From Mate-in-3 to Mate-in-1 Here is my new problem in 3 steps - the key designates here the piece that moves first in the solution. An other way to reach a mate in one I hope you'll enjoy...
    [Title &quot;From Mate in 3 to Mate in 1&quot;]
    [FEN &quot;5R2/p2k1p1P/1P1P1PPr/bPpKBN1p/1pR3n1/7B/2P2N1P/1b6 w KQkq - 0 1&quot;]

Here is the starting position. White to move (at each step)
Step 1 : Mate in 3
Step 2 : Remove the key then Mate in 2 (same starting position with one piece less)
Step 3 : Remove the key then Mate in 1 (same starting position with two pieces less)
Edit. For Step 3, say White R8 hasn't capture any piece at last move.
Spotting potential mates isn't too difficult. There are a couple of obvious ones involving the b and h pawns which aren't far off from promoting. An immediate b7 threatens b8=N# for example. It can only be stopped by Bc7 but then after dxc7 black has Nxf6+ and the mate is delayed beyond 3 moves.
Pushing the h pawn is more promising. If it weren't for the black rook on h6 then it would be mate in 2 with Rxf7+ followed by h8=Q#. That can be fixed by starting with Nxh6.
Going back to the initial position it's worth looking at possible black spoiler moves, moves which either deliver check or threaten check which delay the mate. We've already seen that Nxf6+. Also Ne3+ and Ba2 threatening Bxc4+ if black is given the opportunity. If the mate is going to involve the Bishop on h3 delivering the coup de grace then we also have to watch out for Rxg6 followed by Rg4 blocking the check.
That said the key move of Nxh6 solves a lot of these problems. It gets rid of the pesky rook and it pins the pesky knight on g4. Hence I think the first part goes like this:
[Title &quot;First part - mate in 3&quot;]
[fen &quot;5R2/p2k1p1P/1P1P1PPr/bPpKBN1p/1pR3n1/7B/2P2N1P/1b6 w KQkq - 0 1&quot;]

1. Nxh6 Ba2 {threatening to disrupt with Bxc4+} 2. Rxf7+ {thank goodness this comes with tempo and black has no time for the disruptive check} Ke8 3. h8=Q#

So, remove the f5 knight and try and mate in 2. The b7 push is obviously too slow so the obvious line to take is mating with the h3 bishop.
The obvious first try is Nxg4. The problem is that black is not obliged to take with hxg4 allowing Bxg4#. Instead black has the tricky Rxg6 followed by Rg4 to block the mate. Mate in 2 is denied.
So, how to stop this defence? The cunning answer is to take with the rook. Then Rxg6 can be answered by Rxg6# or, to rub it in, Rg5#. What about Ba2+? That was one of black's disruptor moves. Does that stop us? No! We can just move the rook back to c4, blocking the check and delivering a discovered mate.
So:
[Title &quot;Second part - mate in 2&quot;]
[fen &quot;5R2/p2k1p1P/1P1P1PPr/bPpKB2p/1pR3n1/7B/2P2N1P/1b6 w KQkq - 0 1&quot;]

1. Rxg4 (1. Nxg4? Rxg6 2. Ne3+ Rg4) hxg4 2. Bxg4# (1...Rxg6 2. Rxg6#) (1...Ba2+ 2. Rc4#)

That then leaves this as the final problem. As @wimi points out the key here is to spot the possibility that black's last move could have been c7-c5 meaning that bxc6 en passant would be checkmate. Great spot by @wimi.
[title &quot;Third part - mate in 1&quot;]
[fen &quot;5R2/p2k1p1P/1P1P1PPr/bPpKB2p/1p4n1/7B/2P2N1P/1b6 w KQkq - 0 1&quot;]

1. bxc6#