Originally posted by BigDoggProblem
White is missing only a Rook; it must have been captured on the c-file by a bP.
Black is missing 7 units; all were captured by white pawns.
Black's last move was not -1...d7xRc6?? because this would require both white c-pawns to uncapture to let Pc2 go home, and there aren't enough captures available to do it.
Black's last move was not -1... ...[text shortened]... 6 square changes everything!), permitting White to mate in 2 by:
[b]1.axb6e.p.+ Kxe3 2.Qe2#[/b]
Here is some background and a more elaborate/complete explanation of the retrograde part:
Problem by Ems Mundt, published in 'De Schaakwereld' 1937 and solved by 13 yr Johan Barendregt (1924-1984). Republished by Leo Lewin.
What was black's last move?
a) not the knight on e1, would have come from f3, but then white was in check with black to move.
b) the black king couldn't have come from c4 because of the following: c4 is covered by the knight on e3. That would have been white's last move then, interrupting the check from blacks bishop on d2. But black cannot have given this check with the bishop, because he is completely blocked. And discovery is also not possible if the black king was on c4, because then there is no piece to do the discovery.
c) other moves by the black king are also impossible.
d) Let's count the pieces: 15 white and 9 black. Black has captured once, and white 7 times.
The original pawn on b2 has captured towards the a-file.
The c-file is special: two times did a pawn of one side get behind a pawn of the other side. If the white pawn that was originally on c2 went to c7 without capturing, then black must have captured with both his pawns. But he captured only once. Hence, the pawn on c7 arrived there by capturing d6xc7. With the pawns on c7, d6, e5 and e4, white must have captured 6 times. Plus the capture from b- to a-file equals the total of 7. Hence, white has not captured with the pawn towards c3, so the pawn was originally on c2. Hence the black pawn got on c2 through capturing d3xRc2 (the original b-pawn is still there on the a-file, and the only missing piece by white is a rook). So the rook was captured on c2 and not on f5 or on c6. But d3xRc2 wasn't black's last move because black's king is on d3. So, only one black piece could have made the last move: the pawn which is now on b5, either by b6-b5 or b7-b5. With that pawn move, black canceled the check from the bishop on a6. But how did white make that check? The bishop could not have come from b7 because then black had no moves at all. It must have been a discovered check. The only possible move by white was Nc4-e3+. Before making that move, his king was in check by the bishop on d2, and also that check was a discovered check, only possible with the move Ke3-d3+. This black king was in turn in check himself by the knight on c4. That knight can have come only from one square: b6.Meaning that there was no black pawn on b6, eliminating b6-b5 as black's last move alternative. Hence, black's last move was b7-b5.
Now the puzzle is trivially simple, white can take en passant: 1.axb6+ Kxe3 2.Qe2#