Bishop, Knight & King versus King.

@Squod -
Thank you for that quick reply.
Somehow - your post appeared under my second post -
instead of between the two.
So I think what I'll do is delete my second one -
and then repaste it at the bottom.  It has a link.

There's three Deletang triangles.  And you very very much -
DO let the target King off the edge of the board.
It appears the edge of the board thing is a nasty red herring -
that makes the whole thing much more difficult than it has to be.

Okay - its Deletang's Triangles - plural.  Or Deletang method.
Very well explained here -

https://www.youtube.com/watch?v=PWZ7h2yrJME

There's not really an issue of "edge of the board"
which apparently causes a lot of the trouble with this.
This method apparently takes a few more moves -
than some other methods.  But is easier.  Easier to understand and do.
It involves three triangles.  Big - medium - small.
There's also a Philidor's method.  And a "W" method. 
They're the same?
Okay - now I see Squod's post indicating they are.
But Deletang's looks much better.
And if you understand the big triangle -
then it looks like you're going to understand the other two also.

Got to log into V3 to try it out.
But I think its going to be a relief - to consider this in terms of zones.
Instead of the edge of the board.
Just hope I can now find the COMPUTER WORKOUT feature in V3.
I think I found it here -

https://www.chess.com/drills/practice/bishop-and-knight-mate

but that was tough to find! 

Ruben, you may find this easier and quicker if you control the squares where the king wants to move. Check is almost always a mistake in these endings.

I get to move 6 postion becuase from there I know every thing.

Maybe it is possible to do it one or two maybe three moves quiker. But I don t care about this. I can do all positions in about 36 moves in one or two minutes.

Judit Polgar managed to checkmate an opponent with a knight and a bishop and a king in a blindfold game. That is impressive. You can find that game in this article:

https://www.chess.com/article/view/the-embarrassing-endgame

[COMMENT DELETED] (Double post by mistake.)

No it's not. One of my party tricks is knight + bishop checkmate blindfolded. non-chess players love it and it's not hard at all.

I mean, before I could play entire games blindfolded, I used endgames like this as a training exercise, since there are a minimum number of pieces but I still have to keep track of an 8x8 board. It helps to train the blindfold chess player's imagination for the bishop diagonals. I still need a board to play a complete blindfolded chess game, but an endgame like this is very easy with my eyes closed.

This is certainly not true. Consider the diagram below

:

If the pieces are all in the yellow tinted sub-board and the black king cannot escape the d10-j4-j10 triangle, the the position is also won on the full board, because such positions can be (optimally) won on that sub-board without letting the black king escape the triangle. Similarly for the green sub-board.

But the won positions are by no means restricted to such positions. In the position shown, for example, White can mate in at most 33 moves against any Black defence. There is, of course, a significant  percentage of similar positions.

On the other hand I think the percentage  of drawn positions on a 10x10 board would be higher than on an 8x8 board. For instance, if the position of the kings in the diagram were interchanged I believe that if Black plays correctly the position would necessarily  be drawn under the 50 move rule, or, if the 50 move rule were waived, by repetition. Draw by repetition is not an option for Black on an 8x8 board.

I would guess that any position that could be won on a 10x10 board could also be won within the 50 move rule. 

Not easy at all. 

It's usually easy if you play against a strong computer program and particularly against an EGTB, but this is because they're designed to play accurately, i.e. to maximise the distance to mate. With accurate play by Black from most positions he will finish up in the corner ready for his opponent to play Philidor's method to deliver mate.

While accuracy is a correct strategy for the side with the pieces in this endgame, it is not a correct strategy for the side with the lone king unless the position can be drawn. Even if the machine does play accurately it will not invariably walk obediently to the wrong corner. E.g. in the following Black to play position, which, had White played accurately on move 69, might have occurred in this game https://en.wikipedia.org/wiki/Bishop_and_knight_checkmate#Rob_van_de_Plas_vs_Philip_Cody_.28EM.2FM.2FGT.2FA001_1996.29 , any king move is accurate, so your program may accurately (and in this case correctly) play the Black king back to the 5th. rank instead of h7.  White will be able to mate in 21 moves either way, but  with the Black king on the 5th. rank he will have to think of all the moves himself, rather than just remembering 20 of them from Philidor's method in his endgame manual.

Here is an example played against the Nalimov EGTB, where driving the king to the corner is very easy. This is because the program is playing accurately (the reasons are shown in the comments). But correct (and inaccurate) play would have been 5...Ke7. Now White can mate in 23 instead of 24 with the move played. But he will need to think of the 23 moves himself instead of trotting out Philidor's 20 move sequence by rote.

The rules of chess are not concerned with the number of moves to mate; only if mate occurs or not. If you make an inaccurate move as Black in this endgame you may be mated 3 moves earlier than you might have been and lose a point, instead of lasting three more moves and, er, losing a point.

It is apparent from reading posts on the internet that many players have learned Philidor's method by rote without actually acquiring the technique of playing the ending. In practising "forcing" the king to the wrong corner against a computer, the computer, by playing accurately may also be doing the thinking for them By making moves such as suggested in the previous examples, you force them into their own analysis and this will give you a much better chance of a draw.

I spent some time looking for perfectly accurate play in this ending, either in handbooks or games, for a Wikipedia update, without finding one. So far as I can see this is such an one.

The only change I could make would be 89. Bg4, which is no more accurate, but the move made forces Black to the square he should in any case go to.

Harking back to my previous post, Black's move 90...Kb7 was inaccurate but I would say correct.

But Nakamura left it too late. Better would have been 80...Kc1 (81.Kc3 Kd1).

@MARattigan
I couldn't find my post where I stated a 10 x 10 board idea.  I vaguely remember it though.
And obviously if the lone King was already trapped in the corner - then the mate is possible.
But would the mate be possible - in 50 moves -  if the target King was in the center?
Was that implied?  If so - maybe I should have  qualified that when I said it the first time.

Regarding driving the lone King to the side of the board - then again - qualification.
Its RELATIVELY easy to drive the lone King to the edge of the board.
And Relatively difficult to force him from there - to one of the two correct corners.
Why is it Relatively easy to drive him to the edge?
Because the attacking King can get zugzwang whenever he wants it.

But the whole edge of the board idea is misleading I'm saying.
Much better is the idea of Deletang's triangles.

Playerafar is right. Deletang's triangles is so simple, that after studying it, it took me a few minutes to master Bishop + Knight checkmate.

I can now checkmate anybody.

I admit that the beginning part of the checkmate is where I'm most inefficient. Generally, I simply place my pieces in the center and then use that positioning to drive the king to the edge of the board, but not in a completely organized way. However I have memorized some patterns, and I've seen people who run to the same color square as the bishop anyways and get checkmated in less than 10 moves.

DELATANG'S TRIANGLES 

Nobody has any problem AFTER that position is reached goodknight0badbishop, the problem is getting to the position where the black king is in the corner.

That is the point. One must have the skill to do so.