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Jun-09-25
 | | beatgiant: "Game ending in 5...NxR mate" continued.
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. Example of failed attempt: 1. f3 h5 2. Kf2 h4 3. Qe1 h3 4. d3 hxg2 5. Be3 gxh1=N+, which isn't mate because of 6. Kg2. |
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Jun-09-25
| | beatgiant: Further on "game ending in 5...NxR mate."
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. Examples of failed attempts: 1. f3 Nf6 2. Kf2 Ng4+ 3. Kg3 Nf2 4. Nh3 g5 5. e3 Nh1+. There wasn't enough time to cover g4. Or 1. f3 Nf6 2. Kf2 Ng4+ 3. Kg3 Nf2 4. d4 d5 5. Bf4 Nxh1+. There wasn't enough time to cover h4. Or 1. Nf3 Nf6 2. Nh4 d5 3. f3 e5 4. Kf2 Ng4+ 5. Kg3 Nf2 and Black needs one more move to play 6...Nxh1#. |
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Jun-09-25
| | beatgiant: "Game ending in 5...NxR mate" continued.
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.
Examples of failed attempts: 1. f3 Nc6 2. Kf2 Nd4 3. d3 Nxe2 4. Be3 Ng3 5. Qe1 Nxh1+ is not mate. Or 1. d3 Na6 2. Be3 Nc5 3. f3 Ne4 blocks 4. Kf2. Or 1. f3 Na6 2. Kf2 Nc5 3. d3 Ne4+ and now White has to lose moves responding to the check. Or 1. f3 d6 2. Kf2 Nd7 3. Qe1 Ndf6 4. d3 Nh5 5. Be3 Ng3, and
Black needs one more move to play 6...Nxh1#. |
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Jun-09-25
| | beatgiant: "Game ending in 5...NxR mate" continued.
Claim: If the KN gives the mate, its only possible paths are Nf6-e4-g3-h1, Nf6-h5-g3-h1, Nh6-f5-g3-h1. As before, the last move has to be Ng3xh1#. To reach g3 in three moves, the knight has to come via e4, f5 or h5. Each of those is reachable in two moves only along the paths listed. |
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Jun-09-25
| | beatgiant: "Game ending in 5...NxR mate" continued.
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. Examples of failed attempts: 1. d4 Nf6 2. e4 Nxe4 3. Bg5 Ng3 4. f3 h6 5. Bxe7 Qxe7+ and it takes one more move to play 6.Kf2 Nxh1#. Or 1. f3 Nc6 2. Kf2 Nd4 3. Qe1 Nxe2 and now White isn't able to play 4. Qxe7+ yet so the mate will again take a move longer. Or 1. d4 Nh6 2. Bg5 Nf5 3. Bxe7 Ng3 4. f3 Qxe7 5. Kf2 and it will take another move for White to play 6. Qe1, allowing 6...Nxh1#. |
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Jun-09-25
| | beatgiant: I'll probably revisit here in about a week to see if anyone has posted counterexamples or refutations of any of my claims about "game ending in 5...NxR mate" and based on what we've learned, post my accounting. |
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Jun-13-25
| | beatgiant: I found a counterexample for my first claim above: 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. So we have to consider solutions with a rook not captured in the corner. This is already my second oversight on this problem, so the invitation is still open for anyone to break any of the other claims above. |
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Jun-16-25
 | | WannaBe: Saw this one online, so...
I have two ropes, different length, material, thickness, (but the ropes do burn...) and it takes an hour/60m to burn the ropes. With this information, can you measure out 45 minutes? |
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Jun-16-25
| | beatgiant: <WannaBe> In your puzzle, are we allowed to measure and cut the ropes? |
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Jun-16-25
| | WannaBe: <beatgiant> No, but the ropes do burn |
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Jun-16-25
| | beatgiant: <WannaBe> To answer this question, I think we need a more detailed description of what it means to say "it takes an hour/60m to burn the ropes." How are we burning them, and what happens within the 60 minutes? Do we mix them together in a jumble and light a bonfire? Just before the 60 minutes are up, is there a little bit left of each of the ropes, or might one get finished very early and the other one take a long time? Or do we lay them end-to-end and start a fire from one end of the first rope, and 60 minutes later it will reach the second end of the second rope? Again, could it happen that the first rope gets burned through very early and the second one takes a long time? |
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Jun-16-25
| | WannaBe: <beatgiant> Light the first rope on both end, it will burn out in 30m At the same time lit the second rope at one end. When rope 1 is done (burnt through) light the other end of the 2nd rope, so the remainder will only burn 15m Hence 45m total.
Does that explain how it's done without going down to the atomic level of how burning and fire and heat works? I was able to understand the puzzle while watching a 20s YouTube Short. |
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Jun-16-25
| | beatgiant: <WannaBe> That solution depends on assuming that if the two ropes take an hour, each one alone would take 30 minutes, but just given what you told us, it's not at all obvious that would be true. Isn't it possible that the first one alone could take 1 minute and the second one alone could take 59 minutes (which was the point of the questions I posted above)? In the latter case, I don't think your solution would work, but correct me if I'm wrong. |
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Jun-16-25
| | beatgiant: <WannaBe> It also depends on the assumption that the original 60 minute burning was by lighting the pair of ropes at one point, so that we can achieve a speedup by lighting a rope at both ends. In the bonfire case, we typically would light them at multiple points already (which was the point of asking about jumbling them in a bonfire versus burning them end-to-end). |
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| Jun-16-25 | | stone free or die: <<beat> Do we mix them together in a jumble and light a bonfire?> OK, that got a chuckle. |
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Jun-16-25
| | beatgiant: "Game ending in 5...NxR mate": While going over my list of claims above and looking for ways to break them, I found a new type of solution. 1. Nc3 Nf6 2. Nd5 Ne4 3. Nxe7 Nxf2 4. Nd5 Qh4 5. g4 Nxh1#. I'll update my list of claims and give it another week before posting my count. There might still be some more solutions out there. |
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Jun-17-25
| | beatgiant: "Game ending in 5...NxR mate": And yet another type: 1. h4 Nc6 2. Rh3 Nb4 3. d4 c6 4. Rd3 Qa5 5. Nf3 Nxd3# |
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| Jun-17-25 | | stone free or die: Oh, I wish I had found that last one - very, very nice! |
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Jun-17-25
| | beatgiant: Here's my updated list of claims about "game ending in 5...NxR mate." Claim: The rook must be captured on h1, d3 or f3. Claim: There's no solution with a pawn promotion. Claim: The king must be mated on e1 or f2.
Claim: For the mates with the rook on d3 or f3, either Black's knight must capture the e-pawn, or Black must set up a double check with the knight and queen for Nxd3#. Claim: If the knight delivers the mate to a king on f2, either White must develop the bishop to e3 and the queen to e1, or White must open the e-file and draw Black's queen to e7, and Black's knight must maneuver to h1. Claim: If the knight delivers the mate to a king on e1 by capturing a rook on h1, it will be a variant of the discovered check solution with a queen on h4. Which claims are most likely to be broken? Maybe the king can be mated on c2, b3 or g3, or maybe the rook can be captured on g2. All those are possible in 6 moves, as is a solution with a pawn promoted to knight. Is there a clever way to save a move in any of those scenarios? |
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Jun-17-25
| | beatgiant: As a side puzzle, how long is the shortest game ending in ...NxR mate where the white king is mated on d2? I managed to find a solution in 8 moves; can anyone beat that record? But I'd be amazed if it can be done in as few as 5 moves. |
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Jun-18-25
| | beatgiant: Following up on the side puzzle, I found a 6-move solution for a game ending in ...NxR mate with the white king mated on d2. I still don't think it's possible in only 5 though. |
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Jun-20-25
| | beatgiant: "5-move game ending in ...NxR mate"
So have we found all the solution types? I haven't found any new ones lately, we have pretty good reasons to believe we're done, but I'm not up for running an exhaustive search for absolute proof.Instead, I've decided to catalog here in one place all the currently known solution types, plus the frontiers of other situations where I've only found ways in more than 5 moves and we'd need a speedup. I'm hoping this might help other kibitzers bring new insights to decide whether we really have found all the solutions. |
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Jun-20-25
| | beatgiant: "5-move game ending in ...NxR mate"
Known Solution Methods
Type 1: White K on f2, Q and B blocking e1 and e3, Black N on h1
Example: 1. d4 Nf6 2. Be3 g6 3. f3 Nh5 4. Kf2 Ng3 5. Qe1 Nxh1#Type 2: White K on f2, Black Q on e7, N on h1
Example: 1. e4 Nf6 2. f3 Nxe4 3. Qe2 Ng3 4. Qxe7+ Qxe7+ 5. Kf2 Nxh1# Type 3: White K on e1, Black N on f3 or d3 after capturing e-pawn
Example: 1. g3 Nc6 2. a4 Nd4 3. Ra3 Nxe2 4. Rf3 Nd4 5. Ne2 Nxf3# Type 4: White K on e1, Black N on d3 and Q on a5 with double check
Example: 1. d4 c5 2. Nf3 Nc6 3. a4 Nb4 4. Ra3 Qa5 5. Rd3 Nxd3# Type 5: White K on e1, Black N on h1 and Q on h4 with disc. check
Example: 1. Nc3 Nf6 2. Nd5 Ne4 3. Nxe7 Nxf2 4. Nxc8 Qh4 5. g4 Nxh1# |
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Jun-20-25
| | beatgiant: "5-move game ending in ...NxR mate"
Frontiers of other situations - the challenge of 5 movesWe've probably already found all the ways with capturing the rook on h1, d3 or f3, but can it be taken somewhere else? Here are some tries, but they take 6 moves. Capture on g2: 1. g3 Nf6 2. e3 b6 3. h3 Nh5 4. Rh2 Nf4 5. Ba6 Bxa6 6. Rg2 Nxg2# Capture on c2: 1. c4 Nf6 2. b3 c5 3. d3 Nc6 4. a3 Nb4 5. Ra2 Qa5 6. Rc2 Nxc2# Capture on g4, h3 or e4: 1. h4 Nf6 2. g4 Nxg4 3. Rh3 e6 4. Rg3 Nxf2 5. Rg4 Qxh4 6. Nc3 Nxg4# (Note: similar method also works for rook on h3 or e4) Capture on a2: 1. d3 c5 2. Kd2 d5 3. a3 Nc6 4. Kc3 Nb4 5. Bd2 Qb6 6. Ra2 Nxa2# Similarly, can the king be mated on other squares besides the ones we've seen? Here are some tries. Mate on g3: 1. f3 g5 2. Kf2 d5 3. Kg3 Nf6 4. b3 Ng4 5. Bb2 Nf2 6. Bxh8 Nxh1# Mate on c2: 1. d3 Nc6 2. Kd2 Nd4 3. c3 Nb3+ 4. Kc2 e5 5. Nf3 Qg5 6. g3 Nxa1# Mate on b3: 1. d3 c5 2. Kd2 Nc6 3. Kc3 Nb4 4. Kb3 Qa5 5. c4 Nc2 6. Nc3 Nxa1# Mate on d2: 1. d3 Nf6 2. Nc3 Ne4 3. h4 Ng5 4. Rh3 g6 5. Rf3 Bh6 6. Kd2 Nxf3# Mate on c3: See example above with rook capture on a2 Mate on e3: I couldn't even manage to find a way in 6 moves, but I'll post a couple of 7-move examples in case anyone wants to take on the daunting challenge of making it faster. 1. h4 Nf6 2. h5 b6 3. Rh4 Bb7 4. f4 c5 5. Kf2 c4 6. Ke3 e5 7. Rg4 Nxg4# 1. d4 e5 2. Kd2 Qh4 3. Ke3 Nc6 4. a4 Na5 5. Ra3 e4 6. Rc3 Nf6 7. Rc4 Nxc4# |
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Jun-20-25
| | beatgiant: "5-move game ending in ...NxR mate"
For completeness, I should mention I also looked for a mate on e2, but didn't find one shorter than 7 moves. Example: 1. a4 b6 2. Ra3 Ba6 3. e3 Nc6 4. Rc3 Nd4 5. Nf3 Nb5 6. Ke2 Nf6 7. Qe1 Nxc3# |
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