Thus: Put a maximal number m of pieces on the board such that the position is legal, no obviously promoted pieces occur (I will be gracious with retro-forced promotions) and each piece has exactly n legal moves (obviously 0<=n<=8).
- n=0: solution is well known, m=30 is possible.
- n=1,2: My m=32 can't be beat, try to find it.
- n=3: This is the "king discipline". Even Rewan will get a headache! (And since I can only accept one answer, I'll prefer to accept that record.)
- n=4: Note you can't just pile up 16 promotable pawns on both-)sides as the position would be illegal!
- n=5,6,7: My m=16 can't be beat (pawns can't occur), try to find it.
- n=8: This is the other "king discipline". Promotable pawns with a capture object can occur!
- n>8 is impossible - the kings.
Thus I suggest that you try m=3,4,8 first (I will not reveal my m there).
EDIT: Here is an update.
- 0 moves: maximum, see here Stalemate situation with all pieces on the board
- 1,2 moves: maximum, see Rewan's link
- 3 moves: see Rewan's link
- 4 moves: see this thread
- 8 moves: see here https://puzzling.stackexchange.com/questions/110360/chess-construction-challenge-d-8-moves
- 5,6,7 moves: maximum, I give them now (please check, as a mathematician I can't count :-)
[FEN "4rq2/rk2Nn2/6NQ/6nR/8/B7/b5K1/1Bb3R1 w KQkq - 0 1"]
[FEN "8/3KNN2/6n1/6n1/RBb3k1/1r1B4/q1Rb4/1Q1r4 w KQkq - 0 1"]
[FEN "1r5q/7r/1k3N1b/5N1B/b1n5/B1n3K1/R7/Q5R1 w KQkq - 0 1"]