Tactics
Removing the defender
The bluntest way to win a defended piece is to take the thing defending it. Everything that looks solid in a position is held up by something, and that something is usually capturable, often at a cost worth paying, because the piece it was holding is worth more.
How to spot it
Find the piece you would take if it were not defended, then find its defender and ask what happens if it disappears. Trade into it, capture it, chase it. The arithmetic is the whole tactic: give up three to remove a guard and win five, and the position does the rest.
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
13...Qxd5 14.exd5 Bxg5
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2. Black to play Play it out →Answer
33...Qxh4 34.gxh4 Bxh3
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3. White to play Play it out →Answer
8.Nxc6 bxc6 9.Qxb4
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4. Black to play Play it out →Answer
18...Qxd4 19.exd4 Bxf4
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5. White to play Play it out →Answer
12.Bxg7 Kxg7 13.Qd4+ Nf6 14.Nxf6 exf6 15.Qxg4
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6. White to play Play it out →Answer
11.Bxf6 Qxf6 12.Qxg4
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7. White to play Play it out →Answer
28.Bxf6 Qxf6 29.Bxd5+ Kg7 30.Bxc4
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8. Black to play Play it out →Answer
12...Nxe5 13.dxe5 Qxc4+ 14.Qe2 Qxe2+
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9. White to play Play it out →Answer
9.Ndxe7+ Nxe7 10.Qxd8 Rxd8 11.Nxe7+
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10. White to play Play it out →Answer
30.Qxd5 Qxd5 31.Nxf6+ Kg7 32.Nxd5
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11. Black to play Play it out →Answer
13...Nxf2 14.Rxf2 Qxg3
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12. Black to play Play it out →Answer
19...Nh3+ 20.Kh1 Qxe4 21.Qxe4 Nxf2+ 22.Kg1 Nxe4
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.