Here you need to calculate three white moves and all of Black's defences. Tip: first work out what the mating position might look like, then think about how to get there. Play on the board: the program answers for Black with the most stubborn defence. Every puzzle has been verified by a computer: mate is unavoidable, the key is unique and there is no faster mate.
1. Anastasia's mate
White to move. Mate in 3
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1. Ne7+ Kh8 2. Qxh7+ Kxh7 3. Rh3# — the knight drives the king into the corner and covers g8 and g6. The queen is sacrificed on h7, and the rook mates along the h-file.
2. Philidor's legacy
White to move. Mate in 3
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1. Nh6+ Kh8 2. Qg8+ Rxg8 3. Nf7# — a double check by knight and queen, then a queen sacrifice and smothered mate.
3. A quiet rook move
White to move. Mate in 3
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1. Re4 gxf6 2. Qxf6 Kg8 3. Re8# — White threatens Re8# — the knight guards e8. If Black takes the knight, the queen breaks in on f6 and mate is unavoidable: Re8# or Qh8#. On 1… g6 or g5, 2. Qh6# follows at once; on 1… h6 or h5, 2. Re8#.
4. A quiet queen move
White to move. Mate in 3
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1. Qe5 f6 2. Qc7 Kg8 3. Rb8# — the queen cuts the king off from e7, and White threatens mate with the rook — Rb8#. If Black defends with f6 or f5 (giving the king the f7 square), the queen goes to c7, covers that square, and the rook mates on b8.
The last step is endgame studies: there you do not have to mate but to win or save the game.