Continuing my previous puzzle.
If both players cooperate, what is the quickest stalemate in chess that includes a Queens exchange, in a legal game?
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Black can stalemate in move
11 (22 plies):
White's job here is moving their pieces such that black can capture almost every move while bringing their pieces into the final position.
1.h3 d5 2.c4 Bxh3 3.g4 Bxg4 4.Rxh7 Bxe2 5.Nf3 Bxf3 6.Rh1 dxc4 7.Qb3 Qxd2+ 8.Nxd2 cxb3 9.axb3 e6 10.Ra4 Rxh1 11.Rb4 Bxb4 *
13 moves (25 ply):
1. e3 a5 2. Qh5 Ra6 3. Qxa5 h5 4. h4 Rah6 5. Rh3 f6 6. Qxc7 Kf7 7. Rf3 d5 8. Qxb8 Bf5 9. Qxb7 Bh7 10. Qxd5+ Qxd5 11. Rf5 Qc4 12. Bxc4+ Kg6 13. Be6 1/2-1/2
I started off with the well-known stalemate (without queen exchange) in 10 moves, composed by Sam Loyd (see this chess.com post). In that final stalemate position, the role of the queen is three-fold: block the f5 and f7 squares, prevent the e7 pawn from moving, and pin the f6 pawn.
The black queen obstructing the rooks needs replacement, with the easiest being black's white bishop. All three of the white queen's roles need to be satisfied unless the final position is substantially altered, and I trust Loyd that this final black piece configuration is roughly optimal.
I replaced the white queen with a white bishop and rook, which can be moved into place in 5 moves. Conveniently, the rook is given space to move by the initial h-pawn push by white. Like the rook, the white knight can get to that square in 3 moves, providing an alternative solution, but the rook is more fun. :P One of the bishop moves is used to recapture the black queen.
It would be uneconomical to promote a pawn, since that takes 5 moves + at least one to get into position, while the bishop and rook combination I use takes 5 moves. I imagine there is a clever solution in 12 moves, but 11 moves would definitely surprise me.