Capture defender of fork (3 moves)
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Explanation
Let's look at the most common ways to remove a defender of a fork:
In this diagram, the black knight on b6 defends d7, preventing White from winning the b8 rook with the fork Nd7+. This defense is insufficient, because White can remove the defender in a few ways:
White can capture the defender of d7 with 1.Bxb6! This also captures the defender of the bishop on d5, so Black’s best response would trade that first with check: 1...Bxg2+ 2.Kxg2. However, after Black recaptures on b6, d7 is undefended; for example, 2...axb6 3.Nd7+.
Because the black knight is overloaded, White can deflect it with 1.Bxd5! If Black recaptures with 1...Nxd5, the knight no longer defends d7, and White can win with the fork 2.Nd7+. If Black tries a counterattack with, for example, 1...Bd6, White can just play 2.Bxb6 to win a piece.
White can also attack the defender of d7 with 1.a5, but here Black can play the counterattack to lose only a pawn: 1...Bxg2+ 2.Kxg2 Bd6 3.axb6 Bxe5 4.bxa7. That looks pretty good for White too, but the other options are much clearer.
Examples
White could win with the knight fork Nxc7+, giving check and attacking the rook on a8 at the same time. However the immediate Nxc7 does not work, because the knight on a6 protects c7. White can win by capturing the defender with 1.Bxa6. If Black recaptures with 1...bxa6, White can win with 2.Nxc7+, because c7 is now unprotected. If Black does not recapture, White will have won the knight a6.
Black could win with the queen fork Qc5+, giving check and attacking the unprotected bishop on b5 at the same time. However the immediate Qc5+ does not work, because of Nxc5. Black can win by capturing the defender with 1...Bxb3. If White recaptures with 2.axb3, Black can win with 2...Qc5+, because c5 is now unprotected. If White does not recapture, Black will have already won the knight b3, and threaten Bxd1 and Qc5+.
Sometimes capturing the defender can involve sacrifices and other motifs, as in the following example:
White has just played Nd5, attacking the pinned knight f6, but this was a terrible mistake. Black won with the surprising 1...Nxd5, capturing the knight which was defending e3, and attacking e3 with the knight. Black moved the knight out of the pin, but if White captures the queen with 2.Bxd8, Black can play the knight fork 2...Ne3+, and for example after 3.Kg3 win the queen back with 3...Nxd1, winning a knight at the end.
How to Spot an Opportunity to Capture the Defender
When you are threatening a fork, but the forking square is protected, always think about ways to remove the defence.
Related combinations
There are various ways to remove the defence. In addition to capturing the defender, other ways to remove the defence include deflection, and attacking the defender.