I had an idea for a type of stalemate: One side is stalemated with 8 pins in a legal position with as many pieces as possible, with all of the stalemating side’s units being “necessary.”
”Necessary” is defined as either pinning a piece, blocking a piece, or covering a king’s flight square. The king must be involved of course.
As a bonus optional question, what is the maximum number of units in a legal position with 4 “natural” pins, which means no promoted pinning pieces?
[FEN "b3r3/1R1p1p1q/1p1P1P1b/1Pk1BR2/1rN1K1Nq/p7/P1P1B1P1/1qn1q1nq w - - 0 1"]
Here is an (ugly) suggestion wth 29 units.
White is stalemated with 8 pins; removing any Black unit would break the stalemate. Black has promoted all of their four missing pawns (all into queens).
The three captures (white queen and 2 white pawns) ensure that this position is reachable. The three missing White pieces ensure that the position is legal and that Black could have promoted all four of their missing pawns.
The Black g-pawn captured the White h-pawn to produce doubled pawns that both promoted. The Black c-pawn captured the White queen to bypass the White pawn structure and promote. Finally, the White e-pawn was captured and then the Black e-pawn marched down and promoted.
This is optimal because 4 pawns must be promoted to create 8 pins, with requires a minimum of 3 units, thus leaving 29 left for the final position.
Bonus question : 4 pins stalemate without promoted piece.
Here is a maximal solution with all 32 units :
White's Rc2, Bd5, Nf2 and Ng6 are pinned. The king has no move and all the other white pieces are stuck.
Removing any black piece, or moving the black King away, would break the stalemate.
Removing the bPg4 would be particularly unfortunate because White, in spite of having almost all their army stuck, would deliver mate with their only legal move, g3-g4#.
Regarding the bonus question, for 4 “natural” pins, here is a position with 30 units.
[Title "me, chess.stackexchange.com 8/9/2019"]
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