Here are two examples by the retro specialist Dr. Karl Fabel. I think they are not too difficult but still suitable for demonstration purposes. By the way, there is a project "UltiMate Fabel" with the aim to publish his complete opus. Maybe, it will be finished this year. Watch out for this book!


Now, let's have a look at the diagrams. A mate in 1 move? Solved in a second! Really?
The solution to diagram 1 is 1. Kf2#. Or is it 1. 00#? Are there even two solutions? To find out, we must look for hints.
For example, we see that the three white pawns on e3, f3 and g3 captured the three missing black pieces. We can't determine the details, but evidently, e3 came from f2, f3 from g2 and g3 from h2. Also, only a rook could be captured on f3. as the bishop from f8 moves on black squares.
Then we have the Na8. First the knight moved to a8, then there was the capture a7xb6. This means, none of the black rooks could leave the 8th rank via the afile. Either, there was the pawn on a7 or the knight on a8. Now, that we know this, we further conclude that f7f5 had to be played before the capture on f3. The captured rooks could only get out via f6! Furthermore, g7g6 had to be played prior to the capture: either to enable Rh8g8g7f7f6 etc. or to let the bB leave f8, so that Rh8f8f6 etc. was possible.
So, we know that the pawns already were on f5 and g6 when g2xRf3 was played. Now, we arrived at the crucial point. How did the bishops reach their respective squares in the diagram? Obviously, the bB stood on h3 when g2xRf3 happened, this was the only square. After the capture, he blocked the path of the wB. Thus, he had to move to h1, in order to let the wB get to h5. This was the only square to make way.
Wow! We just proved that the wR already had left h1 and later returned. Therefore, White can't castle anymore. Only 1. Kf2# solves the first problem.
How does it feel to be a chess detective? Come on, it is just great! Oh, there was a second diagram. We will crack this puzzle, as well. It also asks to mate in 1 move, but there is something added. "RV" stands for "retro variants". This means, that a certain position does not simply consist of the pieces on their squares but also includes the possible moves (especially castling and e.p. captures) and the history of the position. In such problems, the history of a position cannot be determined with certainty, but each of the alternative histories demands a different solution.
OK, we already know we have to examine Black's last move(s) and must figure out whether White can castle or capture the pawn dxe6 e.p. Both moves deliver the desired mate (and only those).
It is impossible to determine what move Black played last, but clearly it must have been either e7e5 or something else (law of excluded middle, tertium non datur). So, the solution has two lines.
Let's assume, Black just played e7e5. Then, bBh4 is a promoted piece, coming from e1 or g1. Therefore, the wK had to move. Promotion on e1 is the trivial case. If Black promoted on g1, then the bishop must have moved to f2 giving check and forcing the wK to leave e1. We see, castling is not allowed. But 1. dxe6 e.p.# solves, as we can prove (we explicitely assumed it) that the last move was e7e5.
What, if Black did not play e7e5, maybe just e6e5 or moved the king or the bishop? Then, the bBh4 can be the original piece and 1. 000# solves, as we cannot prove that castling is not possible.
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