Notice that the white bishop a3 controls f8 and that if Black could castle he would mate the white king.
Tries: 1.Kg7? but 1…hxg5! 1.Be7? but 1…Rf8+, 1.Qxg6+? but 1…hxg6, 1.Bxg6+? or 1.Rxb7? or 1.Qxh4? or 1.Bd6? but 1…Kd8!
Solution: 1.Rb4 (thr. 2.Rb8#) Rf8+ 2.Kg7 (thr.3.Rb8#) d6 3.Rxf8+ Kd7 4.Qd8+ Kc6 5.Qa8+ Kd7 6.Qe8#
The function of the wBa3 is a mere trick to induce the solver to think that 1.Rb4 enables black castling and look for alternative tries. Simple retroanalysis proves that Black can never castle as his last move must have been either with the bK or bR.
Notice also that the set check 0...Rf8+ by Black is provided for (followed by 1.Kg7 and mate in three more moves).
Retroanalysis: The position is legal. There are five missing white pieces that must have been taken by the bPs d3,h6, and h4. Hence, no other black piece has made any capture. Black's last move was not made by any pawn (both e6 and gxh6 would have had the wK in check). Black's last move was not made by the bS. It couldn’t be a capture on g6. Thus the bS move would have had to be in response to a white check by the wBh5. That check is impossible to make, the wB could not move there in its previous move. It could neither be a discovered check by the wK as it couldn’t be on f7 or e6 nor a discovered check by the wQ (it would have been also checking the bK in g6). So Black’s last move was either with the king or rook and thus he cannot castle!
A new theme is shown: Black castling is clearly illegal in the diagram position, making any retroanalitical study unnecessary. Then the key move seems to allow the strongest possible black defense: mate (pseudoBerlin theme)!
