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Hey all. So I 'm looking at this KBN v K endgame, and I think I've got it mastered, but I had an unusual question about this part....
So, it looks comical to me. It appears that the bishop is over extending itself and by some miracle the bishop, knight, and edge of the board cover just the squares they need to to keep the king from escaping. So my question is this:
If chess were played on a 10x10 board, would KBN v K be a draw?
I don't think so, but there's a way to find out.
Set up a ten by ten board.
well, following the traditional method, I think it would in fact be a draw, I don't see how I can "find out"
On an 8x*8 board the technigue I follow is as follows.
1. ... Kg8 2. Bg6 Kf8 3. Bh7 Ke8 4. Ne5 Kd8 (After 4 ... Kf8 White uses the standard winning technique with 5. Nd2+) 5. Ke6 Kc7 6. Nc7 Kc6 7. Bd3 Kb7 8. Kd6 Ka7 9. Kc6 and the king is trapped.
You can easily see that this technique would not work on a 10x10 board.
I know there are other techniques to corner the king, and one of those may work.
Assume Black is the player with the bare king. Whether or not KBN vs K is drawn on a 10x10 board depends on whether Black's king can successfully run from one "wrong corner" to the other, I figure. There is a winning technique for White that makes me think Black can't do this. It goes like this:
Let's start with the standard KBN vs K position BK on a1, WK on c3, WN on d4, WB on d5. The winning procedure I have in mind starts with 1. Nc2+ Kb1 2. Bc4 Kc1 3. Ba2 Kd1 4. Nd4 Ke1 (4...Kc1 5. Ne2+ Kd1 6. Kd3 makes progress) 5. Bd5 Kf2 (making his break for the other corner) 6. Nf5! This sets up the key barrier that makes me think KBN vs K is won on a 10x10 board. With the g2 and g3 squares unavailable to the Black king, he has to play 6...Kg1 to try for the opposite corner, but then 7. Kd3 Kh2 8. Ke3 Kh3 9. Bf3 Ki4 (we'll call the ninth and tenth files i and j) 10. Kf4 Ki5 11. Kg5 Kj6 12. Kh6, and the Black king cannot reach j10.
So, I'm thinking the ending is still winnable on a 10x10 board, but not a 12x12. Of course, the way to prove this conclusively would be to build a 10x10 tablebase for KBN vs K. Surely this has been done - wish I could find one on the Internet!
11/28/2014 - Inch By Inch
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