# Maximum number of legal moves after n moves in a chess game [closed]

Given the chess start position :

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!

• Just to clarify: can the first $n$ moves of the solution for $n+1$ be different from the moves of the solution for $n$? Apr 4, 2020 at 12:07
• 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). Apr 4, 2020 at 12:51
• "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.
– Deusovi
Apr 4, 2020 at 14:59
• 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. Apr 4, 2020 at 15:10
• 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.
– Deusovi
Apr 4, 2020 at 16:35

The bold entries correspond to the positions where White is to move, so those are your answers for $$n = 0$$ to $$4$$.