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I know how to win the endgame with bishop and knight, but it is a slippery process and seems to be just barely a win, with the enemy king very nearly escaping. For this reason, I am curious about the endgame on other board sizes and if it will still be possible in the general case of an MxN board. For example:

  • Is there a forced win on a 10x10 board?
  • Is there a forced win on a 7x7 board, with a bishop of the 'wrong' colour? (i.e. a bishop which can't attack the corner squares)

Let's assume the 50 move rule doesn't apply.

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1 Answer 1

Let's start with the 7x7 question:

Is there a forced win on a 7x7 board, with a bishop of the 'wrong' colour?

This seems to be the easier of the two questions to answer. First, convince yourself that this is the only mating pattern (the black king could also be on the dark square immediately to its left):

Mate with wrong color bishop

The key point is that it is not possible for white to force this position. Black's king would have been stalemated on the previous move. Alternatively, if black's king is moved one square to the left, the only legal move that white could have just played would be to move the bishop onto that diagonal, delivering mate. If this were the case, where was black's king before that? It would have been on f2 (two to the left, one up). So black was not forced to move into the corner and could have instead avoided the mate. To conclude, there is no way to force mate in the wrong corner, shortening the board doesn't change this fact.

Now the first question:

Is there a forced win on a 10x10 board?

In this case, white will have a proper corner, but let's assume that white can force the black king into the wrong corner. On the standard 8x8 board, white has to release the king from the side for a few moves in the process of driving the king to the mating corner (see wikipedia for a complete tutorial). Here is the normal position when black flees the edge (temporarily):

enter image description here

Black usually plays ...Kc6 and then after Bd3! the king has no escape. On a 10x10 board, however, black could play ...Kb7 followed by ...Ka7 and finally ...Kz6 (let's call the first file to the left "z"). White has no way to get the king and knight over to help prevent the black king from escaping the bind. So again, it's a good thing that the board is only 8x8 otherwise the bishop and knight would never get to mate the king!

Disclaimer: I have not proven any of my assertions with tablebases

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Is there a problem with your answer in the second part? Supposing there are files y and z to the left, wouldn't the "release from the edge" then have occurred with the black king on a7, rather than c7? (So there wouldn't actually be more room than normal to escape in that direction.) I may not be clear on the scenario you are meaning to describe ... –  ETD Oct 18 '12 at 21:29
    
@EdDean, no, the black king started on h8 and was driven along the back rank using the standard W technique. I copied the position from wikipedia if that helps. Black escapes from the edge 5 files away from the starting corner. –  Andrew Oct 18 '12 at 21:30
    
You know what, I didn't look and thought you had a dark-square bishop in the position. So I thought you were talking about driving the king out of the "y8" corner. –  ETD Oct 18 '12 at 21:32

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