What is black's idea in this puzzle? This is one of Lichess's puzzles, taken from a real game. Black is on the move.
[FEN &quot;1r1q1rk1/5pbp/3pbnp1/1p1Np1B1/1p1nP3/P1N2P2/2P1B1PP/R2Q1RK1 b - - 0 1&quot;]

1... Nxe4 2. Bxd8

The suggested move is Nxe4. However, I don't see how Black gains an advantage after White plays Bxd8.
Correction: The suggested move is Nxd5
You are quite right Nxe4 makes no sense. However Nxd5 does because then Qxd8 ends up losing 4 pieces for a queen:
[fen &quot;1r1q1rk1/5pbp/3pbnp1/1p1Np1B1/1p1nP3/P1N2P2/2P1B1PP/R2Q1RK1 b - - 0 1&quot;]

1...Nxd5 2. Bxd8 Nxc3 3. Qd2 Ncxe2+ 4. Kh1 Rfxd8

However if white doesn't take the queen and instead plays the next most obvious move, taking the knight with the e pawn, white still ends up losing a piece:
[fen &quot;1r1q1rk1/5pbp/3pbnp1/1p1Np1B1/1p1nP3/P1N2P2/2P1B1PP/R2Q1RK1 b - - 0 1&quot;]

1...Nxd5 2. exd5 Qxg5 3. dxe6 bxc3 4. exf7+ Rxf7 5.