What are the quickest helpmates, which is where both sides help administer checkmate to one side, delivered by the rook, knight, bishop, and pawn? The said piece has to deliver the checkmate on the final move.
Here's what I came up with.
Bishop took me 3 moves:
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"] 1.e4 f5 2.Be2 g5 3.Bh5# *
Knight took me 3 moves as Black:
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"] 1.e4 Nc6 2.Ne2 Nd4 3.g3 Nf3# *
Pawn took me 5 moves:
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"] 1.e4 Na6 2.Bc4 Nb4 3.e5 Na6 4.e6 Nc5 5.exf7# *
Rook initially took me 6 moves, with 1.h4 f5 2.Rh3 f4 3.d4 f3 4.Rxf3 g6 5.Bh6 Nc6 6.Rxf8#. But I found a quicker way to checkmate with a rook - promote to one! This results in a checkmate in 5:
[FEN "rnbqkbnr/pppppppp/8/8/8/8/PPPPPPPP/RNBQKBNR w KQkq - 0 1"] 1. g4 Nh6 2. g5 g6 3. gxh6 Bg7 4. hxg7 Na6 5. gxh8=R#
I actually discovered a mate by a rook in 4.5 moves WITHOUT promotion! My source is this page with lots of other interesting shortest checkmates.
This made me wonder what the fastest bishop and knight checkmates by promotion are (DM’s answer has that for the rook. It works just as well for a queen!) I also saw these and an even shorter pawn checkmate then what @DM has on the same site.
Shortest Non-Promoted Rook Checkmate:
[FEN ""] 1.a4 c5 2.a5 Qxa5 3.Rxa5 c4 4.Rc5 c3 5.Rxc8#
Shortest Pawn Checkmate (Many varieties exist):
[FEN ""] 1.e3 e5 2.Ke2 Qh4 3.Kf3 f6 4.Be2 e4#
And just because pawns have other moves (This site is the source and it has tons more!) here’s the fastest double step pawn mate:
[FEN ""] 1. c4 d6 2. Qb3 Kd7 3. e3 Kc6 4. Qxb7+ Kc5 5. d4#
Shortest Knight Promotion Checkmate
[FEN ""] 1.h4 d5 2.h5 Nd7 3.h6 Ndf6 4.hxg7 Kd7 5.Rh6 Ne8 6.gxf8=N#
Shortest Bishop Promotion Checkmate
[FEN ""] 1.d4 g6 2.d5 Bh6 3.d6 Kf8 4.dxe7+ Kg7 5.Qd6 Qf8 6.exf8=B#
Then again, those are checkmate with promotion. If you want it without promotion, here you go.
Shortest Promoted Bishop Checkmate, By @DM
[FEN ""] 1. d4 c5 2. dxc5 Qb6 3. c6 Nh6 4. cxb7 Nc6 5. bxc8=B Nd8 6. Bxd7# [FEN ""] 1. e4 f5 2. exf5 g5 3. fxg6 Nf6 4. g7 Ne4 5. g8=B h5 6. Bh7 h4 7. Bg6#
Shortest Promoted Knight Checkmate, By @DM
[FEN ""] 1. g4 Nc6 2. g5 e6 3. g6 Nce7 4. gxh7 g5 5. hxg8=N Rh3 6. Nf6#
Shortest Promoted Rook Checkmate
[FEN ""] 1. g4 h5 2. gxh5 g6 3. h6 Bg7 4. hxg7 Nf6 5. gxh8=R+ Ng8 6. Rxg8#
It's not in your original question, but the game here can be adapted for a mate-in-5 by the king:
[FEN ""] 1. f3 e5 2. Kf2 Ke7 3. Kg3 d6 4. Kh4 Kf6 5. g3 Kg6#