qe5 19. f4 Qg5 20. fxg5 f5 21. gxf6
As for the little bonus question, the two helpmates (also in the same lichess study) are:
(a) 1. Rf1 Ne6 2. Rxf5 Nd8 3. Nc7# and
(b) 1. gxf6 Kf7 2. Rf1 Kg8 3. f7#
I must say I am very impressed with this construction! To build such a rich puzzle this "close" to the starting position is quite the achievement!
