REALLY hard puzzle?

So to say, after 4 moves from white.

I don't know exactly which moves you're looking for, but it looks like Black is in a lost position already. (I'm assuming you don't mean checkmate in 4?)

Starting with Bxd7, and if Kd8, Nxb7. Soon the Nd5 and Ba5 will fall too if the black queen doesn't give herself up (which she should have done after Bxd7), and the white queen can easily get into the action with Qf5 or something.

Here is the solution: 

I don't see why the "solution" is better than, say:

when White has a huge material advantage and a raging attack.

I like caveatcanis's version in post #6

Why on earth would black freely take with his king?

Aha. Well, I only post the puzzle. The solution is the situation in where black resigned. If somebody could explain why he did I would be happy, it makes no sense to me!

Ummm, excuse me, but (unless I am drunk) white has several ways to win material, AND initially he is a whole rook (and a pawn, which hardly should make a difference) up for nothing.

What sort of puzzle is this?

black is completely toast regardless of the variation.