Remarkable King Pursuit

While exploring with a student some variations in a game with a dynamic imbalance yesterday, we came upon this position with Black to move.

At first it was surprising that the engine found the position equal. Then, we saw 34...Qa1+ 35.Kg2 f3+ 36.exf3 and Black forces stalemate with 36...Qg1+ (or h1).

Two lines needed more examination:

a) 35.Kh2 fxg3+ 36.Kxg3

Black to move

36...Qg1+ 37.Kf4 Qxf2+ 38.Ke5 and Black's queen should be able to continue checks until there is a repetition or the queen is captured by the king without blocking the bishop's control of h7 nor the rook's control of the g-file.

b) 36.Kxf3 Qh1+ 37.Kf4

Black to move

Here, Stockfish 16 favors 37...Qc1+, but I like the simple aesthetics of 37...Qf3+, which is easier to calculate. Two moves result in stalemate--capturing the queen with pawn or king. 38.Ke5 drops the rook and shifts the advantage to Black. That leaves 38.Kg5 and Black has a mate in two: stalemate, that is.