Here is a chess puzzle I recently created with help from Stockfish (my calculation skills past five moves are inadequate ). There are probably better examples of this concept out there, but here's mine:

Black to add a unit and win as quickly as possible. Given that no side is in check, what is the unit, where does black add it, and what is the first move? Note that white moves first unless it can be proved this is not the case]

[Edit: To clarify, black to move and place a unit without check such that he wins with best play. Also, what is the first move of the solution?

Solution in white text:

Black adds a black bishop on e8 and the key is Bd7!

Black to add a unit and win as quickly as possible. What is the unit, where does black add it, and what is the first move?

Solution in white text:

You didn't copy the white text, unless you didn't want to

Nice! But did you figure out why white couldn't move first? I never specified who to move

Good retro-setup!

Btw, your instruction suggests there might be more than 1 workable black placement but the others mate slower. Probably clearer to state: "Place a black unit without check such that black provably wins. What's the 1st move of the solution?"

@Arisktotle Thanks! The question is, did I write that on purpose I will edit the sentence.

@texaspete Who said it was black to move?

Couldn't it have been white to moves? 

Like couldn't black's last move be a pawn move like g5 or am I missing something really simple here...

I missed that bit of the question!

So the first part is to place a black piece somewhere which means that white must have moved last i.e. rule out Ng7, Ng8 and dxe6 while also ruling out the new piece having made the last move...

Edit: I think I have it as I can’t see more than one identity/location for the piece without even thinking about the checkmate...

Texaspete answered that. You must place the black unit in such a way that black last moves are impossible or lead to an illegal position. For instance the last move g7-g6 makes the black pawn formation illegal.

Its still white to move with or without the bishop,I get the pawn can only come by capturing so the last move cannot be pushing,but the capturing is still available

What is wrong with this?this is a legal position

That's still a serious point. Formulating "retrograde questions" is an art in itself unless they are of a standard type. Suggesting there might be more winning placements actually diminishes the value of your construction. Amongst the legal placements in any chess position there will always be a winner in the "fastest mate" category but your real achievement here is that there is only one winning placement. And well hidden in the retro outpost!

Things would be different if the legal placement choices were integrated in the content scheme. For instance when you design 2 legal placements where black cleverly checkmates in 3 moves with white on move, while the retro-based solution checkmates in 2 because it is provable by retro-analysis that black is on move. That would perfectly justifty asking for a "fastest mate placement".

How did the black bishop get to e8?

Ah, it's chess960

Fair point @Arisktotle.  I did phrase my OP a bit incorrectly. It is not a checkmate in n moves, but rather black to win. Also, there is only one solution that wins for black.

@tacticspotter The bishop couldn't have gotten to e8 in your diagram as the pawns on d7 and f7 are blocking it.

@texaspete Sorry for not making my instructions more clear.

My reading skills - your instructions were clear if I’d have actually read them properly!