The objective of both the puzzles is mate in two. In the first one Kd6 is the best move to a mate in two. If Qe5 then black blocks it with c5. The theme is cross-checks. That is white blocks a check while giving a check at the same time. It's the case in the second one too. I guess now its easy.

Anyone got the second one? Come on Turm have a try.

I studied the engine's moves and here is what I think it does.
The obvious problem is that White has two rooks pointed at a3, which means the Black King can't take it, and if KxR after c7-c5 White hasn't time to move both rooks and it's stalemate.
So White needs to stop the checks, to move a rook.
1.Bf5 Qh7+ 2.Bxh7 Rxh7+
Now the King heads to the 2nd rank. 3.Kg6 Rh6+ 4.Kf5 Rf6+ 5.Ke4 Re6+ 6.Kd3 Rd6+ 7.Ke2 Rd2+
Notice that after Re6+ White can block the check with the rooks.
8.Ke3 Re2+ 9.Kf3 Rf2+ 10.Ke4 Rf4+ Black can't check to Re2 cause White can block.
11.Kd5 Rd4+ Not Rf5+ cause then gxf5 and the g5 pawn can move. That's why White doesn't take g5 with the king initially.
12.Kc5 Black can't check with Rd5+ cause 13.cxd5 c6 Rc4#.
So Black must either make a waiting move with the rook or move c6. But now White can move one of the rooks away from the 2nd rank. The rest is easy I think. Let me know if there are other better moves for Black.

11. ... Rd4+(?) is pretty weak. Black plays 11. ... c6+. How do you continue?

Sutirtha11, my solution to your 2nd problem (post n0.13) is 1. Rb1! (Threat 2. Bg4/h5#) If 1...exd2+ 2. Nd3. Or 1...e2+ 2. Bb3#. Finally 1...exf2+ 2. Bf3#. 

The problem features crosscheck and self-block. Black pawn made a self-block to his own King. Squares d2, e2 and f2 is blockaded which White exploited brilliantly!

Amazing Puzzle!

But what if black simply checks on d6?

1. Rb1(!?) Qd6+ and I don't see any mate.

Asorski is correct.

Because of what? There are 3 ways to mate in 3 moves.

1. Rb1 is by far the most beautiful, but the crude Rxe3 and Bxg3 will also work.

Did you miss a piece? Maybe a black pawn on d6?!? Or maybe better put the rook to f4, not f5?

Turm, I did not miss a piece but I think Black won't do anything foolish. Both these puzzles came out in a magazine and this was the answer provided and the main theme like I and Asorski said is cross checks and self blocks. 

Regarding the second puzzle by Turm Breuburg... It is indeed a very tough puzzle, and I could at max force a draw against Shredder (Depth 16). Thank you so much for posting the puzzle...

Now I am very anxious to know the solution that would lead to the score of 1-0. Please do let us know :)

Then it was printed wrong. This is clearly meant as a mate in 2 problem, but like this it is a very bad mate in 3.
The first one is certainly a correct and quite nice #2 (yet not too difficult).
But the second one will like this not even count as a real problem as it has multiple solutions and the main line is rather poor.

I might guess the rook has to be on f4 instead of f5. Then it is would be a correct problem (but very easy, as Rb1 is a rather obvious key)
This also allows the defence Qg6 which is countered with Ng4#, blocking the g-file against Qg3. This might make it a real chess problem and I guess this was how it was meant to be.

Like this:

The second puzzle from me is way more easy than the first one and any good engine can find the correct solution.

yours are excellent puzzles turm and I think rook was on f4 I don't have the magazine now