Should have been easy

Late Thursday night during a thunderstorm, I played a few three minute games online. Two draws from dead lost positions became lessons in elementary endings to end the spring term for my before school elementary chess club.

In the first, my opponent ran desperately short of time in a bishop ending where he had an significant advantage in number of pawns. After promoting a pawn and using the queen to capture my last pawn, he had a forced checkmate in seven, but only 4.0 seconds left.

Black to move

50...Qe4+

My opponent opts for a sequence where premoves can be played to save time.

50...Qc7 leads to mate in six moves more.

51.Kd6 f4 52.Kc5 f3 53.Kd6 f2 54.Kc5 f1Q

A second queen appears and 3.0 seconds remain on the clock. It is also forced mate in three. Black has done well and should win.

There are two schools of thought on this position:
1) a decent player resigns in such a hopeless position, and
2) its is foolish to resign in a blitz game when your opponent has three seconds left.

55.Kd6

Black to move

What should a player know about checkmate with two queens?

From the hardest position against a lone king, it is mate in four. Checkmate can occur in the middle of the board. Checks are often the best move.

Most players, however, seem to think the ladder mate is simplest and therefore best. In this case, it can be accomplished with premoves.

55...Qff5

This move does not get closer to checkmate. That is, mate is still three moves in the future.

55...Qf6+ executes an easy ladder mate unless White opts for 56.Kc5 Qfc6#
I would probably play 55...Qa6 in a heartbeat.

56.Kc7 Qee6 57.Kd8

Black still has 2.8 seconds remaining and a simple mate in one.

57...Qff7??

The premove squandered half a point.

The second undeserved draw had gross errors by both players. Time was a less critical factor with both having more than half a minute on the clock.

White to move

58.Rh6??

I would play 58.Rb6

One of very few non-winning moves in this simple rook ending.

58...Rh1+ 59.Kg4 Rg1??

Black here is stuck in unimaginative routine moves that were wholly unnecessary when simply calculation shows that 59...Rxh6 draws easily.

60.Kf4 Rf1+

White to move

I started with this position with the young students, explaining that moving the king towards the rook puts an end to harassing checks. Ke4-d3-e2 (or f6+ if the rook gives up the checks).

61.Ke5 Re1+ 62.Kf4 Rf1+ 63.Kg4 Rg1+

At this point the game is technically drawn, as a repetition is about to occur. It is sad that a player rated in the mid-1900s does not know how to win this ending.

64.Kh5 Rh1+ 65.Kg4 Rg1+ drawn by repetition.

Errors Aplenty

Misplaying a Rook Ending

Why are rook endings difficult? The rook's move is arguably the simplest in chess, yet a rook ending where one side has an advantage of one pawn can prove vexing.

This ending arose in a game-10 Arena on chessdotcom. Both players had slightly more than three minutes remaining after 67 moves. I had Black.

White to move

68.Kf5?

Now, the position is equal. 68.Kd5 leads to a win.

68...Rh1

68...Rd1 also holds the draw.

69.Kg4 Rg1+ 70.Kf5 Rh1 71.Ke4

Black to move

71...Kf6?

71...Rf1 was the only move.

After the subsequent 72.Rf5+, Stockfish gives 72...Ke6 as equal, but an advantage of two pawns after 72...Ke7. However, experience analyzing many rook endings with Stockfish has taught me that anything under +4 should not be considered a decisive advantage. 72...Ke7 likely also hold the draw.

72.d5!+-

White is again winning. The next errors cam after thirteen more moves.

72...Ra1 73.Re6+ Kf7 74.Ke5 Re1+ 75.Kd6 Rf1 76.Re7+ Kf6 77.Re8 Kf7 78.Re7+ Kf6 79.Re4 Rc1 80.Kd7 Kf7 81.d6

Black to move

81...Rc5 82.Re7+ Kf8 83.Kd8 Rxh5 84.d7 Rc5

White to move

85.f5?

85.Re5 secures the full point

85...Rxf5?

Black puts White back in the driver's seat. Four possible moves along the c-file would have held the draw.

Despite Black's failure here, White did not find the right idea, although he came close.

86.Re8+ Kf7 87.Kc7 Rc5+ 88.Kd6 Rc1

White to move

89.Re7+

This move does throw away the win, but White nonetheless missed the right idea.

89.d8Q Rd1 90.Kc5 Rxd8 91.Rxd8 Kf6 92.Rd5 and Black's king cannot help the advance of his pawns.

89...Kf6

Three moves later, the game was drawn by repetition.

Comedy of Errors

It is much better, in fact, to play an objectively less correct game but to be able to win once you've got a winning position.
J.H. Donner, "On the Justice of Chess" (1950), rpt. The King: Chess Pieces (2006), 19.

White to move

White clearly has a winning advantage. After ten moves, the advantage is still there.

White to move

White struggled to find the correct plan to bring home the full point. The correct idea is to march the king to a7 in order to activate the rook. See a discussion of this idea in "Knowledge".

52.f4??=

52.Kg2 Ra2+ 53.Kf1 Kg6 54.Ke1+-

52...Ra2 53.Ra7 Kh6??

Black also seems to be confused about the demands of the position.

53...Kg6 was necessary.

54.f5+- Ra3 55.Ra8 Kh7 56.Ra7 Kh6 57.Ra8 Kh7 

White to move

58.a7??=

This move clarifies matters for Black. Moreover, the correct plan mentioned above is no longer an option. White's only hope is that Black will err.

58...Ra2 59.g4 Ra3+ 60.Kg2

Black to move

60...hxg4??

60...Ra2+ holds. White's king cannot both find refuge from checks and prevent Black from getting a passed pawn that becomes a queening threat. But, Black's haste to get a passed pawn here was a critical error.

To Black's credit, he now knew that his position was lost and what White should do to win. After nine more moves, the critical moment arrived.

White to move

70.f6??=

70.h6 would win. The f-pawn will deflect either the rook from it's critical spot behind the pawn, or the king from security against a skewer.

Although the draw is now assured, the game continued for another 20 moves. I was Black and played this game this morning online at ten minutes for the game.

Two Endings from Informant

Yesterday, I put one of my stronger students through five endings from Chess Informant 163. They proved challenging. This morning, I had three candidate moves for number six. I rejected the move played in the game, but chose the wrong one of the other two.

White to move and win

From Putnam,L.--Kaliksteyn,A., New York 2025

After some discussion concerning efforts that did not work, I found how Anton Korobov drew against Bogdan Deac.

Black to move

 How would you play these two endings?

Think Like a Super-GM

I watched an excessively long video on YouTube concerning the best puzzle books. Nonetheless, the long discussion was often interesting. Most of the books highlighted are useful for masters, but less likely to benefit lower rated players. But there were some, such as the Polgar brick, that also earned accolades.

One of the top choices has proven difficult for strong players, but has improved their game. It reminded me of Jeremy Silman, The Amateur’s Mind because the thoughts of players at different levels were recorded, presented, and evaluated.

That book is Michael Adams, and Philip Hurtado, Think Like a Super-GM (2022). I ordered it Thursday morning, it arrived Thursday evening, and I took it with me on a weekend trip to Montana. I did the first exercise while eating breakfast.

I found the key move, but my analysis needs to go deeper to reach a correct evaluation.

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.

Fighting Spirit

As noted in a couple of recent posts,* I prefer to play on in seemingly equal positions. Yesterday, a player rated above 2000 was not having a good game against me and I won a piece through a fork. Later in the game, my own error allowed a fork that returned the material. Then the exchange of queens gave us an ending with two rooks each after only 28 moves. The game went on another 50 moves.

White to move

Stripes,J (1802) -- Internet Opponent (2011) [D09]
Live Chess Chess.com, 21.03.2025

29.Rxe3 Rf2 30.Rd7 Rxa2 31.Ree7

At this point, I knew that I could not lose and considered forcing a draw.

31...a5 32.Rg7+ Kf8 33.Rgf7+ Ke8 34.Rde7+ Kd8 35.Rd7+ Ke8 36.Rfe7+ Kf8

It may have been here that the first draw offer was made by my opponent. He made two.

White to move

37.Rxh7 Kg8 38.Rdg7+ Kf8

Here, I spent 90 seconds in a ten minute game considering whether I wanted to play on or force the draw, concluding that best play will result in a draw. Nonetheless, with better pawn structure, I risked very little playing on. Moreover, I could force the exchange of one set of rooks.

39.Rxg6 Rd8

Black threatens checkmate in one.

40.Rh8+ Kf7 41.Rxd8 Kxg6 42.Rd5

Black to move

42...Kh5??

Either Ra3 or Rb2 holds things equal.

43.h3?

My failure returns the game to equal. 43.h4 was the winning move.

43...a4?? 44.bxa4+- Rxa4 45.Rxc5 Ra1+ 46.Kg2 Ra2+ 47.Kf3 Ra3+

White to move

At this point, I was fairly certain that my position was winning, but spent another full minute finding the way to proceed.

48.Kf2 Ra2+ 49.Ke1 Rg2

White to move

On the second opportunity to take advantage of the pin, I saw it.

50.h4! Kg4

50...Rxg3 51.Rxg5+ Rxg5 52.hxg5 Kxg5 leads to a position where White must move to the left of the pawn to successfully gain control of the key squares. An example appears on the first page of Dvoretsky's Endgame Manual (2003).

51.Rxg5+

Two connected passed pawns with a rook leading them down the board are nearly always decisive.

51...Kf3 52.c5 Ke3 53.Kf1 Rf2+ 54.Kg1 Rc2 55.Rf5 Rc1+ 56.Kg2 Rc2+ 57.Kh3 Ke4

White to move

58.Rg5

Sensible would be 58.Rf8 Rxc5 59.Kg4 cutting Black's king off from the activity on the g- and h-files. The c-pawn is not important.

58...Kf3 59.Rd5 Rc1 60.Rd3+ Ke4 61.Rd8 Rxc5 62.Rf8 Rc7 63.h5 Rh7 64.Kh4 Ke5 65.g4 Ke6 66.g5 Ke7 67.Rf4

Black to move

67...Rf7

67...Ke6 at least forces me to show that I know what I'm doing. Offering the trade of rooks makes for a simple pawn ending, which the computer finds that I played perfectly. My opponent took the game all the way to checkmate.

68.Rxf7+ Kxf7 69.Kg4 Kg7 70.Kf5 Kh8 71.Kf6 Kg8 72.g6 Kh8 73.g7+ Kg8 74.h6 Kh7 75.Kf7 Kxh6 76.g8Q Kh5 77.Qg3 Kh6 78.Qh4# 1-0

*"Playing Drawn Endgames" and "Equal is not yet Drawn".

When to Trade

This morning in a rapid game, I erroneously rejected the opportunity to exchange rooks into a winning pawn ending. 

White to move

I failed to see that it was indeed a winning ending.

Fortune smiled, and my opponent offered the trade a few moves later.

White to move

This time, however, I needed no calculation to recognize the pawn ending as a win for me.

Playing Drawn Endgames

Do you play out a position that is technically drawn? Twelve years ago, here on Chess Skills, I inquired about an endgame that I was watching live. Levon Aronian made Fabiano Caruana play 36 moves in a dead drawn rook and bishop vs. rook ending. Some research showed that Aronian had won such endings in previous games. When time is short and one side faces no dangers, it makes sense to press and force the opponent to prove their skills in maintaining equality.

Thomas Engqvist makes this point in 300 Most Important Chess Positions (2018) with several examples. First is Duras,O. -- Nimzowitsch,A., San Sebastian 1912.

Black to move

Engqvist writes:

Nimzowitsch tried to make his opponent tired and careless by doing nothing. As a matter of fact there is nothing Black can do. By playing in this "innocent" manner Nimzowitsch managed to fool his opponent. (230-231)

After 24 moves of seemingly pointless shuffling, the players reached this position.

Black to move

Duras erred with 57.Rg5. Nimzowitsch seized the moment with 57...f5 and Duras resigned ten moves later.

Engqvist adds Flohr,S. -- Vidmar,M., Nottingham 1936, where he notes, "In practice it's not easy to defend these kinds of endings and even famous masters and grandmasters make mistakes that eventually lead to a loss" (238). Vidmar resigned 16 moves after his critical error.

Then, in Chigorin,M. -- Tarrasch,S., Budapest 1896, Tarrasch offered a draw in this position.

White to move

"Chigorin correctly declined since White cannot lose anyway", Engqvist observes (240).

After 30.Kf4 Kf8 31.f3 Kg8 32.Ra7 Kf8 33.g4 hxg4 34.fxg4

Black to move

34...Ra1??

34...Kg8 was necessary. Engines need considerable search depth to perceive the error here, but Chigorin appeared to understand the moment.

35.Kf5 Rf1+ 36.Kg6 Rf4 37.g5 fxg5 38.hxg5

In this position, I recognized that White will be able to reach a Lucena position, although that took another ten moves.

My own insistence on playing on when there is the sliver of a chance of eventual victory developed as a consequence of reading Jacob Aagaard, Excelling at Technical Chess (2004) almost twenty years ago. Yesterday, I faced an opponent in online play who pressed when he had a slight edge in a technically drawn ending. I erred. He missed his chance. And then, he took the game almost to stalemate, even offering me opportunities to fall apart in an elementary pawn ending.

White to move

Stripes,J. -- Johnson,F. [A04]
Live Chess Chess.com, 19.02.2025

28.Ra1!= Rxa1 29.Rxa1 Ra8 30.Ke3?!

An inaccuracy

30.Ra6 Rb8 31.Rxa7 Rxb3 32.Rxe7 Rc3 33.Re8+ Kg7 34.Rd8 Rxc4 35.Rxd6=

30...Kf8 31.Ra6 c5

31...Rb8 32.Ra3 a5 33.Kf4 would be more difficult for White.

White to move

32.d5?

32.dxc5 dxc5 33.Kd3=

32...Ke8

32...f5 should win.

33.Kd3 Kd7 34.Kc3

34.g4 is White's best chance. It seems that both of us missed the strength of f7-f5.

34...e6?

34...f5!

I have survived my first error and the game is again drawn with best play.

35.e4 f5 36.dxe6+ Kxe6 37.exf5+ gxf5

White to move

38.Kd3??

38.b4= cxb4+ 39.Kxb4 Rb8+ 40.Kc3 Rg8! 41.c5! Rxg3+ 42.Kd2 Ke5 43.cxd6 Rg7 44.Ke3 Rd7 45.Kf3
38.h5!=

38...h5-+ 39.Ke3 Ke5 40.Ra5

Black to move

40... a6??

Spoils the win.

40...Rb8-+ 41.Ra3 a5
40...Rg8-+

41.b4!=

The rest of the game was played well by both sides and a draw was the appropriate result. I never had winning chances, but there were opportunities for me to go wrong, so my opponent pressed.

41...Rc8 42.bxc5 Rxc5 43.Rxa6 Rxc4 44.Ra5+ d5

White to move

45.Ra8 Rc3+ 46.Kf2 d4 47.Re8+ Kd5 48.Rf8 Ke4

White to move

49.Re8+ Kd5 50.Rf8 Ke5 51.Re8+ Kf6 52.Rh8 Kg6 53.Rg8+ Kf7 54.Rg5 Kf6 55.Rxh5 Rc2+ 56.Kf3 d3 57.Rh6+ Ke5 58.Rh8 d2 59.Rd8 Rc3+

White to move

60.Ke2 Rxg3 61.Rxd2

61.Kxd2 f4 62.Re8+ Kf5 63.Ke2=

61...Rg2+ 62.Ke1 Rxd2 63.Kxd2

And now, in a drawn pawn ending, my opponent goes to remove my pawn, and then makes me show that I know where to place my king.

63...Kf4 64.Ke2 Kg4 65.h5 Kxh5 66.Kf3 Kg5 67.Kg3 f4+ 68.Kf3 Kf5 69.Kf2 Ke5 70.Kf3 Kf5 71.Kf2 Ke4 72.Ke2 f3+ 73.Kf2 Kf4 74.Kf1 Ke3 75.Ke1 Ke4 76.Kf2 Kf4 77.Kf1 Kg4 78.Kf2 Kf4 Game drawn by repetition ½-½

Having lost all four prior encounters against Atlanta coach and chess expert Frank Johnson, I'm very happy with the draw. My work on rook endings with Engqvist's book and Jesus de la Villa, 100 Endgames You Must Know, 4th. ed. (2015) made a difference in navigating difficulties, but did not prevent all errors.

Exhilarating

Blitz chess can reinforce bad habits. It can induce stress. It can produce horrendous play and terrible disappointment. It can be addictive.

Last night, I felt a moment of positive exhilaration when I had a forced mate in three, I won on time with 0.1 seconds left. It was a thrilling moment!

It was a three minute game with no increment. At move 32, White was under one minute remaining. At move 40, Black was under one minute. After move 50, White had 11.4 seconds and Black had 21. The position was equal

51.Kf3 Kf5 52.Ke3? g5?

52...d2 and Black is better.

53.fxg5 Kxg5 54.Rg1+ Kf5 55.Rf1+ Ke5

After 55 moves, White is down to 9 seconds, while Black has 11.8. The position remains equal.

56.Rd1 Ke6 57.Rxd3 Rxd3 58.Kxd3

The exchange restores the material balance, but also makes the game easier, especially for Black.

58...Kd5

Seizing the opposition.

59.Ke3

59.Kd2 Kc4 60.Kc2 d5 was possible and also equal.

It seems that the clock will decide the result.

59...Kc4 60.Kd2 d5

60...Kb3 61.Kd3 and Black almost certainly will err with so little time. 61...Ka3 would be the only move.

61.Kc2 d4

Time for both players is precariously short. White has been moving slightly faster than Black.

62.cxd4 Kxd4 63.Kb3

We have reached a simple pawn ending that should be drawn, although only Black can win on time after White loses the last pawn.

63...Kd3 64.Ka3 Kc3 65.Ka2 Kxb4

66.Kb1??

A game losing blunder, but only if Black does not run out of time. If Black's clock expires, the game is drawn.

66...Kb3 67.Ka1 b4 68.Kb1 Ka3!

Lots of endgame practice and study and I instantly knew that my king had to go to the a-file to preserve the winning advantage. Nonetheless. 0.9 seconds came off the clock.

69.Ka1 b3 70.Kb1 b2 71.Kc2 Ka2 72.Kd3 b1=Q+ 73.Kc4

73...Qf5!

The shortest distance to mate. Again, I found this move instantly because I've been in this position against the computer many times, but it still took 0.9 seconds off the clock.

White had 2.8 seconds to Black's 1.5.

74.Kd4 Kb3 75.Ke3 Kc3

White's last 1.9 seconds expired while Black had 0.1 remaining. The game could have been drawn. What a feeling to play the last 25 moves in 21 seconds!