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The question concerns chess. I call a move forced if, in a given position, is the unique move consistent with the rules of the game. I wonder what is the largest integer $n$ such that there exists a legal position in which:

  1. both black and white have forced moves for $n$ consecutive times;

  2. the position is never repeated (with the same color having to move).

Notice that without condition 2., the answer is $\infty$.

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    $\begingroup$ Are you restricting attention to the usual 8x8 board, or do you permit arbitrarily large (perhaps even infinite) boards? $\endgroup$ – Wojowu Apr 19 at 21:29
  • $\begingroup$ I was thinking to an 8x8 board. $\endgroup$ – A.DellaCorte Apr 19 at 21:36
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    $\begingroup$ I think another problem is whether position may be achieved in a game (similar questions might be asked about lot of chess stackexchange positions). $\endgroup$ – Igor Sikora Apr 20 at 13:43
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    $\begingroup$ @IgorSikora In the chess problem world, it is a standard convention that positions must be reachable in an actual game (i.e., "legal"), unless there is an explicit warning that the position is, or might be, illegal. Amateurs might not be aware of this convention, but anything published in a chess problem journal is expected to follow this convention. If it doesn't, that's akin to publishing a theorem in a professional mathematics journal with a fatally incorrect proof. It can happen, but it's not supposed to. $\endgroup$ – Timothy Chow Apr 20 at 21:13
  • $\begingroup$ @TimothyChow Good to know, I wasn't aware of this convention. However, a lot of solutions in chess.SE miss this condition then, or at least miss a proof that the condition is satisfied. For example in the link that you are giving a lot of solutions cannot be achieved in a real game. $\endgroup$ – Igor Sikora Apr 21 at 16:06
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This question has been answered at chess.stackexchange.com. It seems that if you allow promoted pieces, the current record is $n=9$.

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    $\begingroup$ Wow, I didn't even know that there is a chess stack exchange website... (and I am a chess player!). I accepted the answer, although this just shows the current empirical record, not a theoretical bound. I guess the latter is very difficult to achieve... $\endgroup$ – A.DellaCorte Apr 20 at 10:17
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    $\begingroup$ @HymnsForDisco But there the moves are not quite forced, e.g. black has many options for the first move (albeit all strategically equivalent). $\endgroup$ – Will Sawin Apr 20 at 13:40
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    $\begingroup$ @HymnsForDisco What Will Sawin said. If you scroll down far enough in the list of answers to the question I posted a link to, you'll see that Rewan Demontay posted his position there, too. $\endgroup$ – Timothy Chow Apr 20 at 13:47
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    $\begingroup$ @LSpice The $n=9$ example was posted by Hauke Reddmann (but later edited by Rewan Demontay; perhaps that has caused confusion). Rewan Demontay's answer shows several positions; the one I was referencing was the 4th diagram (attributed to Alexey Khanyan), which is the same as the 1st position in his answer to Hymns for Disco's question. $\endgroup$ – Timothy Chow Apr 20 at 21:03
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    $\begingroup$ @LSpice The position by Alexey Khanyan has $n=11$ but it does not satisfy the condition that A. DellaCorte wanted. $\endgroup$ – Timothy Chow Apr 20 at 21:07

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