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In the case where White has King and 3 Knights, and Black has just the bare King, it is known that White can force checkmate. But can White in fact pick any edge square in advance, and force the checkmate to occur with the black King on that square? For example, can White guarantee checkmating on e8, if he so chooses?

There's an answer to a related 3 Knights post

  • mentioning "side mate" as opposed to "corner mate", without going further into it.
  • In the given example the bare king is not obliged to go to the non-corner edge square where the mate occurs.

Edit after answer accepting
The question is answered in the positive: Any edge square can be chosen to become the mate square.

  • 1
    I assume that "the Night party" refers to the side with the three knights, since the question would otherwise make no sense. I suggest editing, if my interpretation is correct. – Scounged May 6 '18 at 12:45
  • I'm not sure I understand the question. – SmallChess May 6 '18 at 12:51
  • @SmallChess Compare the situation with the elementary mate KBN vs K: If your bishop is dark-squared then you'll have to chase the king to a dark-coloured corner since only there the checkmate can be enforced. Thus your choice of the 'mating square' is very restricted. – Hanno May 6 '18 at 14:05
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    When does the "final chase" start? If you have the king on the edge, is it possible to force him to move along the edge in a certain direction? – user1583209 May 7 '18 at 8:41
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    There still seems to be some confusion regarding what is actually asked in the question. For the sake of simplicity, let white be the side with the knights. As I've interpreted the question, you're asking whether white can point at a concrete edge square, say g8, and then force a checkmate where the black king is on g8 in the final position, regardless of the initial piece configuration (excluding some freak cases where a loss of a knight is unavoidable, I suppose). Is this a correct interpretation? – Scounged May 7 '18 at 14:11
5

Here's how to forcibly checkmate the bare King on f8 or e8 (added later: or on g8 or h8) once it's been cornered on h8. First put Kg6 and Nd7, so the other Knights can roam freely as long as they don't accidentally checkmate or stalemate the cornered King; then:

[Title "h8 to f8 (or g8 or h8)"]
[Fen "7k/3N4/2N3K1/4N3/8/8/8/8 w - - 0 0"]

1. Nf7+
(1. Kh6 Kg8 2. Ne7+ Kh8 3. Nf7#) Kg8
2. Nde5 (2. Ne7#) Kf8
3. Kh7 Ke8
4. Nd6+ Kf8
5. Nd7#
(5. Ng6#)

and likewise:

[Title "h8 to e8"]
[Fen "7k/3N4/2N3K1/4N3/8/8/8/8 w - - 0 0"]

1. Nf7+ Kg8
2. Nde5 Kf8
3. Nh6 Ke8
4. Nf5 Kf8
5. Nfe7 Ke8
6. Nd5 Kf8
7. Nce7 Ke8
8. Kg7 Kd8
9. Kg8 Ke8
10. Nc8 Kd8
11. Ncb6 Ke8
12. Kg7 Kd8
13. Nc6+ Ke8
14. Nc7# (14. Nf6#)

To answer the question Glorfindel asks in a comment: White can start the same way, then set up the Knights on c5/d5/e5 to form a gauntlet and use the wK to push bK to the a8 corner:

[Title "h8 to a8"]
[Fen "7k/3N4/2N3K1/4N3/8/8/8/8 w - - 0 0"]

1. Nf7+ Kg8
2. Nde5 Kf8
3. Nh6 Ke8
4. Nf5 Kf8
5. Nfe7 Ke8
6. Nd5 Kf8
7. Kh7 Ke8
8. Nd4 Kd8
9. Ne6+ Ke8 (9... Kc8 10. Nc5)
10. Nc5 Kf8
11. Kh8 Ke8
12. Kg8 Kd8
13. Kf8 Kc8
14. Ke8 Kb8
15. Kd8 Ka7
16. Kc7 Ka8
17. Nb4 (17. Kb6 Kb8 18. Ne7)

and then use the previous analysis to force mate on any of a5/a6/a7/a8 (or b8/c8/d8), or push the King further to a1, etc. So once the lone King has been securely cornered it can be checkmated on any prescribed square on the board's edge.

  • Nice! So all we need is a forced mate on g8 (maybe it's too obvious, haven't checked) and proof that it's possible to drive the king in a certain corner? – Glorfindel May 12 '18 at 11:23
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    I thought g8 was already done in a previous answer or comment. From the same diagram, 1 Nf7+ Kg8 2 Ne7#. – Noam D. Elkies May 12 '18 at 13:43
  • Great stuff [+1], thank you Noam. Settles my question to a 100 percent [$\color{green}\checkmark$]. – Hanno May 12 '18 at 15:01
  • Hum, great analysis. That's embarassing for my own (wrong) answer. – Evargalo May 14 '18 at 8:49
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    Thanks. You actually came quite close; I found the forced mate on e8 while trying to carry out your suggestion (in a comment to your own answer) to work backwards from each possible mate to show that it could not be forced, and then finding to my surprise that one of the mates could be set up by force. – Noam D. Elkies May 14 '18 at 20:34
3

EDIT: This answer in incorrect, Noam solved the problem in the accepted answer.

I'm not sure if I'm supposed to delete this answer or to leave it as an unsuccessful attempt. If anyone can precise it in comments...


After the question has been edited, I will try and demonstrate how White fail to mate on e8 starting from the same random position I used before (conveniently, it is not too far from e8).

The main point is that if you use a knight to block one escape square along the edge (say, d8), then you are basically reduced to mate in the hence created 'artificial corner' (here, e8) using a king and two knights. And just as in the KNNK ending, due to impossibility to attack the last black square (f8) and the last light square (e8) with the same knight, White fails and stalemates.

[FEN "6kN/8/4NK2/8/6N1/8/8/8 w - - 0 1"]

1.Nf7 Kh7 2. Nfg5 Kh8 3.Kg6 Kg8 4.Ne5 Kh8 5.Nd8 Kg8 6.Ndc6 Kh8 7.Ngf7 Kg8 8.Kh6 Kf8 (8.Kf6 {the other plan: the king blocks in front and the two knights try to attack g8, then f8, then e8.} Kf8 9.Ng6 Kg8 10.Nge7 Kf8 11.Nfe5 Ke8 12.Ke6 Kf8 {and you cannot force the king to e8 and deliver mate.} 13.N7g6 Ke8 (13...Kg7 {escapes}) 14.Ng4 {and if it was White's turn the next move would be Nf6#, but this is stalemate.}) 9.Kh7 Ke8 10.Nd6 Kf8 11.Nf5 (11.Ng6 {mates on the wrong square}) Ke8 12.Nfd4 (12.Kg7 {stalemate}) Kf8 13.Ne6 Ke8 14.Nc7 {The knight can control either f8 or e8, but not both.} 

Of course, if the square you have picked is in a corner, White succeeds. I feel like he would fail on g8, but I leave it for someone else to check.


Previous answer: trying to mate on the original square of the knight.

I am not totally sure I understand what you are looking for, but is that it ?

[FEN "6kN/8/4NK2/8/6N1/8/8/8 w - - 0 1"]

1.Nf7 Kh7 2. Nfg5 Kg8 (2...Kh8 3.Kg6 Kg8 4.Nf6 Kh8 5.Nf7) 3.Ke5 Kh8 4.Kd4 Kg8 5.Nf6 Kh8 6.Nf7

White's knight starts on h8 and ends mating the black king on that very square.

  • The question is if White can decide in advance where to checkmate the Black king, hence your answer covers a particular instance. The OP has been edited in the meantime, and should be clearer by now. – Hanno May 7 '18 at 15:00
  • +1, sounds convincing. A definite 'proof' will probably require somebody to write an endgame database generator starting with Black mated on e8 as the only winning position. – Glorfindel May 7 '18 at 17:24
  • I think a solving program like Popeye could do that too. Or we can try a 'mathematical' proof, like: to build a cage and restrict the bK to two squares, we need to control six squares. Any wN can control at most two of them, the wK two or three, etc. – Evargalo May 7 '18 at 17:42
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    Your answer remains useful IMO, but right now I haven't got the time to share my thoughts; will do it within 36h. Pls don't delete. – Hanno May 14 '18 at 11:44
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    Here are my thoughts why your post is worthwhile and should be kept IMO: (1) Your 'main point' mentioning the 'artificial corner' fully accounts for thwarting white's intentions if the white King had to stay on the 6th rank. Passing the guarding job to the Knights and going towards the edge allows the white King to make tempo moves which are needed indeed. (2) As evidenced by the comments, you triggered Noam's e8 solution. – Hanno May 19 '18 at 18:01

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