Another mobility puzzle

This is the "opposite" of A mobility puzzle - I held it back because it could have interfered with Schwalbe 2021 construction challenge.

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).

Remarks:

• 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.

5

``````[FEN "4rq2/rk2Nn2/6NQ/6nR/8/B7/b5K1/1Bb3R1 w KQkq - 0 1"]
``````

6

``````[FEN "8/3KNN2/6n1/6n1/RBb3k1/1r1B4/q1Rb4/1Q1r4 w KQkq - 0 1"]
``````

7

``````[FEN "1r5q/7r/1k3N1b/5N1B/b1n5/B1n3K1/R7/Q5R1 w KQkq - 0 1"]
``````

Sorry Hauke, but I can't get a headache this time. I've already commissioned n=1 n=2, an n=3 over on the Puzzling Stack Exchange. However, here the positions from the accepted answers for reference. The way I see it, I technically am the one the who got the gears rolling that produced them. Thus, I "found" them in my archives. Either way, I've beat you to the punch once more, you old bird (Hauke=Hawk, giggles)!

`````` [Title "n=1 by  @loopy walt, 32 piecews"]
[FEN "1nrqkbnr/3bp3/1pp3pp/p3Pp2/2Pp3P/PP3PP1/3PB3/RNBQKRN1 w Qk - 0 1"]
``````

``````[Title "n=2 by  @loopy walt, 32 pieces"]
[FEN "rq1nnk1r/1pp2pp1/7b/p2pp2p/1PP2PP1/1BB5/PN1PP1NP/1R1QKb1R w K - 0 1"]
``````

`````` [Title "n=3 by  @loopy walt, 30 pieces"]
[FEN "1qr1k2B/pb1p1p1p/1RP3P1/2p1p1n1/1N1P1P2/1p3pr1/P1P1P1BP/b2K1RQ1 w - - 0 1"]
``````

For n=4, you actually can pile up 16 promotable pawns on both sides in a legal position. See the proof here: What is the maximum number of passed pawns in a position?

`````` [Title "n=4 by  me, 24 pieces"]
[FEN "8/PPPPPPPP/BRK4n/8/8/N4krb/pppppppp/8 w - - 0 1"]
``````

Regarding n=8, knight and bishops can't be on the last rank. This leaves the wide ranged, and therefore difficult to contains, queens and rooks to be fresh meat for the pawns. It also restricts the number of useable pawns. So far, the knights have also proven too tricky for me.

`````` [Title "n=8 by  me, 10 pieces"]
[FEN "r7/1P6/r2k4/1b6/6B1/4K2R/6p1/7R w - - 0 1"]
``````

An improvement for n=4, building on Rewan Demontay's answer:

`````` [Title "n=4, 26 pieces"]
[FEN "2Rnr3/PP1PP1PP/RKQ3rk/7B/5n2/6Bb/pp1pp1p1/3NN2q w - - 0 1"]
``````

The trick is to replace some pawns by knights or rooks to keep the position legal with less necessary captures.

This version has more pieces and less pawns, so I think building a proof game will be easier. However adding a wPb7 would probably make it illegal :

`````` [Title "n=4, 26 pieces"]
[FEN "2Rnr3/P2PP1PP/RKQ3rk/8/1N3n1b/6Bb/pp1pp1p1/3BN2q w - - 0 1"]
``````
• You can probably test it with Jacobi: wismuth.com/jacobi/latest/solver.html - I'm not yet convinced about legality, as most pawns must promote to become victims, which needs new captures. Jun 8, 2021 at 20:16