<
1.f3 Nf6 2.e4
({Kanamori} 2.Kf2 Nh5 3.d3 Ng3 4.Be3 a6 5.Qe1 Nxh1# )
2...Nxe4 3.Qe2 Ng3 4.Qxe7+ Qxe7+ 5.Kf2 Nxh1# *
>

Clearly there are many permutations, and even superfluous waiting moves (4...a6 in Kanamori's solve).

I think adding the condition that both sides check each other before the final mate restricts the solve quite a bit.

But I'm not sure it's so interesting, beyond being a counting problem. You have the White pawn move permutation, and the various paths the Black knight can take to h1.

With intervening check, the only solutions I found were the one given, and the same with White's first two moves reversed. I agree that's boring, except for the cleverness of the solution itself (like I said, I had missed that until you said it was possible).

< 1.f3 Nf6 2.d3 Nh5 3.Be3 Ng3 4.Kf2 e5 5.Qe1 Nxh1# >

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I'm being trying to see if I could do it with the QN on the Black side, but the best I've gotten is this M-3:

<1.e3 Nc6 2.Ne2 Ne5 3.g3 Nf3#>

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The hardest part is showing that we've found the complete set of solutions. Although I'm not sure we have, I'll make the case by posting a series of claims about possible solutions. Some of these claims are ironclad, others more like "there's no such solution because I couldn't find one after trying hard" so I won't be shocked if I got some of them wrong. So it's a new side puzzle for other kibitzers to try to refute these claims.

Claim: The rook has to be captured on its original square.

It would take at least two moves to develop the rook to a different square. If the king also has to be moved, that's another two moves and there won't be enough time to set up conditions for the mate.

So we have to consider cases with the rook on d3 or f3, which takes three moves to reach. We also need to prevent one of White's pawns from recapturing the rook, which can be done either by moving them or pinning them.

With the king on e1, only the e-pawn can potentially be pinned but it takes at least three moves to get Black's queen in position for that, because White is too busy to set up a Qxe7+ Qxe7 type exchange here. If Black has to play three knight moves to reach d3 or f3 and three moves to get the queen to the e-file, that's already more than five moves.

Or with the pawn moves, it's three moves to get white's rook to d3 or f3 and two to move the pawns protecting the rook, but the rook would still be guarded by another White piece and it would take a sixth move to prevent that.

Claim: The mate has to take place on h1 and not on a1.

It takes white four moves to clear a path to c2 and move the king there, or five to reach b3, and that doesn't leave enough time to cover the squares to set up a mate.

Claim: The mate must be given by an original knight and not a pawn promoted to knight.

It would take 5 moves to promote a pawn. Assuming we stretch the problem statement to allow the last move to be 5...PxR=N mate, the last move must be 5...g2xh1=N, but that cannot give mate because White would have 6. Kg2.

Claim: The white king can only be mated on f2 and not on g3.

It takes White three moves to clear a path to g3 and move the king there, leaving two spare white moves plus one spare black move, and that doesn't allow enough time to cover all the squares around g3.

Claim: If the QN gives the mate, its only possible path is Nc6-Nd4-Nf5-Ng3-Nh1#.

If the king is on f2 as we claimed above, the last move has to be Ng3xh1#. The knight can't reach g3 via e2 because that would remove the e-pawn and break the mate. It can't go via e4 because of interference with White's king on f2. It can't go via h5 because the pawn on d7 blocks the 3-move path there. So only f5 remains, and the only path there within the available time is the one claimed.

Claim: If the KN gives the mate, its only possible paths are Nf6-e4-g3-h1, Nf6-h5-g3-h1, Nh6-f5-g3-h1.

Claim: Our known solutions with the queen checks are the only ones that don't require White to play Be3 to block this escape square for White's king.

Black has at most one spare move and that move will have to cover at least the e3 square unless White is doing so, and ...Qxe7 is the only way to do that in just one move. Alternate solutions for accomplishing this all take more moves than our known ones.

1. a4 Nc6 2. Ra3 Nd4 3. Rd3 Nxe2 4. c3 Nf4 5. Ne2 Nxd3#.

The key point I previously missed is we can save a move for White by having the knight capture the e-pawn. I think solutions with capturing the c-pawn or g-pawn still won't work because that would be check.

I have two ropes, different length, material, thickness, (but the ropes do burn...) and it takes an hour/60m to burn the ropes.