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Puzzle 5044

I have 2 questions about this puzzle (http://en.lichess.org/training/5044).

In the initial position Qxh6 (Stockfish: ~7) is the best move, but Nf6+ is just as winning (Stockfish: ~4). Why does lichess not say "Good move, but you can do better" as in other puzzles?

Also, after Kf8 the puzzle only accepts Ba3+, while claiming that Nxh7+ is a "good move, but you can do better", even though Stockfish actually prefers Nxh7+. Why is that?

Hello,

For the first question, I don't know, maybe the game is harder to win with the move you propose.

For the second one, it happened to me several times that Stockfish plays a different move than the one proposed in the solution. And I do not see any reason not to take the pawn because if the king try to escape in E7 instead of G8 to save the rook, the white bishop can go to F6.
It isn't a "good move" that's why. 1. Nf6+ is objectively a mistake in this position. It doesn't allow Black to equalize by any means, but it is not as forcing and gives up White's very powerful knight. It's just objectively better to get the queens off the board and remove the defender of the f6 square in the process, allowing the knight to check with impunity and win enough material that Black's situation is entirely resignable. After 1. Nf6+ Black can keep playing, even if he gives up the queen, since he has the bishop pair, knight, and rook are fairly well coordinated against White's king. White can easily hold, obviously, and the queen will eventually be enough to overpower, but it's much harder to find a winning solution.

As for why the bishop check may better? Well it leads to a few forced mates. It seems like after Nxh7+ may allow the king to sidestep for awhile, so it may take a little longer to find the win, but the engine can likely calculate a plan that wins from there as well, since Black's king safety is very compromised even after moving toward the center. But, pushing the king back into the corner and grabbing the rook guarantees Black doesn't have enough material to defend, and also pushes Black's king into a position where the bishop pair can't defend against anything. From a human perspective at least, Ba3+ is the more natural way to win. Every move in the solution line is a forcing move, so it's simple chess.
Yes, Nf6+ might now win as easily, but I think an evaluation of +4 should be enough to at least make it a 'good move' and offer the solver to find the better move, Qxh6.
And if it isn't a 'good move', I would like to know by what criteria a suboptimal move is still considered a 'good move', because in some cases you get a retry and in others you don't.

As for the second question, Stockfish prefers Nxh7+ because it literally transposes into the Ba3+ line, yet with an extra pawn. Nxh7+ is the better move.
I'll restate in simpler terms...

Nf6+ is a mistake because it loses 3 points in evaluation...it would be annoted as mistake. Therefore it cannot be conisidered a "good move, but you can do better." I'm fairly certain the "good move, but you can do better" dialog only occurs when the evaluation does not lose enough points to even be considered an inaccuracy...i.e. if you have a mate in 3 and you go for a mate in 5, it may very well fail. If you have the chance to go up a pawn but instead you go for line that doesn't win that pawn, you'll likely fail. That's the basic criteria, I'm fairly sure. It's why some endgame puzzles will give you "good move, but you can do better" for virtually every move other than the correct one, because the evaluation doesn't really change, it's maybe the difference of 5 centipawns. Losing 300+ centipawns is going to fail any puzzle, not give you a pat on the back and a push to try a little harder.

As for the transposed line, yes, under proper play it would indeed transpose into that line. But as I stated, there are options. It's not a forced line, as Black could sacrifice the queen and possibly get a small amount of counterplay. Granted, that would be even worse in terms of evaluation for Black, but it's definitely not forcing the mating sequence, so that is likely the reason that while a computer would obviously choose the line that wins an extra pawn, as a human it's objectively a little better to play the forced line. Make sense?
Whoops!

Ignore that second paragraph as I was still looking at the Nd6+ line I guess. The Nxd4+ versus Ba3+ is probably just a matter of depth used in the evaluations and really can't be explained any other way. But I do still think the Ba3+ line is easier for humans to see a forced mate and so it makes better sense for the solution, but Nxd4+ may be slightly better. I really don't know.

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