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Twin Problem 2: Mate in 2 Moves

(White: Ka3, Rb3, Bc8, Bf6, Black: Ke8, Rh8, f7, h5.)

White to move and mate in 2

Now, remove black pawn h5:

Note: In the first position black castling can take place, while in the second it is illegal, since it can easily be proved that either the black king or the black rook has moved last.

(White: Ka3, Rb3, Bc8, Bf6, Black: Ke8, Rh8, f7.)
 
 
White to move and mate in 2
 

Solution:

(a) 1. Bf5! O-O 2. Rg3# 
(b) 1. Rb8! Rh5 2. Be6#