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Miscellaneous Problems

Solution

Raymond Smullyan
The Chess Mysteries of Sherlock Holmes, 1979

[8/3p1p2/8/7p/88/k3P1P1/1N1QKB2]

6+4. Monochromatic chess. White to move.
Prove that a promotion and an en passant capture occurred

 

Solution: The bK reached a2 by O-O, h7, g6, f5 ... Therefore Black's could not have just played h7-h5 or g6xh5. Rather Black just played Kb3-a2. But on b3, the bK was in check by the wQ which can only be explained as a discovery check. Last move was 0. .. Kb3xRa2! after the mandatory -1. Rc2-a2.

This wRa2 was a promoted R because a R from h1 cannot reach a2 without violating the monochromatic condition. Therefore a wP did reach the 8th rank, requiring at least 4 captures. But only three black men could have been captured on light squares: bQB, bQR and bQNP (as a bP or as a promoted Black officer). The bKN could only be part of the 4 captures if the White promotion took place on g8, which would require 6 captures and is therefore ruled out. (Incidentally, we see that White's last move was -1. Ra2-c2 without capture.)

The only conclusion is that one of the 4 captures by the wP took place on a dark square: it was the en passant capture of a bP on a5 or c5.

In the original version, the bishop on f1 wasn't present. However, then the position is illegal, since no black piece could've captured this bishop! Ra8 could only go to the even rows, Bc8 and Ke8 could never reach f1, and Pb7 needs too many captures to promote to a rook or a queen, and capture Bf1. (White pieces available for capture towards the promotion line are only Rh1, one of the pawns from a2 and c2, and another pawn which was capture en passant, one piece too little)