10

I believe that a King and two Knights are able to force stalemate against a lone King (though obviously not checkmate)... but what about a King and one Knight against a lone King?

I first thought that forcing stalemate would be impossible. So I set up a random position with King+Knight vs King where the lone King was at the edge of the board, and I tried to analyze it.

The result: White can force a stalemate! The trick is the move 3. Kd2!!

[fen ""]  
[White "Stalemate in 9"]
[Black "-"]
[SetUp "1"]
[FEN "8/8/8/8/8/k1KN4/8/8 w - - 0 1"]
[PlyCount "17"]

1. Nb2 Ka2 2. Nc4 Ka1 (2... Kb1 3. Kd2 Ka1 (3... Ka2 4. Kc2 Ka1 5. Na3 Ka2 6. Nb1 Ka1 7. Nc3) 4. Kc1 Ka2 5. Kc2 Ka1 6. Na3 Ka2 7. Nb1 Ka1 8. Nc3) 3. Kd2 Kb1 (3... Ka2 4. Kc2 Ka1 5. Na3 Ka2 6. Nb1 Ka1 7. Nc3) 4. Kd1 Ka1 (4... Ka2 5. Kc2 Ka1 6. Na3 Ka2 7. Nb1 Ka1 8. Nc3) 5. Kc1 Ka2 6. Kc2 Ka1 7. Na3 Ka2 8. Nb1 Ka1 9. Nc3

This doesn't prove that a King and a Knight can always force stalemate against a lone King, but it at least shows that it's not completely inconceivable that King+Knight could force the stalemate.

I obviously don't want a "yes/no" answer without any evidence to back it up. I would like either an irrefutable proof or at least some very strong evidence.

One idea is to build an endgame tablebase which takes into account stalemate as a win, which is equivalent to say that White wins when he captures Black's King. There would only need to be 64x63x62 = 249984 positions.

A second idea would be to get a basic engine and modify its code so that it takes into account stalemate as a win, and you can probably also throw away most of the code of the engine to make it calculate faster. Then make it calculate King+Knight vs King in a few positions where the lone King begins at an edge of the board (but not too close to a corner). But this idea would be less convincing than the tablebase.

8

An exhaustive computer search shows that as expected K+N cannot in general force stalemate against a lone K.

In fact, the defending King can avoid stalemate as long as it's not on one of the six-square triangular neighborhoods of the corners shown in the following diagram

[Title "Danger Zone"]
[fen "kkK2Kkk/kK4Kk/K6K/8/8/K6K/kK4Kk/kkK2Kkk w - - 0 0"]

even on the long edge of each triangle (the twelve White Kings in the diagram) stalemate can only be forced in a few special positions. Namely, Kb2 to move can be stalemated by force only from this position

[Title "Mutual Zugzwang"]
[fen "8/8/8/2N5/8/8/1k1K4/8 w - - 0 0"]

and its reflection about the a1-h8 diagonal; and Ka3 to move is stalemated only if facing a Kc3 and a Knight on one of the squares b2, c5, b6 that control a4 (the first of these arises at the start of the stalemate-in-9 shown by Petrosian).

It follows too that each of these four positions is mutual Zugzwang: the defender get stalemated by force only if on the move. There are few enough other Zugzwangs (up to board symmetries) to list them all. One is the unique losing position of maximal length:

[Title "Mutual Zugzwang"]
[fen "8/8/8/2N5/8/8/8/1k1K4 w - - 0 0"]

Two more are obtained from it by moving the Kings from b1/d1 to a1/c1 or a2/c2. Finally, Ka2 against Kc2 and Nb1(b5) are the shortest mutual Zugwangs.

As Glorfindel suggested, some of these positions are relevant to the Troitzky ending KNN/KP. For example, this is a mutual Zugzwang (in the usual sense of a win BTM and draw WTM), as are each of the three BTM positions in the main line 1...Kg7 2 Kg5 Kg8 3 Kg6 Kf8 4 Kf6:

[Title "Mutual Zugzwang"]
[fen "8/7k/3N4/1p5K/1N6/8/8/8 w - - 0 0"] 
[Title "Mutual Zugzwang"]
[fen "8/7k/3N4/1p5K/1N6/8/8/8 w - - 0 0"]
1

The question if king and knight against king alone can force stalemate is rather theoretic. In practice it is always a draw since the king and knight are unable to mate the king alone.

Stalemate cannot be forced. It can only be achieved when the king alone is in one of the corners so the king alone must help:

[FEN "6k1/8/7K/8/4N3/8/8/8 b - - 0 1"]

1. Kh8 Nf6
  • 2
    It is not entirely theoretic. In the endgame KNN vs KP, which under some conditions is won for the side with the knights, the winning procedure relies on blocking the pawn with a knight, stalemating the lone king with king+knight, and allowing the pawn to run while the blocking knight delivers checkmate. – Glorfindel Dec 17 '15 at 21:37
  • The OP asked if a forced stalemate was possible in KN vs K. – Marco Dec 17 '15 at 21:59
  • 2
    Glorfindel's point is that if you have a procedure for stalemating in KN vs K, you can use that procedure to win KNN vs KP. – dfan Dec 17 '15 at 23:35
  • 2
    @dfan Winning Troitzky is not the same as forcing stalemate in KN/K. Even though only one N moves, another N strongly assists in the process. – user58697 Dec 18 '15 at 0:25
  • 3
    @user58697 Yes, agreed. If it were possible to reliably stalemate a king with just a king and knight, KNN vs KP would be a lot easier! – dfan Dec 18 '15 at 1:46

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