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Given the chess start position :

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What is the maximum number of legal moves white can play after $n$ full moves?
We will assume black is helping white in this task.

I want the best answer you could find with $n$ varying from 2 to 20 (too high and it becomes too long to solve, too low and it will be trivial). Since I do not know the answer for high values of $n$, your goal is to produce the 'most perfect' answer. The 'score' of an answer will be calculated by adding the number of moves for each value of $n$.

I know the answers for n=0 to 4. Computing the answer for low values of n is trivial, but becomes almost impossible for high values.

For clarification, here are the answers for $n=0$ and $n=1$:

$n=0$:

20 moves (16 pawn moves and 4 knight moves)

$n=1$:

31 moves after 1.e4 d5 or 1..f5 (16 pawn, 5 knights, 5 bishops, 4 queen, 1 king)

Remember: you do not have to find the perfect solution. The accepted answer will be the overall best if no improvements are found for a long period of time. Computers are allowed. Good luck!

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  • $\begingroup$ Just to clarify: can the first $n$ moves of the solution for $n+1$ be different from the moves of the solution for $n$? $\endgroup$
    – melfnt
    Apr 4, 2020 at 12:07
  • $\begingroup$ Yes. The intended answer is 19 variations of length 2-20 full move. The 19 variations are independent from each other. Otherwise it would be too easy to compute ( you are still encouraged to compute it anyways, as it can be used to determine the lower bound for each value of n). $\endgroup$ Apr 4, 2020 at 12:51
  • $\begingroup$ "the answer with the highest overall number of legal moves will be marked as accepted after ~1 week" -- This does not appear to be a puzzle then, but a game that people compete in. This question seems to me to be off-topic. $\endgroup$
    – Deusovi
    Apr 4, 2020 at 14:59
  • $\begingroup$ Because it is impossible to compute and verify what is the perfect answer, I had to set an arbitrary timer. It can be removed if it bothers you. What I meant is, the 'most perfect' answer will be chosen after 1 week. I'll remove the 1 week delay, but I think optimization questions with an unknown perfect answer are within the rules. Correct me if I'm wrong. $\endgroup$ Apr 4, 2020 at 15:10
  • $\begingroup$ This is near exactly the type of question that this meta post was meant to rule out. In my eyes, it's not a puzzle, because it isn't designed to have a solution -- it's a question you're interested in, but there's no point at which you can say it's definitively solved. $\endgroup$
    – Deusovi
    Apr 4, 2020 at 16:35

1 Answer 1

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I found

OEIS sequence A278830 'Maximal number of possible moves at the n-th ply of a chess game'
whose entries are: 20, 20, 31, 32, 46, 48, 52, 55, 61, 63
The bold entries correspond to the positions where White is to move, so those are your answers for $n = 0$ to $4$.

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    $\begingroup$ Yes, these were my answers for 0-4. After 5 it becomes really long to compute. This is where the real challenge starts. $\endgroup$ Apr 4, 2020 at 11:18

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