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In a recent blitz game, I was playing as White and faced the Lasker-Pelikan variation of the Sicilian. Things were not really going according to plan, I was down a Knight and facing obliteration. In the position below, Black has just played Qxe5 :

[FEN "2r2rk1/5ppp/1p6/p3q3/1n5R/1P1P2RP/1P1Q2P1/7K w - - 0 25"]

In this position, both the Stockfish engine that I have on my phone and the engine from lichess.org give Rxg7+ as the best move for White. Not only that, but they do not even consider the move I played, which is Qh6. The reason why the engine does not think that Qh6 is a good move is that, apparently, Black can respond with Qe1+. However, I cannot see what the problem is for Black in such a case.

To conclude, I think that Black is doomed after Qh6 but the engines do not even recommend it as a correct move. Is there something I am missing here?

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I think that Black is doomed after Qh6 but the engines do not even recommend it as a correct move. Is there something I am missing here?

Yes. The only reason black can't take your queen on h6 is because the g7 pawn is pinned by your rook on g3. If black could take that (even losing the queen to do so) with tempo (i.e. so you don't have time to play Qxh7#) then your queen is lost and with it the game.

Here's how:

[FEN "2r2rk1/5ppp/1p6/p3q3/1n5R/1P1P2RP/1P1Q2P1/7K w - - 0 1"]

1. Qh6 Qe1+ 2. Kh2 Qxg3+ 3. Kxg3 gxh6

and you are a rook and a knight down and completely lost.

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    Also note that 1 Rxg7+ does not simply drop a rook: Black must reply Qxg7 (not Kxg7? 2 Qh6+ and 3 Qxh7#, let alone 1 . . . Kh8?? 2 Rhxh7#) and then 2 Rg4 regains the material. Black should still win, but it's harder because Black's King is now more exposed and the dominant Qe5 is gone. Mar 1 at 22:02

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