How can I get a checkmate in three moves?

[FEN "8/2R5/1B1k4/2R5/3N4/1K6/3p4/8 w - - 0 1"]
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    That's why we're here. – Tony Ennis Jan 12 '14 at 15:13

[FEN "8/2R5/1B1k4/2R5/3N4/1K6/3p4/8 w - - 0 1"]

1.Kc4 d1=Q 2.Rd5#

So two moves, not three.

The first thing to note is that the Black King cannot move. The second thing to note is that (most) non-forcing moves by White will allow Black to Queen the pawn - with check. This fails the challenge.

Experimentation will show that moving the Rook on c7 (for example R7c6+) allows the King to escape to d7 or e7. The other frcing move is Rd7+ but of course that isn't helpful. Since the c7 Rook is protected by the Bishop, we can try moving the other Rook. We have to find forcing moves, so the first one I see is R5c6+. But this lets the King escape to d5 or e5. Another forcing move is Rd5+ but the Black King would eat it. But it's so close to checkmate. There's no forcing move with the Bishop. The Knight has 2 forcing moves, but neither seems to help.

So we move the White King to c4 to support the Rook we intend to put on d5. This has the additional benefit of not costing us a tempo if Black queens the pawn. That realization is fundamental here. The White Knight blocks the Queen from capturing the Rook on d5.

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    If the puzzle/problem is to do the mate in exactly three moves, would that be possible? – supercat Mar 18 '14 at 1:47
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    @supercat I have not checked it with an engine, but I doubt it. The reason is that after the black pawn queens, white needs a forcing move. Rd5# is a dandy move, but it a mate in 2. For a mate in 3, some other forcing move is needed. If the Knight moves, the king escapes at e5.The rook on c7 has no check. The bishop has no forcing move.This lets the Black Q throw a check and the mate-in-3 is spoiled. I have not checked this with an engine. – Tony Ennis May 29 '18 at 18:53

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