Hint: key move is Ng6!

I'm a nit: there are "two solutions" so technically this isn't really a
"proper study" ( Nb6 and Nc5 are equivalent ).

Still, it has aesthetic appeal having nice themes: stalemate, underpomtions
and of course, mate!

Pretty sweet. Took me forever to figure out why it wasn't really straightforward.

White obviously marches the h-pawn to promotion. Black's king is too far away to stop it, and he can't promote his own a-pawn, because black can just block the a- or b-file with his own king. What Black tries to do is stalemate his king on a4 with Kc6, Kb5, Ka4, a5 and b5. It takes White 5 moves to promote, so he has one move to avoid stalemate, which he does by promoting his pawn to a knight and sacrificing with 6. Ng6. Black is forced to capture, then white marches his pawn and promotes on the 8th move. White can't actually break through the pawns around Black's king with a queen, so he underpromotes again, and mates with Nd7, Nb6#.

At first I was like, "wtf, just queen the pawn". My next thought was that Black was going to march down the board and do a king and pawn mate, but he doesn't have enough time. Then I noticed how few pawn moves he has, so I started to look for a stalemate. I spent about 10 seconds trying to end up with a position where Black would have no moves right before White queened, but I couldn't find one. The necessary leap is to realize that Black can actually put himself in the stalemate position right after White queens, and White can't do anything about it. So the stalemate line goes 1. h4 b5 2. h5 Kb6 3. h6 Ka5 4. h7 Ka4 5. h8=Q?? a5 6. Any Stalemate. But White can force Black to regain some pawn moves: 5. h8=N! a5 6. Ng6! fxg6 7. f7 g5 and here I spent a good twenty seconds stupidly promoting to a queen and trying to get the queen to a1 in time, but then I realized that the Black king is stalemated, and the move should have been dead obvious...8. f8=N! g4 9. Nd7 g3 10. Nb6 mate. Nice double under-promotion. Really, the only reason that this would be hard for someone is if they failed to come up with a good defense from Black. If you don't hit on the idea of the Black king burying himself you'll just be incredulous because it looks like White can just queen.