Mate in one, two, three
Mate in four
Four moves deep, and almost always a king hunt: a sacrifice, a check, the king dragged into the open, and mate. These are hard, and the skill they build is the one that matters most in real games, which is holding a position in your head several moves from the one on the board.
How to spot it
Work with pictures rather than move lists. Find where the king is being driven to, and check the mate there before you calculate the route. If a line runs out, the mistake is usually early: an order of moves that lets them interpose, where the other order does not.
The positions
Easiest first. Every one is from a real game, and the rating under each board is what it costs the players who have tried it on lichess.
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1. Black to play Play it out →Answer
15...Rh1+ 16.Kxh1 Qh8+ 17.Kg1 Qh2+ 18.Kf1 Qh1#
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2. Black to play Play it out →Answer
56...Qxh2+ 57.Qh3 g5+ 58.fxg5+ hxg5+ 59.Kg4 Qe2#
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3. White to play Play it out →Answer
27.Rxe7+ Qxe7 28.Qg6+ Kh8 29.Qxh6+ Qh7 30.Ng6#
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4. Black to play Play it out →Answer
35...Ng4+ 36.Kd2 Rxc2+ 37.Ke1 Rb1+ 38.Rxb1 axb1=Q#
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5. White to play Play it out →Answer
31.Qe8+ Qxe8 32.Rxe8+ Nf8 33.Rxf8+ Rxf8 34.Rxf8#
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6. Black to play Play it out →Answer
29...Qf4+ 30.g3 Rh1+ 31.Kxh1 Qxf3+ 32.Kg1 Qg2#
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7. White to play Play it out →Answer
29.Rxg7+ Kh8 30.Rh7+ Kxh7 31.Qxg6+ Kh8 32.Qg7#
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8. White to play Play it out →Answer
28.Bg6+ Kg8 29.Qh7+ Kf8 30.Qh8+ Bg8 31.Qh6#
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9. Black to play Play it out →Answer
14...Qxf2+ 15.Kh1 Ng4 16.Bg2 Qg1+ 17.Rxg1 Nf2#
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10. Black to play Play it out →Answer
37...Qe1+ 38.Kh2 Be5+ 39.g3 Qxg3+ 40.Kh1 Qh2#
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11. White to play Play it out →Answer
20.Bxh7+ Nxh7 21.Ne7+ Kh8 22.Rxh7+ Kxh7 23.Qh5#
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12. Black to play Play it out →Answer
38...Re2+ 39.Kb1 Qe4+ 40.Ka1 Re1+ 41.Rxe1 Qxe1#
Reading about a pattern is the easy half. Motify finds the positions where you missed one, by running your own lichess and Chess.com games through Stockfish, and puts each of them behind two or three puzzles on the same idea so you meet it having just found it three times. See how Motify works.
Positions from the lichess puzzle database, released under CC0.