Taking the knight on f8 instead of e4 is acceptable for black. White it's still up the exchange after retreating the e4 knight.
I can't understand today's daily puzzle (A Well-Calculated Leap) 10/1/2022

In puzzles, there's a very big difference between your "best move", which you must find to correctly solve, and the opponent's "best move", which actually doesn't exist! Why?
When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others. An engine may rate an opponent's move A as better than the puzzle move B, but you'll notice that the evaluation for A is that it's also completely losing (higher than 3 for your side).
In a practical game, the engine move A might offer more chances to avoid defeat (that's what the engine is suggesting), but that's irrelevant in a puzzle. A puzzle assumes perfect or best play on your side, which means defence A has zero chance of saving the game, just like B. Hence the puzzle move B cannot be viewed as a mistake, and there's no "best move" for the opponent. The puzzle picks B as the defence because it's more instructive than A, or because it forces you to find a unique winning reply. If the puzzle had used the engine move A, you could have multiple winning replies, which would result in a faulty puzzle.

In puzzles, there's a very big difference between your "best move", which you must find to correctly solve, and the opponent's "best move", which actually doesn't exist! Why?
When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others. An engine may rate an opponent's move A as better than the puzzle move B, but you'll notice that the evaluation for A is that it's also completely losing (higher than 3 for your side).
In a practical game, the engine move A might offer more chances to avoid defeat (that's what the engine is suggesting), but that's irrelevant in a puzzle. A puzzle assumes perfect or best play on your side, which means defence A has zero chance of saving the game, just like B. Hence the puzzle move B cannot be viewed as a mistake, and there's no "best move" for the opponent. The puzzle picks B as the defence because it's more instructive than A, or because it forces you to find a unique winning reply. If the puzzle had used the engine move A, you could have multiple winning replies, which would result in a faulty puzzle.
I'm sorry, this doesn't really make much sense to me. Since you said
"When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others."
then would that mean, puzzles are not meant to be able to be solved on the first try? If you are really supposed to explore incorrect lines then wouldn't that defeat the purpose of learning from puzzles? I came in assuming I would be trying to make the best move against an opponent who would make the best move they could as well. If the puzzle doesn't give the best play possible as the opponent, why should I or anyone use it as a learning tool? Just to study something that is incorrect? Is the reason it made the bad move because it wants the player to find the mistake? If so wouldn't it be more productive to just show the best move by the opponent? When you play a practical game, do you play the best move you can or do you make mistakes to help your opponent learn that you make mistakes (just an example, not meant towards anyone in particular)? I don't know I would love to hear someone's opinion on this. It would be greatly appreciated.

In puzzles, there's a very big difference between your "best move", which you must find to correctly solve, and the opponent's "best move", which actually doesn't exist! Why?
When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others. An engine may rate an opponent's move A as better than the puzzle move B, but you'll notice that the evaluation for A is that it's also completely losing (higher than 3 for your side).
In a practical game, the engine move A might offer more chances to avoid defeat (that's what the engine is suggesting), but that's irrelevant in a puzzle. A puzzle assumes perfect or best play on your side, which means defence A has zero chance of saving the game, just like B. Hence the puzzle move B cannot be viewed as a mistake, and there's no "best move" for the opponent. The puzzle picks B as the defence because it's more instructive than A, or because it forces you to find a unique winning reply. If the puzzle had used the engine move A, you could have multiple winning replies, which would result in a faulty puzzle.
I'm sorry, this doesn't really make much sense to me. Since you said
"When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others."
then would that mean, puzzles are not meant to be able to be solved on the first try? If you are really supposed to explore incorrect lines then wouldn't that defeat the purpose of learning from puzzles? I came in assuming I would be trying to make the best move against an opponent who would make the best move they could as well. If the puzzle doesn't give the best play possible as the opponent, why should I or anyone use it as a learning tool? Just to study something that is incorrect? Is the reason it made the bad move because it wants the player to find the mistake? If so wouldn't it be more productive to just show the best move by the opponent? When you play a practical game, do you play the best move you can or do you make mistakes to help your opponent learn that you make mistakes (just an example, not meant towards anyone in particular)? I don't know I would love to hear someone's opinion on this. It would be greatly appreciated.
His point was that the losing side doesn't have a correct move. Sometimes they will have less losing moves but they're still losing. But yes, you should have calculated what happens if black doesn't take the knight before you decided on your move. How can you know that recapturing the knight is the best defense by black if you didn't calculate other possibilities as well? That is the reason the solution veers off because you have to see that you win the rook in the end here or else your advantage disappears. If black recaptures the knight on f8 you're being rewarded even though you didn't see the entire solution. 1... f5 isn't even the best line for black according to the engine but it's the line you need to see to know that you have a significant advantage. If the rook was on a safe square instead this line would be blacks best line.

You're right. If Black wanted the best chance of recovering they would take the other Knight. The puzzle just adds a bit more complexity by playing an inferior move. Your opponent doesn't always have to play the best move but so long as you have an answer to all possible responses then you win the exchange.

In puzzles, there's a very big difference between your "best move", which you must find to correctly solve, and the opponent's "best move", which actually doesn't exist! Why?
When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others. An engine may rate an opponent's move A as better than the puzzle move B, but you'll notice that the evaluation for A is that it's also completely losing (higher than 3 for your side).
In a practical game, the engine move A might offer more chances to avoid defeat (that's what the engine is suggesting), but that's irrelevant in a puzzle. A puzzle assumes perfect or best play on your side, which means defence A has zero chance of saving the game, just like B. Hence the puzzle move B cannot be viewed as a mistake, and there's no "best move" for the opponent. The puzzle picks B as the defence because it's more instructive than A, or because it forces you to find a unique winning reply. If the puzzle had used the engine move A, you could have multiple winning replies, which would result in a faulty puzzle.
I'm sorry, this doesn't really make much sense to me. Since you said
"When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others."
then would that mean, puzzles are not meant to be able to be solved on the first try? If you are really supposed to explore incorrect lines then wouldn't that defeat the purpose of learning from puzzles? I came in assuming I would be trying to make the best move against an opponent who would make the best move they could as well. If the puzzle doesn't give the best play possible as the opponent, why should I or anyone use it as a learning tool? Just to study something that is incorrect? Is the reason it made the bad move because it wants the player to find the mistake? If so wouldn't it be more productive to just show the best move by the opponent? When you play a practical game, do you play the best move you can or do you make mistakes to help your opponent learn that you make mistakes (just an example, not meant towards anyone in particular)? I don't know I would love to hear someone's opinion on this. It would be greatly appreciated.
His point was that the losing side doesn't have a correct move. Sometimes they will have less losing moves but they're still losing. But yes, you should have calculated what happens if black doesn't take the knight before you decided on your move. How can you know that recapturing the knight is the best defense by black if you didn't calculate other possibilities as well? That is the reason the solution veers off because you have to see that you win the rook in the end here or else your advantage disappears. If black recaptures the knight on f8 you're being rewarded even though you didn't see the entire solution. 1... f5 isn't even the best line for black according to the engine but it's the line you need to see to know that you have a significant advantage. If the rook was on a safe square instead this line would be blacks best line.
But how can it be said that "the losing side doesn't have a correct move"? Can a move that brings you closer to winning or to a draw when you are in a losing position not be considered the best move?

This daily puzzle did not make sense, because of the hanging pawn on e4 which the puzzle considered the correct move. My first time trying to play the puzzle I made the assumption, that king takes or bishop takes the knight on f8... so why is hanging the pawn better in this case?
I certainly don't understand what you are trying to say here. Does anyone else understand?

I'm sorry, this doesn't really make much sense to me. Since you said
"When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others."
then would that mean, puzzles are not meant to be able to be solved on the first try? If you are really supposed to explore incorrect lines then wouldn't that defeat the purpose of learning from puzzles?
I'm just giving you the theory behind puzzles to explain why the opponent only SEEMS to make "mistakes" in them. @PopcornSC is right – once you have played the correct move for your side, there are no "correct" moves for the opponent.
In practice, you need to find the correct move for your side that would win against ANY move by the opponent, whether it's the engine move A or the puzzle move B. If you had calculated that right, then it doesn't matter if the puzzle played A or B, because you'd be ready with the right responses against both moves. Yes, chess is hard like that!

In puzzles, there's a very big difference between your "best move", which you must find to correctly solve, and the opponent's "best move", which actually doesn't exist! Why?
When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others. An engine may rate an opponent's move A as better than the puzzle move B, but you'll notice that the evaluation for A is that it's also completely losing (higher than 3 for your side).
In a practical game, the engine move A might offer more chances to avoid defeat (that's what the engine is suggesting), but that's irrelevant in a puzzle. A puzzle assumes perfect or best play on your side, which means defence A has zero chance of saving the game, just like B. Hence the puzzle move B cannot be viewed as a mistake, and there's no "best move" for the opponent. The puzzle picks B as the defence because it's more instructive than A, or because it forces you to find a unique winning reply. If the puzzle had used the engine move A, you could have multiple winning replies, which would result in a faulty puzzle.
I'm sorry, this doesn't really make much sense to me. Since you said
"When you play the correct move for your side in a puzzle, that wins against ALL moves by the opponent. That means the opponent only has losing moves to choose from, and none of them are actually "better" than others."
then would that mean, puzzles are not meant to be able to be solved on the first try? If you are really supposed to explore incorrect lines then wouldn't that defeat the purpose of learning from puzzles? I came in assuming I would be trying to make the best move against an opponent who would make the best move they could as well. If the puzzle doesn't give the best play possible as the opponent, why should I or anyone use it as a learning tool? Just to study something that is incorrect? Is the reason it made the bad move because it wants the player to find the mistake? If so wouldn't it be more productive to just show the best move by the opponent? When you play a practical game, do you play the best move you can or do you make mistakes to help your opponent learn that you make mistakes (just an example, not meant towards anyone in particular)? I don't know I would love to hear someone's opinion on this. It would be greatly appreciated.
His point was that the losing side doesn't have a correct move. Sometimes they will have less losing moves but they're still losing. But yes, you should have calculated what happens if black doesn't take the knight before you decided on your move. How can you know that recapturing the knight is the best defense by black if you didn't calculate other possibilities as well? That is the reason the solution veers off because you have to see that you win the rook in the end here or else your advantage disappears. If black recaptures the knight on f8 you're being rewarded even though you didn't see the entire solution. 1... f5 isn't even the best line for black according to the engine but it's the line you need to see to know that you have a significant advantage. If the rook was on a safe square instead this line would be blacks best line.
But how can it be said that "the losing side doesn't have a correct move"? Can a move that brings you closer to winning or to a draw when you are in a losing position not be considered the best move?
All of the puzzles here end in a decisive advantage. The main point I'm making is that 1.Nh7 only gives you that decisive advantage if you see this line and recognize that the rook is hanging. As I stated before, 1... f5 isn't the "correct" move in the first place so if you want to have beef with the problem you should start there anyways.

"All of the puzzles here end in a decisive advantage. The main point I'm making is that 1.Nh7 only gives you that decisive advantage if you see this line and recognize that the rook is hanging. As I stated before, 1... f5 isn't the "correct" move in the first place so if you want to have beef with the problem you should start there anyways."
Yes, but then the question still stands. Why would the puzzle be showing incorrect moves in the first place? The later move was just the one I caught at first glance. Just because there was an incorrect move earlier in the sequence doesn't change the question at hand.

"All of the puzzles here end in a decisive advantage. The main point I'm making is that 1.Nh7 only gives you that decisive advantage if you see this line and recognize that the rook is hanging. As I stated before, 1... f5 isn't the "correct" move in the first place so if you want to have beef with the problem you should start there anyways."
Yes, but then the question still stands. Why would the puzzle be showing incorrect moves in the first place? The later move was just the one I caught at first glance. Just because there was an incorrect move earlier in the sequence doesn't change the question at hand.
The point of the puzzle is that you have to win no matter what the opponent responds later. Did you consider the possibility of fxe4? If yes, then all is fine, you have already calculated the answer. If no, then why not? You have to be ready for all possible moves the opponent can make.

"All of the puzzles here end in a decisive advantage. The main point I'm making is that 1.Nh7 only gives you that decisive advantage if you see this line and recognize that the rook is hanging. As I stated before, 1... f5 isn't the "correct" move in the first place so if you want to have beef with the problem you should start there anyways."
Yes, but then the question still stands. Why would the puzzle be showing incorrect moves in the first place? The later move was just the one I caught at first glance. Just because there was an incorrect move earlier in the sequence doesn't change the question at hand.
Because the line only works if you see that you're winning in the "incorrect" line. If black just takes back on f8 you'd never see the theme of the undefended rook. You'd go on thinking you fully solved the puzzle when you actually didn't.
What I don’t understand is why the solution isn’t just Kf6+, black takes with either the bishop or pawn to eliminate the check or the king moves h8, the white mates with Qh7#. Am I missing something obvious?
What I don’t understand is why the solution isn’t just Kf6+, black takes with either the bishop or pawn to eliminate the check or the king moves h8, the white mates with Qh7#. Am I missing something obvious?
For the record, I didn't get the puzzle at all, but I think I can answer your question. I assume you meant Nf6+ btw. This won't work because black's knight can take which defends against mate.

New to chess, and have been doing puzzles everyday. I don't really get the point? Yes I see the need for them, but are you supposed to look for the best moves? Are you supposed to just assume a specific line of play via theory? What is the goal? This daily puzzle did not make sense, because of the hanging pawn on e4 which the puzzle considered the correct move. My first time trying to play the puzzle I made the assumption, that king takes or bishop takes the knight on f8... so why is hanging the pawn better in this case? I also ran the position through chess.com's analysis board, and it came to the same conclusion I did. Is there something I'm on seeing?
Black to move, pawn takes knight on e4, then white queen takes the hanging pawn in response freeing up the queen and putting the rook on a8 in danger. Why is king or bishop takes knight f8 not played here?
To answer your last question first: King or bishop can take knight on f8, but that still results in White gaining materiel (+2 via the exchange) without giving Black a chance to equalize. After capturing on f8, White can move the e4 knight out of danger (for example, to c3 or d2), leaving Black down the exchange.
As to your earlier questions, tactics puzzles are about finding the best move in a given position. Usually (but not always) they are designed so that you are guaranteed to achieve mate or a significant materiel advantage, no matter how the opponent responds.
"Significant materiel advantage" usually means "at least an exchange", but there are exceptions. Also, the puzzles don't always continue all the way to the final capture, since there may come a time when you have multiple equally-good options to secure an advantage. When that happens, they end the puzzle rather than allow multiple solutions.

Yes, but then the question still stands. Why would the puzzle be showing incorrect moves in the first place? The later move was just the one I caught at first glance. Just because there was an incorrect move earlier in the sequence doesn't change the question at hand.
The question has already been answered though. The task is to play the best move for (in this case) white. Yes Black could have taken back on f8 but the position would still be ‘solved’ - white would have won material. Taking on e4 for black actually makes more sense as a defence from a puzzle point of view as it gives white the chance to go wrong with Nxe6. It’s not necessary that black makes the number 1 computer move, often there’s an inferior but trickier defence that should be considered.

White is a complete rook up so what's the use of the puzzle? I would just play Ng6 with a completely won game for white. What's the point? I don't care if there's a quicker way to win since Ng6 is winning. I don't think we can learn from puzzles, like we can learn from Daily chess if done properly, or by analysing our rapid games.
Didn't even notice Nxe6 at first glance. In a blitz game I'd play what I first saw was winning without looking for another move. In a slow classical game my opponent should have resigned by now and anything sensible wins for white. What use are puzzles, if they do not reflect game conditions?
You are talking about the position from the first post? It is set up incorrectly, actually it is black to move in that position. The OP's question was why black plays fxe4 instead of Bxf8.
New to chess, and have been doing puzzles everyday. I don't really get the point? Yes I see the need for them, but are you supposed to look for the best moves? Are you supposed to just assume a specific line of play via theory? What is the goal? This daily puzzle did not make sense, because of the hanging pawn on e4 which the puzzle considered the correct move. My first time trying to play the puzzle I made the assumption, that king takes or bishop takes the knight on f8... so why is hanging the pawn better in this case? I also ran the position through chess.com's analysis board, and it came to the same conclusion I did. Is there something I'm on seeing?
Black to move, pawn takes knight on e4, then white queen takes the hanging pawn in response freeing up the queen and putting the rook on a8 in danger. Why is king or bishop takes knight f8 not played here?