# Kbnk

David113
Posers and Puzzles 29 Mar '08 17:38
1. 29 Mar '08 17:38
Can K+B+N win against K on a square chessboard of any size? (maybe with some exceptional small boards)

What about a quarter-infinite board? (a board which has only left and bottom border)

Is there a fast algorithm which will make the K+B+N win from any starting position in a reasonable number of moves?
2. 29 Mar '08 20:14
I think it would be drawn already on a 9x9 board. On an 8x8 board the lone king gets very close to escaping while being driven to the 'matable' corner. The extra step the shepherding king has to do on a 9x9 seems to be too much of a delay...

For each step of the lone king, the mating posse has to organize 3 pieces (2 of which are short range and have to pursue the lone king) and is thus slower in moving.
3. 29 Mar '08 21:26
When having an odd number of squares you have the four corner squares in the same color.
If a K vs KBN can be mated or not has to do with the colour of the bishop, the same or not the same.
4. 29 Mar '08 21:26
On (odd)x(odd) boards you have to worry about whether there is a corner the same color as the bishop. I would guess, though (based on no evidence) that mate can be forced on any finite rectangular board with an appropriately colored bishop. On a quarter-infinite board I bet the lone king can escape in a large fraction of positions.
5. 29 Mar '08 23:13
Originally posted by apawn
I think it would be drawn already on a 9x9 board. On an 8x8 board the lone king gets very close to escaping while being driven to the 'matable' corner. The extra step the shepherding king has to do on a 9x9 seems to be too much of a delay...

For each step of the lone king, the mating posse has to organize 3 pieces (2 of which are short range and have to pursue the lone king) and is thus slower in moving.
If I am not mistaken, can't you force a king into mate with just a K+B+N? If anybody has a game in which this happened please post it. Because I am pretty sure that it is possible.
6. TheMaster37
Kupikupopo!
30 Mar '08 17:431 edit
Originally posted by kfennessy
If I am not mistaken, can't you force a king into mate with just a K+B+N? If anybody has a game in which this happened please post it. Because I am pretty sure that it is possible.
It is possible, though very hard (there is little to no room for mistakes).

On any finite even-sided board it's possible, save perhaps the very small boards (4x4 and 6x6).

On odd-sided boards it depends on the colour of your bishop.

On a quarter finite board it's only possible if the corner is the right colour for your bishop.

On an infinite board it's not possible.

(Pretty much what is said above, but hey, I dont'want to stay behind :p)

For a guide:

http://chesstactics.wordpress.com/2006/08/06/achieving-harmony-concepts-and-techniques-in-the-bishop-and-knight-mate/
7. 30 Mar '08 20:34
Nice link you have there TheMaster37 - thanks!
You all are right of course about the (odd)x(odd) board, and with the right bishop the mate should be easier than on an 8x8 board.
On a 10x10 board I suspect the B+N+K vs K is a draw. Proving it is another matter...
8. 30 Mar '08 23:45
Originally posted by apawn
Nice link you have there TheMaster37 - thanks!
You all are right of course about the (odd)x(odd) board, and with the right bishop the mate should be easier than on an 8x8 board.
On a 10x10 board I suspect the B+N+K vs K is a draw. Proving it is another matter...
Yes, very good link, I agree. Also, if anybody has a B+N+K vs K winning game please post it here.
9. 31 Mar '08 00:52
kfennessy, here is a win by Judit Polgar playing blindfold!
The K+B+N vs K ending starts at move 83.

[Event "Monaco Amber blind"]
[Date "1994.??.??"]
[Result "0-1"]
[White "Ljubomir Ljubojevic"]
[Black "Judit Polgar"]

1.e4 c5 2.c3 d6 3.d4 Nf6 4.Bd3 g6 5.h3 Bg7 6.Nf3 O-O 7.O-O Qc7
8.Qe2 a6 9.Bf4 b5 10.e5 Nd5 11.Bg3 Qb6 12.Nbd2 cxd4 13.cxd4
Nb4 14.Be4 d5 15.Bb1 a5 16.Nb3 a4 17.Nc5 N8c6 18.Qe3 Na5
19.Bf4 Nc4 20.Qc3 Nc6 21.a3 f6 22.exf6 Rxf6 23.Bg5 Rf7 24.Be3
Rxf3 25.gxf3 e5 26.dxe5 Bxe5 27.Nd3 d4 28.Nxe5 N4xe5 29.Ba2+
Kg7 30.Bd5 Ra6 31.f4 Nc4 32.Bxc6 dxc3 33.Bxb6 Rxb6 34.Bxb5
Rxb5 35.bxc3 Bxh3 36.Rfd1 Rb3 37.Rd4 Rxc3 38.Rad1 Bf5 39.Re1
Nb2 40.Re3 Rxe3 41.fxe3 Be6 42.e4 Bb3 43.Rd7+ Kf6 44.Rxh7 Nc4
45.e5 Ke6 46.Rg7 Bc2 47.Rc7 Nxa3 48.Rc3 Bb3 49.Rc6+ Kf7
50.Rc7+ Kf8 51.Rc6 Kg7 52.Rc7+ Kh6 53.Rc6 Kg7 54.Rc7+ Kf8
55.Rc6 Bf7 56.Ra6 Be8 57.Ra7 Nc2 58.Kf2 a3 59.Ke2 Nd4+ 60.Kd2
Nb5 61.Ra6 Ke7 62.e6 Nd6 63.Kc2 Nc4 64.Kc3 Bb5 65.Ra7+ Kxe6
66.Kb4 Nd6 67.Rxa3 Be8 68.Ra5 Nb5 69.Kc5 Kf5 70.Ra8 Nc7 71.Rc8
Ne6+ 72.Kd6 Ba4 73.Ra8 Bd1 74.Ra5+ Kf6 75.Ra1 Be2 76.Ra4 Bd1
77.Re4 Ng7 78.Re1 Bf3 79.Rf1 Be4 80.Re1 Kf5 81.Ke7 Nh5 82.Rg1
Nxf4 83.Rxg6 Nxg6+ 84.Kd6 Kf6 85.Kc5 Ke5 86.Kc4 Bd5+ 87.Kd3
Nf4+ 88.Ke3 Be4 89.Kd2 Kd4 90.Kc1 Kc3 91.Kd1 Bc2+ 92.Ke1 Kd3
93.Kf2 Ke4 94.Kg3 Bd1 95.Kf2 Nd3+ 96.Kg3 Ke3 97.Kh4 Kf4 98.Kh3
Ne1 99.Kh4 Ng2+ 100.Kh3 Kf3 101.Kh2 Kf2 102.Kh3 Be2 103.Kh2
Bg4 104.Kh1 Ne3 105.Kh2 Nf1+ 106.Kh1 0-1

You can find it at
http://www.chessgames.com/perl/chessgame?gid=1092636
10. 01 Apr '08 23:33
Originally posted by apawn
kfennessy, here is a win by Judit Polgar playing blindfold!
The K+B+N vs K ending starts at move 83.

[Event "Monaco Amber blind"]
[Date "1994.??.??"]
[Result "0-1"]
[White "Ljubomir Ljubojevic"]
[Black "Judit Polgar"]

1.e4 c5 2.c3 d6 3.d4 Nf6 4.Bd3 g6 5.h3 Bg7 6.Nf3 O-O 7.O-O Qc7
8.Qe2 a6 9.Bf4 b5 10.e5 Nd5 11.Bg3 Qb6 12.Nbd2 cxd4 13.cxd4
Nb4 14.B ...[text shortened]... 6.Kh1 0-1

You can find it at
http://www.chessgames.com/perl/chessgame?gid=1092636
That triangle cornering technique shown in the link really worked out well. I will have to test this out if I get the opportunity. Thanks for the game apawn ðŸ˜€