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I know popular openings like the Queen's Gambit can be accepted with an initiative for white. Is there any gambit where accepting it would lead to a forced mate for the player that accepted it?

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    This does not exactly answer the question, but there are some gambits which you don't have to decline (in the sense of leading to forced mate), but accepting free material leads to a worse position. An example is the Jaenisch Gambit of the Ruy Lopez 3...f5, where 4.exf5 is considered to be a bad move. – YiFan Dec 3 '20 at 7:54
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    If there is such a gambit, then it is arguably incorrectly named, as the word implies choice by the original definition. dictionary.cambridge.org/dictionary/english/gambit – Stacker Lee Dec 3 '20 at 16:45
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While it's not generally given as part of the explicit definition, the term "gambit" usually refers to an offer of material that can be accepted. If it can't be accepted, then it's a trap.

You can look through chess traps and see that many of them involve apparently giving up material.

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    Are there any lines which were originally thought of as gambits, but which have evolved into traps because forced winning lines have been discovered for the player whose piece was taken? – supercat Dec 3 '20 at 17:50
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    @supercat, this sounds like a good question for a separate topic. – Akavall Dec 3 '20 at 20:17
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    @Akavall: I think it's probably what the OP intended to ask. If one defines a "gambit" as an offer of material that can be accepted without major disadvantage, then the answer to the question as asked would be "no", by definition. The only way the question would be interesting would be if "gambit" was intended to have a broader meaning, and I think extending it to include historical gambits would be the most logical one. – supercat Dec 3 '20 at 21:26
  • @supercat, That very well might be true, but I still think the best way to proceed is to ask this (clarified questions) a separate question. – Akavall Dec 3 '20 at 21:30
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Yes. In this line in the Dutch Defence black cannot take the offered bishop

[fen ""]

1. d4 f5 2. Bg5 h6 3. Bh4 g5 4. e3 gxh4?? 5. Qh5#
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    Does that count as a gambit though? – David Dec 3 '20 at 9:25
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    @David Well what do you think counts as a gambit? To make sense of the question, we have to include in the definition of a gambit things which lead to a forced mate when accepted. – James Martin Dec 3 '20 at 14:31
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    @JamesMartin If it's the case that you have to define "gambit" to mean things that aren't really gambits to answer the question with a "yes", maybe the answer isn't "yes". – Eliza Wilson Dec 3 '20 at 16:45
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    One could argue that losing a bishop in one ply and winning in the next ply qualifies as short term and long term gains. I would consider a gambit one where the long term gains are more informally defined like "gets a kingside attack". If this answer was a correct answer, then any move which threatens mate in 1 with a piece at risk would qualify as a gambit. – Cort Ammon Dec 3 '20 at 23:44
  • @CortAmmon Great point. Indeed, you wouldn't even need to threaten mate in 1. After 1.e4 d5 2.exd5 Qxd5 3.Nc3, White can choose between three different unacceptable gambits! – David Dec 4 '20 at 7:18
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I would add that there are situations where one side "offers" the material that cannot be taken, but it is not really intended as a trap, it happens in the course of a logical development of the opening. The most "mainstream" example is perhaps the Berlin defense:

[fen ""]

1.e4 e5 2.Nf3 Nc6 3. Bb5 Nf6 4.O-O Ne4 5.d4 Nd6 (5... exd4 6. Re1 d5 Nxd4) 6.Bxc6 dxc6 7. dxe5 Nf5 8. Qxd8 Kxd8 

White's 4. O-O and 5.d4 seems to offer black a pawn, however if black accepts, the pin is a problem. White threatens both f3. and Nc6. It is not quite a forced mate but Black will have to give up material without complete compensation. "Everyone knows" that it's better not to take the pawn.

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