Can you checkmate with bishop and knight on a 9x9 bord?

It's actually an interesting question whether there might be a position on a 9x9 board that there isn't a forced mate in under 50 moves whereas there would be one on an 8x8 board. I'm guessing that it doesn't make enough of a difference, but a 10x10 would probably be a different matter.

White to move and mate in 31:

 Yes it could be. If 8x8 is for example possible in 36 moves maybe an 9x9 bord it is posible in about let say 42 moves and a 10x10 bord maybe in 54 moves?  This would not surprise me.  But computer programmers or developers on table bases can prove this thing. In any case is the 8x8 checkmate with bishop and Knight now very easy for me. In this example I would start with Kb2 and Nb6 moves. But anyway I can checkmat this in about 38 moves in about 1 or 2 minutes allready many years. And becomes really boring.  So this 9x9 and 10x10 bord are a new challenge for me! 

The guard moves like an king without royal-attributes attached to it. It's value is close to an knight generally speaking. 

Ah so exactly like a king except it can not be checkmated! Ok now I get it.

Several years I saw a convincing demonstration that checkmate is possible with B+N on any n x n board if n is even.  (The 50-move rule is disregarded, of course.)  Most patterns that execute this mate successfully on an 8 x 8 board have a problem when the board gets sufficiently large, due to possibility of running the defensive king around the bishop and knight and making it to the opposite "wrong corner".   However, with the aid of a chess engine, someone found a pattern that basically confined the king inside a wall, then systematically slid the wall across the board to a good corner.  So the only square boards where the mate is not possible are the n x n boards where n is odd and the bishop is the wrong color.       

 Ok I get it , now this is very interesting! Could you show me something about this methode of a pattern that is slid a wall to the right corner? I am very interested in this.  I allready noticed that a 10x10 and a 9x9 board are winning. The last of course only if the bishop has the collor of the four mate corners. I allready noticed too that in this endgame with the uneven board you can have the so named wrong bishop just as the edge pawns have and in this case it is both true. And also count for the same Bishop. If the four corners are always dark with an uneven nxn board it is the light bishop always.

I would like to see this structure of a wall that is slid to the right corner.

Not sure if knight's W manoeuvre work. 

In post 17 here and in post 3 on the 10x10 threat I showed the  moves. 

Several years I saw a convincing demonstration that checkmate is possible with B+N on any n x n board if n is even.  (The 50-move rule is disregarded, of course.)  Most patterns that execute this mate successfully on an 8 x 8 board have a problem when the board gets sufficiently large, due to possibility of running the defensive king around the bishop and knight and making it to the opposite "wrong corner".   However, with the aid of a chess engine, someone found a pattern that basically confined the king inside a wall, then systematically slid the wall across the board to a good corner.  So the only square boards where the mate is not possible are the n x n boards where n is odd and the bishop is the wrong color.         

This is in response to a post from Ruben H about nine months ago, but better late than never, I suppose...

Ruben asked about the technique to checkmate with B+N on an n x n board where n is even.  I suggest he check out Henry Keiter's post in the following blog:

https://chess.stackexchange.com/questions/1432/kbn-vs-k-checkmate-on-nonstandard-boards

Keiter essentially put this question to bed by building K+B+N vs K tablebases for 10x10, 12x12, 14x14 and 16x16 boards.  He summarizes his results by giving the number of moves to force mate on every even sized board from 4 to 16. 4: 15; 6: 22; 8: 33; 10: 47; 12: 64; 14: 78; 16: 93.  He also gives the 92-move solution for one of the worst positions on the 16x16 board, and the manner in which the Black king is shuffled down the a-file with no breathing room is quite enlightening.  To see this in action, get four chess-boards, line them up to create a makeshift 16x16 board, and play through the analysis! 

Seeing this late again but indeed this are great results.

On the 10 x 10 board I played this and I see that it is precise what I wrote in post 30 and found myself on experience. I am glad you proved it with the table base that you wrote yourself! Impressing,

In fact this is just like the W-shape and also  on the other side of the board. Just what I found myself only a bit adjusted w-shape and the king must walking along with the black king and the king can be stopted at a6.

Incredible that you proved this for 4x4 up till a 16x16 board! And with all the numbers of the checkmate in the longest case. On the 16x16 board I still have to play But I do this later because I need 4 chess boards! But I believe it.  Table bases is a 100% prove. If it counts for all the even boards is still not proven but at least very likely .

I leaves this then for now and assume this is right. That leaves only the odd boards. This seems to be ( of course assuming you have the right corner bishop ) more easy becuase all the corners are right to checkmate the King.  In fact there you only have to do the from the big to the smaller triangel methode. Because there are no wrong corners.

At last if there is a wrong Bishop this is in fact the same as the Two Knights against a pawn endgame. ( or even two panws )  There could also be a Bishop from the wrong collor plus Knight against a pawn endgame!

In this case it is interesting that unlikly the two Knights against a pawn endgame the pawn can be blocked by the Knight but also by the Bishop. I do not know in with case there are more winning possibilities. But in any case it wil influence how far the pawns may stand to be still a winning position. Troitski found at a 8x8 board that the pawns against two Knights may be till a4-b6-c5-d4-e4-f5-g6-h4  squares. And in some cases the pawns may be a bit further for example with the edge pawns they may be one square further but than the king can only be in a specific zone. 

On an uneven board for example 9x9 board with a1 to be dark and a light bishop this can be also the case. A pawn that is blocked for a while when the king plus one piece get the enemy king to a corner sometimes with a bit a help of the blockating piece and then just before the checkmate the blockating piece let the pawn go and when it is running it will be no stalemate and there is a checkmate after all.