[COMMENT DELETED]
Mate in 3 puzzles, for advanced players

The first one is cooked. Bc1 and Ba3 work just as well for mate in 3. Haven't solved the second one in more than 4 ways to tell yet ;)

The first one is cooked. Bc1 and Ba3 work just as well for mate in 3. Haven't solved the second one in more than 4 ways to tell yet ;)
What about 1. Bc1/Ba3 gxh6

The first one is cooked. Bc1 and Ba3 work just as well for mate in 3. Haven't solved the second one in more than 4 ways to tell yet ;)
What about 1. Bc1/Ba3 gxh6
2. Kf7 h5 3. Bb2#

The first one is cooked. Bc1 and Ba3 work just as well for mate in 3. Haven't solved the second one in more than 4 ways to tell yet ;)
What about 1. Bc1/Ba3 gxh6
2. Kf7 h5 3. Bb2#
Cool.

doodinthemood is right, I've made a mistake. In the first one the bishop would have been on the a1 square. Have someone solved the second one?

Correct, black's pawn can transform into Queen, Knight, Bishop or Rook as well. In all solutions, the first move is Kf3.

Ok,
I have a problem with the solution to the second problem.
After 1.Kf3 g1=Q, where does White play without putting his King into check?
He can't move to e3, f2, or g3. And if he moves to another square, Black just interposes with his Queen. White can certainly check with his Knight, but the best White can do is a draw.
Barefoot_Player

Ok,
I have a problem with the solution to the second problem.
After 1.Kf3 g1=Q, where does White play without putting his King into check?
He can't move to e3, f2, or g3. And if he moves to another square, Black just interposes with his Queen. White can certainly check with his Knight, but the best White can do is a draw.
Barefoot_Player
Well, after 1.Kf3 g1=Q white should move 2.Nf2 check, and the only defence is Qxf2... But white's king can capture it, so 3. Kxf2#
I think there is only one solution. 1. Ng5 2. Kg1 2. Ke1 h1=any 3. Nf3+ mate
Yes there is (post #8), but it is not yours. Black defends with 2. .. Kh1. The OP does not understand what a "solution" is and counts all 4 black promotion defenses as solutions.
Problemists call different defenses "variations" provided they require different unique white responses. Here is only the main line after 1. .. g1Q since white can mate in several ways after other promotion defenses. Another correct way to look at "variations" is as "partial solutions". All partial solutions together constitute the solution to a problem (or endgame study).
These two puzzles are very common in chess books(connected with endgames), so I think I should show it to you.
First:
Second (4 solutions):
Good luck!