I know you're a FIDE Master :), so I suppose you're more interested in this question from a teaching perspective.
The simplest way to understand a checkmate with King and Rook vs King is the idea of the rectangle of the opposing king. Consider this position-
Here, the Black King is restricted by the White Rook in this giant rectangular area of the ...
I always like to explain this in a visual way.
Basic Idea: Keep the bishops together. They form a large net (restricted area) from which the opponent king cannot escape.
Step 1: Push Opponent's King To Edge Rank or File
Keeping the bishops together and using the king for support, make the restricted area smaller to push the opponent's king back to an ...
Actually, the bishop and knight mate is not as slippery as it appears. I have checked this on a tablebase program I wrote. On a 10x10 board, the side with the bishop and knight (say white) can force mate in at most 47 moves. White can even force mate on a 16x16 board, in at most 93 moves. I believe mate can be forced on an arbitrarily large even size ...
If the enemy king is cut off at the edge of the board, it is possible to mate him using only premoves. For example:
[FEN "2k5/Q7/8/8/8/8/8/K7 w - - 0 1"]
1.Ka2 Kd8 2.Ka3 Kc8 3.Ka4 Kd8 4.Ka5 Kc8 5.Ka6 Kd8 6.Qb7 Ke8 7.Kb6 Kd8 8.Qc7+ Ke8 9.Kc6 Kf8 10.Qd7 Kg8 11.Kd6 Kf8 12.Qe7+ Kg8 13.Ke6 Kh8 14.Kf6 (14.Qf7??=) Kg8 15.Qg7#
Just make sure ...
I present unto you a superbly easy mate in 21 that I found the other day in the Matplus.net forums. Give it a try!
[Title "Filip Bondarenko, Feenschach 1960, Mate In 21"]
[FEN "8/7Q/2r1p3/2rkr3/2rrr2Q/7K/8/8 w - - 0 1"]
This is a mate in 24 that I quite like. You can view the solution in Yet Another Chess Problem Database.
There are many:
[Title "Mate in 20"]
[FEN "3K1B2/1p6/pp6/rk2N3/b1p5/1pP5/1P2P3/8 w - - 0 1"]
You should search puzzles from the 1900. It was a golden era for chess puzzles. I found a mate in 63 some days ago.
Chessmaster also has a nice collection of interesting(and long) puzzles. There are many sites where you can sort puzzles according to ...
In some special cases you can "premove" to a forced mate even against
a King in the center:
8/8/8/8/3k4/7Q/3K4/8 w - - 0 0
1. Qe6! Kc5 2. Kc3! Kb5 3. Qd6! Ka5 4. Qb4+ Ka6 5. Kc4 Ka7 6. Qb5 Ka8
7. Kc5 Ka7 8.Kc6 Ka8 9. Qb7#
(Black's only option is 3...Ka4 when 4 Qb4 is already mate.
The alternative 1 Qf5!? Kc4 2 Qe5 Kb4(b3) 3 Qd5(+) or 3 Qc3+
might not be ...
I think with the black king on f7 it's also possible. For example, in the following position:
[FEN "8/5k2/3Q4/8/8/3K4/8/8 w - - 0 1"]
this sequence of moves for white checkmates always:
Ke4 Kf5 (go to f5 in two moves) Qd7 Kf6 Qh3 Qc8 Qa8 Qh1#
I made this gif to explain why:
In green it is shown where the black king could be (to consider all possibilities)...
Grave Danger in King and Queen vs lone King
It's important for the winning side to avoid this common pitfall in this basic endgame.
The winning side should avoid placing their queen on the c2, c7, f2 or f7 squares if the opponent's King is in the nearby corner, because this leads to stalemate.
Below, I include some other types of stalemates that are not ...
I would teach the slow but safe and easy way first: The rook divides the board in two halves. Then the attacking king tries to get in opposition to the enemy king (sometimes the rook has to swap sides or needs to make a waiting move for this to happen). Then the rook gives a check, hence pushing the king one field closer to the edge of the board. The mate ...
Just to show it's possible, here is a position, without the opponent king trapped at the edge of the board, where you can premove the entire sequence.
[FEN "8/6k1/4Q3/8/8/8/8/K7 w - - 0 1"]
Move the king to g5, play Qd7(+), and then continue as in Maxwell86's answer.
[Edited] The winning plan is to force the king into either of the corners which have the same colour as your bishop.
This is done in a few steps.
Force the king to one side of the board so he is stuck on either the a/h file or 1st/8th rank
Use the W manoeuvre to force the king to the corner of your bishop, then use the motif of knight check on (e.g. Na3+ ...
While it's possible to force this mate, before you go getting interested in it, take a look at these facts:
It's a very rare ending. Most GM's never see one.
There are only 263 such endgames in my database of 1,172,256 games in which at least one player was at least 2450 ELO. That's only 1 such endgame in nearly 4,500 games.
Of the 263 games, 24 (9%) were ...
Yes it is. Check out this video!
The general rule is to get the opponent king in the same corner as your bishop. Use the knight and king to control the squares of opposite color to the color your bishop controls. Then, drive the opponent king to one of the two corners with the correct color!
Here's an easy mate in 20 with a unique solution
(which becomes a very easy mate in 19 after the first move):
[Title "Noam D. Elkies, Chess Stack Exchange 2020"]
[FEN "6bn/p4p1k/p4Ppp/p7/p3K1P1/P4N2/1P6/7n w - - 0 1"]
The basic idea is to constrain the enemy king with the bishops working together on a pair of diagonals, just as with a rook you work with ranks and files to push the enemy king to the side of the board.
With two bishops you need to drive the king to one of the corners.
We'll first start with the ending moves to know where you want to go.
Lol, this question is not as silly as it sounds as once I was in time trouble and ended up stalemating my opponent by accident! Most players resign when their opponent is up a queen so such blunders can happen if you occasionally come across a player who is stubborn enough to play till the end. Anyway, after that incident I went back to studying the Queen ...
I would suggest -- Play this out with an opponent. That's the best way to get the feel of it. Otherwise just google and you will see lots of explanation -- with some nice graphics.
www.chess.com and chesstempo.com even have some practice board where you can learn.
One link that covers good theory and explains slowly is
There are obviously many forced wins on any boards where M and N are at least 8 (including M or N or both infinite), so long as there is a corner of the same colour as the bishop's square.
If the pieces are all in the yellow tinted sub-board and the black king cannot escape the d10-j4-j10 triangle, the position is also won on the full board, because such ...
The simplest answer is to restrict the enemy king's movements by "walling off", not attacking, with the queen, move by move, driving it to the edge and then bringing up the king to help administer the mate. You must remember to always leave the king at least one free square once it's in a corner to avoid stalemate. This is basically the same method used ...
It is possible, but hard.
Memorizing it doesn't help (it rarely occurs over the board). But, if you try to do it own your own, it will help you with learning how to make better use of minor pieces in the end game.
If you can confine the king to a corner of the same colour as the bishop, then it is over (try it!).
The method most people know is composed of ...