Quest for Understanding

I have been trying to hold a draw against Stockfish from a position that Jan Timman failed to hold against Anatoly Karpov (Belfort 1988). The position was brought to my attention when I began rereading Jacob Aagaard, Excelling at Technical Chess (2004). Nearly twenty years ago, I read this book quickly and superficially, spending very little time with the instructive positions. I did read all the text and it had a profound impact on my game, particularly in my attitude towards offering or accepting draws when there is even a faint chance of battling for a win. In "Excelling at Technical Chess" (2011), I presented an annotated game where I prevailed in a minor piece battle, somewhat inspired by Aagaard's work.

Aagaard's annotations of the game elaborate the importance of activating your strongest piece. Timman does this well early in the endgame, but then seems to lose the thread and "play without a plan" (Aagaard 13).

White to move

Here, Aagaard writes, "everything has gone wrong for White", noting the passivity of White's rook compared to Black's active rook. However, according to Stockfish and tablebases, White's position remain a technical, however difficult, draw. A comment on chessgames.com credits Karsten Müller and Yakov Konoval, Understanding Rook Endgames (2016) with pointing out that the position is drawn.

Drawn according the computer is one thing, however, and holding such a position as a human is quite another. Aagaard points out in the next chapter of the book, "an advantage does not have to be decisive in order to win" (24). The defender must continue to find the best moves, while the side that is pressing for a win can try different things.

When Timman played his next move, 53.Ra3, the position was then technically lost. However, Karpov's technique was not without error. Five moves later, Timman again had chances to draw and missed the correct moves.

Likewise, playing 53.Ke3, 53.Rc4, or 53.Rd4 does not immediately secure the draw. Many subsequent positions offer a single move that must be found. Others offer two options, but no more. I played the position against Stockfish 18, needing help and takebacks at least half a dozen times to hold this position. Then I studied my efforts with the engine running, and practiced the ending again. After several efforts, I was able to hold the position against a weaker version of Stockfish on my iPad.

Alas, errors were made both by the weaker Stockfish and by me. So, more time was spent playing the latest Stockfish on a computer with the determination to reach such an understanding of the position that I can draw the machine with no errors.

One such exercise continued:

53.Rd4 h5 54.Ke3 h4 55.a4 Rf1 56.Rd8 Ra1

White's rook has become active, while Black's rook's mobility has been reduced.

White to move

I am beginning to understand that in this position (and likely similar ones) Black's rook must balance continuous checks with threats to capture the a-pawn should it be undefended. The king must get in front of the h-pawn.

57.Rd5+ Kg4 58.Kf2 Rxa4 59.Kg2

Each of White's last three moves were the only move to maintain a draw.

59...Ra2+ 60.Kh1 Kg3 61.Rd3+

The side check is the only chance.

61...Kf2

White to move

62.Rd5

This move both restrains the mobility of Black's rook with a threat on the pawn and prepares checks from the rear.

62...Ke3 63.Rc5

63.Re5+ Kd4 64.Rf5 also holds.

63...Kd4 64.Rf5 h3 65.Rb5 h2

White to move

66.Rb5+

The next 50 moves were efforts to get the engine to capture my rook for stalemate.

A Longer Battle

Starting from the initial position at the top of the post with a different move still led to a position in the line above.

53.Ke3 h5 54.Rd4 h4 55.a4 Rf1 56.Rd8

56.Rd7 also holds

56...Ra1 57.Rd5+ Kg4 58.Kf2 Rxa4 59.Kg2

Black to move

59...h3+

59...Ra2+ was played in the line above.

60.Kh2 Ra1

White to move

In this position, White has two viable moves, but for most of the next thirty or so, only one move holds.

61.Rd4+

61.Rc5 was the other option. Would it make subsequent moves simpler?

61...Kf3 62.Rd5

A check allows the king to attack the rook, so cannot be played. After 62.Rd3+ Ke4, White's rook can neither check the king nor attack the a-pawn. Hence, Black's rook becomes more active and the a-pawn becomes a credible queening threat.

At least, that has been my understanding. However, if Black attacks the rook with 62...Ke2, White is lost nonetheless. 63.Rd5 attacks the pawn 63...a4 is now the only winning move. 64.Rd4 a3. Again, Black had to push the pawn to maintain the win. After 65.Ra4 Kd2 66.Rf4, White appears close to setting up Vancura's defense, but 66...Rc1 refutes White's plan (Black also might slide the king back to the e-file temporarily). 67.Ra4 Rc3 Both of Black's pawns are secure and the king is free to move to b2 where all possible checks can be blocked.

62...a4 63.Rd4

Why does 63.Rc5 fail? 63...Re1 64.Ra5 Re4 65.Kxh3 Ke2 and Black's king will reach b2 while his rook is also more active.

63...Ra2+ 64.Kh1

64.Kxh3 was possible here, but not in many other positions that I reached during my training exercises.

64...Kg3 65.Rd3+

Of course, the mate threat must be stopped.

65...Kh4 66.Rd4+

66.Rc3?? The loss of a single tempo allows Black to reposition the rook with a winning advantage. 66...Rb2 (or Rd2 etc.) 67.Ra3 Rb4. An instructive line continues 68.Kh2 Kg4 69.Rg3+ Kf4 70.Rxh3 Rb2+ 71.Kg1

Black to move

Analysis diagram

71...Rb3 is the only winning move, but it is not difficult to find.

66...Kd3 67.Rd3+ Kf2 68.Rd4

Again, White must eye the a-pawn.

68...Ra1+ 69.Kh2 Ke3

White to move

70.Rc4

A rare instance when White has four viable options, all along the fourth rank.

After dodging many checks, Black advanced the pawn to a3 on move 78. White continued the checks from the side without difficulty as the position is essentially the same as found in the Vancura defense.

Then, 90...Ke4 offered White some choices.

White to move

Again, any move along the third rank that does not hang the rook holds the draw.

91.Rc3 Kd4 92.Rf3 Kc5 93.Rf5+

This check was unnecessary with Black's king temporarily on holiday.

93...Kb4 94.Rf4+

This check, however, was the only move.

94...Kc5 95.Rf5+ Kd6 96.Rf3

Either check from f6 or return to the third rank.

Both players shuffled their pieces for many moves until Black set a tactical trap: a discovery that wins a critical tempo.

White to move

After 120...Ra1

121.Kh2

121.Rf3 Kb2-+

After another two dozen moves:

Black to move

After 145.Kh2

145...a2

This move was what White had been anticipating, for now the ending has become a simple one that I have played many times.

146.Ra3 with an easy draw, but the engine took it out to move 195 when the game was drawn by the 50-move rule.

Has this exercise increased my knowledge in a way that might be applicable to new positions? The central theme of this ending, as analyzed by Aagaard, and as played out against the computer in the effort to improve on Timman's play, is the mobility of the strongest piece--the rook.

Difficult Breakthrough

I had Black and the move in this position, found my way to a sacrificial breakthrough that led to a win, and ended feeling good about my skill. Stockfish was critical and brought me back down. My breakthrough idea was flawed and only my opponent's defensive errors gave me the win.

Black to move

30...Kf8 31.Kh2 Ke8 32.Kh3 Rc1 33.Kg3

My moves so far have been fine, although lots of deep computer analysis and dozens of games played against Stockfish on my iPad have convinced me that it would have been more accurate to place my rook on the first rank before moving my king towards the queenside.

Black to move

33...Rd1

This move does not yet squander the advantage, but would have been an appropriate moment for the correct sacrifice.

34.Rf4 Rd3+ 35.Rf3

Black to move

35...Rdxd4??

This sacrifice, which I was proud of during the game, was a blunder that should have led to a draw.

36.Bxd4 Rxd4 37.Rb3

Black to move

My thinking may have been that my centralized king's ability to help my center pawns made them more powerful than White's queenside pawns.

37...Rg4+ 38.Kh3??

This unfortunate error restored my advantage. Had White's king stayed near my central pawns, soon to advance, they would have been stopped.

39...Re4?

39...d4 retains the advantage

39.Rxb5 Rxe5 40.Rb8+ Ke7 41.a4 d4 42.Rb3 Rd5

Placing my rook behind a passed pawn.

White to move

43.Rd3 e5 44.Kg3 e4 45.Rd1 Ke6

White to move

46.Kf4??

The final error.

46.b4 Ke5 47.Kf2 e3+ 48.Kf3=

46...e3-+ 47.Ke4 e2 48.Re1 d3 49.Kf3 Re5 White resigned 0-1

Computer Training

After running quick analysis using the website's analysis feature, I realized my play was not as brilliant as it had felt while playing. But, it was not clear to me what I should have played to convert the clear advantage Stockfish found in the position.

Running the engine and playing its favored moves seemed to involve a great deal of shuffling and often unforced weakening moves by White.

I was intrigued.

The key idea that I missed was a pawn sacrifice to open the a-file with the idea of getting both Black rooks on the first rank. Black then aims to force the exchange of rooks.

Having learning the correct idea from computer analysis, I began playing and replaying the position against the machine. I spent one evening playing more than a dozen games against Stockfish on my iPad. White's resources were revealed in games ending in a draw and even one loss. I won a few, too.

Over the course of several weeks, I analyzed these games on my computer. Then, while writing this article, I played Stockfish 17 on my computer from the position the top of this post.

30...Rc1 31.Kg3 Kf8 32.Kf2

The beginning of a maneuver that the iPad version of Stockfish never found.

32...Ke8 33.Ke3 b4

The correct idea, but here, perhaps, timing is off. Possibly better is 33...Rg1 34.Rf2 Rh1 35.g3 Rg1 36.Rf3 b4

34.axb4

Black to move

34...Rh1

My first effort continued 34...Rc7 35.Kd2 Rg1 36.Kc2 Rxg2+ 37.Kb3 and I did not find the only move that holds an advantage: 37...Rg1

35.g3 Rc7 

Better than 35...Rc8 due to White's attack on the f-pawn.

36.Kd2 Ra7

36...Rh2+ is better 37.Kc1 Ra7

37.Kc2 Raa1 38.b5 Rhf1

The f-pawn must be protected so that the king can get in front of White's passed pawn.

White to move

39.Re3

Perhaps 39.Rf6 was a better choice.

39...Kd7 40.Bb4 Rf2+ 41.Kb3 Kc7 42.Bc5 Rb1 43.Ba3 Kb6 44.Rc3 Kxb5 45.Rc5+ Kb6 46.Kb4

Another slight inaccuracy, according to Stockfish when it has more time to analyze this position than it did to play it.

46.Ka4 is more stubborn.

Black to move

46...Rbxb2+

In this position, my original exchange sacrifice is now correct.

47.Bxb2 Rxb2 48.Ka4

48.Kc3 drops the g3 pawn, but it seems to me that the d-pawn is more important and it soon falls.

49.Rb5+ Kc6 50.Rc5+ Kd7 51.Ra5 Rxd4+ 52.Kb5 Rc4

White to move

53.Ra7+ Rc7 54.Ra3 Rc2 55.Ra7+ Ke8 56.Ra8+ Ke7 57.Ra7+ Kf8 58.Ra8+ Kg7 59.Ra3

Black to move

I stopped playing against Stockfish here because I had places that I needed to go, but I was confident that after 59...Re2 I would have the sort of rook ending that I can win against the engine. Maybe I will continue at a later datew.

Knight over Bishop

I had this position this morning. I was White.

Black to move

41...Bd3??

It is clear that 41...Be2 would have prevented, or at least delayed my plans to remove Black's g-pawn, but 41...Ba2 also keeps the draw in hand. To wit, 42.Nf3 Kc4 43.Ne5+ Kb4 44.Nxg6 Kxa4 45.Ne5 Bg8 46.Kxf5 c4 and White will be forced to give up the knight for Black's last pawn.

42.Nf3+- c4 43.Ne5 Bc2 44.a5!

This move was necessary to the win.

44.Nxg6 Bxa4 45.Ne7+ Ke6 46.Nxf5=

Black to move

Analysis diagram

Black's only drawing move may not be so easy to find here. What would you play?

44...Kc5 45.Nxg6 Kb5 

White to move

46.Ne5

Stockfish favors 46.a6! The lines following are pure engine.

46...Bb3 47.g6 Kxa5 48.g7 c3 49.bxc3 Bg8 50.Kxf5 Kb5 51.Kf6 Kc5

White to move

52.c4

White wins more quickly with a simple technique: Ke7 or Nf7 with the idea of trapping the bishop. 52.Nf7 Kc4 53.Kg6 Kxc3 54.Nh6 Be6 55.Nf7 and the pawn promotes.

52...Kd4 53.Kf5

I appeared to be oblivious to the technique mentioned above.

53...Kc5 54.Ke4 Kd6 55.Kd4 Ke7

White to move

56.c5!

I found the only winning move.

56...Kf6 57.Nd7+ Kxg7 58.c6!

And again, only one move wins.

58...Be6 59.Ke5

59.Kc5 also wins

60.Nc5! Be2 61.c7 Bg4

White to move

62.Ne6+

Black's last few seconds of time ran out.

1-0

Working through Dvoretsky

Dvoretsky's Endgame Manual by Mark Dvoretsky, and now revised by Karsten Mūller, is reputed to be too difficult for most players below master. I was a C-Class player (~1500 USCF) when I bought and started studying the first edition of Dvoretsky's Endgame Manual (2003) the year after it came out. Several years later, perhaps about the time that I became A-Class (above 1800), I created cards for reviewing and training pawn endings--all of the blue diagrams in the first chapter. I would be lying if I claimed that I can instantly play correctly all 48 positions despite many hours with these cards, the print copy of the first edition, and the e-book edition of the second edition. However, I do know well at least 75% of these positions and this knowledge has served me well in many games.

I also created similar cards for the first 24 blue diagrams in the rook endgame chapter. These cards served as prompts for some endgame lessons with some young chess students during the spring break chess camp earlier this month (see "Stockfish's Curve Ball"). My difficulty with some of the positions convinced me that it was time to revisit the rook chapter of Dvoretsky's Endgamne Manual. 

I do not disagree with those who opine that the book is not aimed at players of my strength. However, I must insist that it depends on how you read. It might take me a month or longer to get through the 24 blue diagrams (now gray in the fifth edition [2020], the one revised by Mūller after Dvortsky's death). Some of my lessons with a student the past year had us working thorough all of the rook endings in Jesus de la Villa, 100 Endgames You Must Know, 4th ed. (2015) and then I worked through all the positions in Thomas Engqvist, 300 Most Important Chess Positions (2018)., which included a fair number of challenging rook endings. I believe that I'm ready for the advanced work in Dvoretsky.

I do not expect to get through the entire chapter on rook endgames. For the time being, however, I am pushing through carefully and slowly. Some days I study one or two positions. I read Dvoretsky's analysis and then practice many of the positions against Stockfish. 

Tuesday, I worked on exercise 9/4. First, I studied the diagram. Then, I set it up on my iPad and studied it longer. The clock started when I made my first move.

Playing against an engine usually presents information that play against humans never reveals. The engine presents its evaluation of the position. 

Black to move

I first considered 1...Ra6, but saw that White would force an exchange of rooks or win my pawn for no cost and leave me in a hopeless position of passive defense against a passed f-pawn.

Then, I considered putting my rook behind the king or the pawn, but saw that would fail.

The key was understanding that my king needed to move to the short side.

1...Kf8!

At first Stockfish on the iPad gave the evaluation as 2.6, but that quickly dropped and after several seconds, settled at 0.1. I knew my move was correct.

2.Rb6 Ra1 3.Rxf6 Kg8

Short side is the only move that holds.

4.Re6 Kf8

Rg1+ is also okay. Occupation of the promotion square should be sought when possible, moving to the short side when forced by checks or mate threats.

5.Rb6 Rf1

Alternately, Black can play 5...Rg1+ 6.Kf6 Kg8

6.Rb8+ Ke7 7.Rb7+

The point of 5...Rf1 was that f6+ would be met with Rxf6+.

8.Rb2

Black to move

Matters remain critical: Black must hold the rook on the f-file, move the king to e7 again, or throw in a check that leads to a mate threat and then retreat to the short side.

8...Rf3 9.Rb8+ Kf8 10.Rb7+ Kf8 11.Rf7+ Kg8 12.Rg7+

Black to move

12...Kh8

This move holds, but the defense might be easier after 12...Kf8. For instance, 13.Rh7 Rg3+ 14.Kf6 Kg8

13.Re7 Rg3+

Only move.

14.Kf7 Ra3

Now, I was forced to return the rook to the long side.

15.Rc7 Ra1 16.Kg6

Black to move

16...Rg1+

Again, there was only one move that held.

17.Kf6 Ra1 18.Kf7

The pawn's advance can no longer be prevented, but it cannot yet advance far enough for victory.

18...Ra2 19.f6

Black to move

19...Kh7

This move was necessary with an important idea that is revealed in the next few moves.

20.Re7 Ra3 21.Rd7 Ra2 22.Ke6+

Black to move

22...Kg6

This move--a vital defensive resource--was only possible because of 19...Kh7. 22...Kh6 also holds, but I did not consider it as threatening the pawn should the king step away seems best.

22...Kg8 loses to a technique that is easy to learn. 23.Rd8+ Kh7 24.f7. With the rook controlling the promotion square, White's king deals with checks from the rear by marching towards Black's rook. When the checks run out, the pawn promotes.

23.Rd6 Re2 24.Kd5 Kf7

24...Rd2+ seems simpler.

25.Rd7+ Kxf6=

Exercise 9/5 proved more difficult. I needed quite a few take backs to find my way.

White to move

The position appears at least twice in ChessBase's Mega 2024. Dvoretsky references Rohde,M. -- Cramling,D., Innsbruck 1977, where Rohde was not successful in winning. 

Dvoretsky also mentions analysis of the position by Nikolay Kopaev. I am able to find this analysis in Harold van der Heijden's Endgame Study Database VI (2020), but not volume 5 of Comprehensive Chess Endings by Yuri Averbakh and Kopaev.

"White wins, but with some hard work," Dvoretsky writes (398).

Dvoretsky's Endgame Manual is indeed challenging, but it is not out of reach for a class player looking for instructive practice against the silicon beasts.

Stockfish's Curve Ball

During chess camp last week in a session ambitiously called "advanced rook endgames", I had the students choose a position from a stack of cards that were face down. These cards have 24 endgame diagrams, some of which specify the side to move. One of those they chose, I had regarded as reasonably simple. I set it up in the Fritz GUI with Stockfish 16 loaded as the main engine, and we (the students with my guidance) played it against Stockfish.

The engine defied Dvoretsky, who wrote, "Black's rook must remain passive, staying on the eighth rank" (Dvoretsky's Endgame Manual. 5th ed. [2020], 161). I explained to the students what I recalled of Dvoretsky's explanation, and drew the arrows visible in the diagram.

White to move

From my lead, the students chose the first move well.

1.Rb7 Rd8 2.Rh7

Only after struggling against the engine's surprising response did we come to understand that 2.Rg7+ is better.

2...Rd6!

3.f7 is not possible. We tried 3.Rg7+ Kf8 4.Rh7 Kg8 and realized something else was needed.
3.Re7 Kf8 confused us, although it should not have.
After 3.Rb7 Rd8, we felt that our progress was still missing something.

After several trials and error, we found the correct idea.

3.Rg7+ Kf8 4.Ra7 Rd8 5.Rh7

Black to move

Now Black must allow either the skewer or f7+ followed by the skewer.

Playing several Lucena positions against Stockfish over the past few years, I've learned that the engine often diverges from the line given in the textbook. Often this complicates the solution and the human must solve a problem.

These curve balls are why computer training is valuable.

Applied Study: Lesson from a Youth Tournament

After spending parts of several days last week on a particular endgame offered in Thomas Engqvist, 300 Most Important Chess Positions (2018), it arose in a game that I was watching during a youth tournament yesterday. I attempted to take a photo of the position, but the student on the move made his move before I snapped the image.

Black to move

The image on my phone combined with my memory allowed me to set up the game on my iPad last night and play it out against Stockfish. I hope some students can learn something from this analysis. An exclamation  mark after a move indicates that the move played is best, or equal best, as measured by the shortest distance to checkmate.

As usual, the moves in bold are from my game against Stockfish. Other moves are suggested improvements.

1...Kg5!

1...Kf5 was played by the child 2.Rc5+ wins, but was not played. Eventually, Black was able to promote the g-pawn, forcing White's rook off the board.

2.Kg2 h4

I remembered this idea from Engqvist, but it is not the engine's top choice as it extends the distance to mate by three moves.

2...Rb4 is best.

3.Rd3 Rf5

3...Ra4

4.Rb3 Rc5

4...Rd5 is slightly more accurate.

5.Rb2

Black to move

5...Rc4

I considered but did not fully calculate 5...h3+ 6.Kg3 Rc3+ 7.Kh2 Kh4 8.Rb4 Rc2+ 9.Kh1.

6.Ra2 Kf4! 7.Rf2+ Ke3! 8.Rb2

Black to move

Rd4!

I'm hoping to exchange into an easy pawn ending.

9.Rb1 Rd2+! 10.Kg1 g3! 11.Rb3+ Rd3!

Six consecutive best moves are the result of book study and practice against the computer.

12.Rb8

Black to move

12...h3

My move is based on an idea in the book: threaten back-rank checkmate, but there was a faster way to accomplish this goal.

12...Kf3 13.Rf8+ Kg4 14.Rg8+ Kh3 15.Rf8 Rd1+ 16.Rf1 Rxf1+ 17.Kxf1 Kh2.

13.Re8+ Kd4! 14.Rd8+ Kc3

14...Ke4 15.Re8+ Kf5

15.Rc8+ Kd2 16.Kf1

Black to move

16...Rf3+! 17.Kg1 Re3

17...h2+ 18.Kg2 Rf2+ 19.Kxg3 h1Q

18.Rd8+ Ke2! 19.Ra8 Rd3! 20.Re8+ Kf3!

The king marches toward the rook until it runs out of checks.

21.Rf8+ Kg4! 22.Rg8+ Kf5! 23.Kf1

Black to move

23...Ra3

23...h2

24.Rf8+ Kg6! 25.Rb8 g2+ 26.Kf2 Ra2+! 27.Kf3

Black to move

27...Kg7

27...g1Q?? 28.Rg8+ and a draw.
27...h2!

28.Rb7+ Kf8! 29.Rb8+ Ke7! 30.Rb7+ Kf6

30...Kd6

31.Rb6+ Ke5 32.Rb5+ Kd4! 33.Rb4+ Kc3! 34.Rb1

Black to move

34...h2! 35.Rc1+ Kd4

35...Kd2

36.Rd1+ Ke5!

Finding shelter behind White's king

37.Re1+ Kf5! 38.Re8 h1Q!

The rest should be easy.

39.Rf8+ Ke6! 40.Re8+ Kd7! 41.Kf4 Qh4+! 42.Kf3 Rf2+

42...Kxe8

43.Ke3 Qf4+

43...Kxe8 44.Kd3 g1Q

44.Kd3 Rd2+ 45.Kc3 Qd4+ 46.Kb3 Rb2+ 47.Ka3 Qb4#

Finishing with the queen and rook roll from Pandolfini's Endgame Course (1988), and excellent endgame resource by chess coach Bruce Pandolfini for young players.

Knight and Bishop Checkmate: How and Why

Is is necessary to know a checkmate that you will never need to execute? Aside from deliberate choices due to underpromotion, I have never had to checkmate with a knight and a bishop in a game. I was on the weaker side in a blitz game on Internet Chess Club 25 years ago and got a draw because my opponent did not know how to perform the mate. I saw one young student try and fail in a youth tournament, and then in the very next round have the ending again, this time with the lone king and another draw.

In one tournament that I won, my last round opponent had failed to execute the checkmate in the previous round, else he could have won the event with a draw against me.

Many players never face it.

In Silman’s Complete Endgame Course: Beginner to Master (2007), Jeremy Silman writes:

I heretically decided not to include Bishop and Knight vs. Lone King because it’s far from easy to master, and it occurs very rarely in over-the-board play. In fact, I only got it once in my entire career, while IM John Watson and IM John Donaldson never got it at all! Is such a rarity really worth the two or three hours it would take to learn it? I say no. (xv)

On the other hand, Thomas Engqvist in 300 Most Important Chess Positions (2018) considers it of such importance that three of the 300 are concerned with this checkmate. Acknowledging its rarity (“it will probably happen once or twice in your life”), Engqvist states:

I used to teach children this endgame at a very early stage and they should know how to do it before they reach an Elo rating of 1500 to get a good feeling for how knight and bishop can harmonise together like a bishop pair. (184)

Practicing positions with few pieces develops understanding and appreciation of each piece’s unique capabilities. As Vasily Smyslov stated in Vasily Smyslov: Endgame Virtuoso (1997), "the properties and peculiarities of the pieces are most clearly revealed in the endgame" (6). Most chess study positions involve knowledge and skills that will be applied to other positions.

Yesterday, using the useful “Table of Computer Database Results for Pawnless Endings” in Karsten Müller and Frank Lamprecht, Fundamental Chess Endings (2001), I set up the knight and bishop ending that is longest distance to checkmate and played it against Stockfish. After my first five moves, I reached this position and noted the wall created on the central squares by the bishop and knight.

Black to move

I also worked through the three positions in Engqvist’s book. The first two involve the technique I have learned, forgotten, and relearned (see "Bishop and Knight Checkmate"). The third position is presented to teach the technique in lessons for beginners on Lichess that I have criticized it as more difficult.

It comes from Andre Chéron, Lehr-und Handbuch der Endspiele (1964). With Engqvist’s encouragement, I decided it was worth learning. It was not as difficult as I had thought and I spent some time practicing against the computer.

Then, I searched ChessBase Mega 2024 for endgames that reached a knight and bishop against a lone king. There are many more than I expected. Finding one that ended in a draw, I started going through the game. At several points where I thought I saw an improvement, I played from that position against Stockfish. Then, I used Stockfish and tablebases to identify areas where my play could be improved.

This exercise showed me that knowing more than one technique for executing the checkmate had practical implications. In a game, one might find one of the techniques is much faster than the other.

Here are some positions from that game. I do not know how much time pressure existed when the player was unable to finish successfully. An internet search turned up tournament information that listed the Black player’s rating as 1850.

Black to move

Black can force checkmate in 8 moves. The game went on another 20.

Nine moves earlier, Black had a mate in 9.

Black to move

A few moves prior, Black had a mate in 15, did not play it perfectly, but made progress as White did not offer the best defense.

Black to move

The closest Black came to finishing was mate in seven.

Black to move

109...Be3 (or Bf4, Bg5, Bh6) should be played, forcing White to the a-file. 110.Ka3 Kc3 111.Ka2 Bc1 and the bishop occupies the critical diagonal. 112.Kb1 Nd3 113.Ka2 Kc2 114.Ka1 Bb2+ 115.Ka2 Nb4#.

Students wanting to learn this checkmate could start with positions where checkmate can be forced in three to six moves and build from there.