The Hardest Mate in 1 ever.

It's mate in 1 with dxe6 e.p. because ...e7-e5+ is the ONLY possible last move. The WK is in check by the e5-P, which has 3 potential last moves, ...f6xe5/e6-e5/e7-e5+. To rule out the first 2 options, we need to do some retro-analysis. First, a piece count: White is missing the a-P only and Black is missing Q, light-squared B, and P. Black's doubled e-Ps are too far away from the a-file for White's a-P to have reached the e-file directly; instead the a-P must have promoted, either to be captured on the e-file or to replace a white piece that was captured. Since Black's a7-P is still on its original file, White's a-P must have captured at least once to promote, and this plus the doubled f-Ps account for 2 of the 3 missing Black units; so there's only 1 spare black unit that could be captured elsewhere.

How did Black's P on e2 get behind the white one on e3? Suppose this e2-P was the original e-P and it marched straight down the file; this would require White's original d- and e-Ps to "cross-capture" and swap files to let the BP through, but this is impossible because there's only 1 spare black unit to capture. Therefore the e2-P came from an adjacent file, capturing directly on e2. That means the e5-P was Black's original e-P, and without spare white units to capture, Black's last move could not have been ...f6xe5+.

Whether Black's last move was ...e6-e5+ or ...e7-e5+, White's previous move was a check given by the d4-B. This check was either Bc3xd4+ (due to the blocking Ps on c5/e3) or Ke5-f4+. Suppose the former - what could the WB have captured on d4? Not the missing Q because it would be giving an impossible check to the WK (e.g. ...Qf6-d4+?? already checking White from f6, or ...Qa4xd4+?? but no spare white unit to capture). Not the light-squared B since d4 is a dark square. And not a P because on d4 this BP and the WP on d5 couldn't have gotten around each other. That leaves Ke5-f4+ as the only possibility, but on e5 the WK would have been in check by the d7-N. The BN delivered that check with ...Nf6-d7+ (or Nf8-d7+), and prior to that, the WR on b7 may or may not have been checking the BK, depending on whether Black's last move in the diagram was ...e6-e5+ or ...e7-e5+.

Consider the former, ...e6-e5+, meaning the WR was checking the BK. How did the WR deliver that check, when there are blocking pieces on b8 and b6? Here's the crucial position.

Only a discovered check could have worked, and there are 3 options. Nf7-g5+ means the WK was in check by the h5-R, but Black had no way of making that check. Similarly, Nf7-d6+ means the b8-B was giving an impossible check to the WK. The third option is c7-c8=Q+. This means White's a-P had made 2 captures, a5xb6xc7 (to get behind the c6-P), and this plus g2xf3 account for the 3 missing black units. Without spare white units to capture, Black's f-P must have been the unit captured on f3 (i.e. e2-P came from d-file). That leaves Black's missing Q and B to be captured by the a-P, but these captures took place on b6 and c7, both dark squares, and the B is a light-squared one. Hence the third option of c7-c8=Q+ is also impossible.

Therefore the idea of Black playing ...e6-e5+ as the last move leads to an illegal position with the WR giving an impossible check on the 7th rank. If Black's last move was ...e7-e5+, however, the WR wouldn't have been giving check and the position is legal. Thus ...e7-e5+ is proved to be the only possible last move and this enables the e.p. mate-in-1.

Excellent retro composition, Arisktotle!

Wow, that is quite a bit of great analysis, covering most of the intended content! 

First of all, I must apologize for falling victim to a liability of quick retro composing, which is overlooking a dual argument. It did not change the correct answer to the question though, which you got wrong! You did find the dual with c7-c8Q without realizing that position was legal! A hint to find the actual retraction proving the correct answer: look carefully at all the options of making the spare black unit reappear; there are a few more than you analyzed.

Note: I corrected the diagram, restoring an earlier version. You can verify that c7-c8Q is now really impossible.

Okay, you've now moved the BN from b6 to f8 and the BP from b5 to b6. The former forces the ...Nf6-d7+ retraction and the latter ensures that White's a-P couldn't have reached c7 after all. Previously, with BP on b5, the a-P could indeed have captured the light-squared B on b5 when the b-P was still on b7, then after WP-b6xQc7, Black could play ...b5 and allow the WR to get to b7 via a6-b6. Hence c7-c8=Q+ was a legal possibility! Now with the BP on b6, a similar sequence is impossible because after WPxBb5-b6xQc7, the BP moving to b6 would have prevented the WR from reaching b7 (or if it got there via b8, the BB couldn't have reached b8 with the WP already on c7). Therefore c7-c8=Q+ is really impossible now. Is that right?

Right! In my terminology, after retracting c7-c8Q the units on a7,b6,b7,b8 c6 and c7 form a cage in lockdown. Nothing in there is retractable given the known capture directions.

Awesome! When I worked out that part, the first term that came to my mind was traffic jam!

After 1. Qf6# Kxh6, what is White's best way to get a second mate?

Can you find the two ways to mate in 2 after your edit?

To which one? I don't see a possible 1. Qf7 in post #1. If you are referring to 1. Qf6 in the first post, that does not checkmate immediately.

En Passant is the solution.

The solution to the diagram in post #69 is checkmate in 2 moves since it is unprovable that white has the right to play en passant. Navigate backward to see the critical moves preceding the diagram. The discussion of the solution follows below.

The general idea of this retrograde puzzle is that it is easy to see that legal proof games exist if blacks last move was Pe7-e5 - giving white the right to capture en passant and thereby mate in 1 move - but very hard to find a proof game where black played a different last move.

Nevertheless such proof games exist if you start with retracting the precise move sequence given with the diagram solution. Every other retraction leads to an illegal game position.

The motif you encounter when making the retrograde analysis is that 1 black unit can be freely uncaptured on one of 4 dark squares: d4, d6, g5 or f4. That unit can be either a black pawn or the black queen but not a black bishop since the missing bishop is of the light squares. Uncapturing a black queen fails on all 4 squares beause it cannot retract on the next move while lifting the check on the white king. The same is true for a black pawn on d6. Uncapturing a black pawn on d4 or g5 makes it impossible for the pawn to return home without making extra uncaptures - with insufficient white material in the box. Remains the uncapture of a black pawn on f4 which is indeed the solution. It is harder to understand than the other uncapture options because the pawn is only activated after intermediate retraction moves by a black and a white knight.

An interesting detail is that the retraction sequence ...Pe6-e5+ .Ke5-f4+ Nf6-d7+ .Pc7-c8=Q+fails because the 6 units on a7,b6,b7,b8,c6,c7 form an unresolvable fortress once you know that Pc6 must return to c7 and Pc7 to a2.

Finding the proof game without Pe7-e5, as described, shows that the position is legal.

Theoretically this is a highly interesting question. Using just the axiomatic chess rules, it is almost impossible to establish that a position is illegal. The rules only reveal which postions are legal and you must collect all of them before you can decide that the remainder is illegal. However, with the help of retraction axioms and some meta-truth - like there can't be 17 units of one color, there is no i-file, retractions must lead to the PAS (game starting position) in a finite number of retraction moves, and so on. - you can construct logical proof that some position under investigation can not arise in a chess game played from the PAS. Therefore, a proof of illegality never consists of a series of chess moves but of a series of deductions accompanied by a pleasing story line with some truths "everybody knows".

We're doing this again? Anyone else here remember the last time this incorrect puzzle came up on this forum and many people tried the wrong answer en passant? The thread seems to have actually been deleted.

In a good retrograde problem, there is not a single item responsible for the (il)legality of the position, but the complete configuration of units on the board. It is a truly holistic experience. The proof may therefore take from 1 up to several pages depending on the skill of the reader. The pawn on e2, like every unit on the board, is placed there on purpose to contribute to the idea of the composition. Would you remove a bishop from an endgame study to make solving it easier?

Well, it all depends on who is reacting to which post. Will people be so kind to specify that?

Wot? You used the thematic e.p. capture as a try!? How Machiavellian is that!

(The last time I saw the OP's puzzle on this site someone (not me) wrote in very large text more than once that en passant was not playable because you could not prove the pawn just made the double push and the correct answer was "no solution" just to try to warn people and get them to stop trying to post the bad answer over and over and I'm pretty sure we wound up getting the whole thread actually removed which seems a pretty impressive feat to me on this forum; I can't think of any other threads I remember that happening to.)

There are only two effective ways of dealing with the immense amount of trolling going on on all social media. Either ignore it or transform it in a creative way. Trying to stop or remove it is a lost cause.