Chess , Maths, Formulae and Endings

GM Akshayraj_Kore
Jun 3, 2014, 1:06 AM |

In order to improve your game, you must study the endgame before everything else. For whereas the endings can be studied and mastered by themselves, the middle game and opening must be studied in relation to the end game.

-          Capablanca, 3rd World Chess Champion

Most of you should know that the following position is a draw.

Studying Endgames is crucial. Especially for younger players. While many of them might know this, I have seen in praxis that most of them choose to neglect this area of the game. I think one of the reasons is that it is boring, tedious and simply uninteresting to study endgames where there are far fewer pieces on the board than in the opening or middle game. And this naturally develops a distaste in younger minds who are easily lured by tactics and traps in openings and middle game. 

I think one of the other reasons, albeit a major one, might be that players just don’t know how to make use of those endgame positions that they have studied. I have heard my pupils complaining that they often not get endings in their games and that’s why they don’t want to study them. 

What I have come to conclude from these grievances is that players find it hard to relate the endgame positions that they study. We all know that pattern matching is one of the strengths of better players. Strong Grandmasters and elite players are able to relate the position on the board from their  memory and thus find the best possible move in the given position. As a result they have to spend lesser time over the board and thus it improves their decision making process. 

My approach studying with endgames is rather a mathematical one.  Mathematics is one of my favorite academic subjects. Back in India, when I was doing under grad level mathematics, one of the important aspect of getting better at Maths was, just memorizing the formulae. I had memorized hundreds of formulae in Trigonometry, Derivatives and Calculus among others. And this made understanding Maths and getting good at it easier. So whenever I could reduce the equation using these formulae, I did it and got a step closer to the final answer. 

The approach is very similar in Chess. I think Chess Endgames are analogous to formulae in Mathematics. There are simple formulae and then there are big and complex formulae. Take for example, the following position –

White to Play. 

I had seen this position some time ago. The first reaction to this position is that white is winning effortlessly. While this empirical observation might seem intuitive, on a closer introspection, we see that things are not so easy for white and that by best play by Black, he is able to draw the game. 

    1.       Kd4 Kd6 2. b5 f6 3.b6 g5 4.b7 Kc7 5.Kd5 g5 6. Ke6 gh4 7.gh4 Kb7 8.Kf6 Kc7 9.Kg6 Kd7 10.Kh5 Ke7 11.Kg6 Kf8 and Black draws


Or White can Play

1.       b5 Kd6! 2. b6 Kc6! (2…f6?? 3.Kb5 and white Queens his b-Pawn with Ka6-a7) 3.b7 Kb7 4.Kd5 f6  5.Ke6 g5 6.Kf6 gh4 7.gh4 Kc7 8.Kg6 Kd7 9.Kh5 Ke7 10.Kg6 Kf8 and Black Draws 

Yes. Now, we see that Black’s best defense here is to exchange the pawns on the kingside to bereft white of any winning material, other than the Rook pawn, when Black’s defense becomes much easier.  All Black has to do after that is to reach the iconic position,  that we saw at the beginning of this post. 

In the second round of the just Concluded Chicago Open 2014, I was able to use this “knowledge” . I was playing against one of the promising sub juniors in US – Awonder Liang and the following position arose in my game.


Kore – Awonder Liang, Chicago 2014.


In the middlegame, I was able to win a pawn, but by stiff resistance from the little guy, I was compelled to convert the extra pawn in a Queen Endgame. 

I want to stress the importance of how the knowledge of the Pawn Endgame I showed helped me in my Calculations and also in my decisions. Here, White has to make an important decision. He can either go 1.Ke3 – saving the pawn on f2 , hide the king and try to find an alternative plan to make things work for him again. 

Or, a serious alternative is 1.Kc4 Qxf2 2.Qc5+, which transposes into a pawn ending that closely resembles the one we just saw. During the game, I was tempted to play 1.Kc4 almost instantly. But, call it sixth sense, intuition or just experience, I somehow remembered the  endgame I just showed you and my radar went off. However, I noticed that Black’s King is not as active as in the diagram and I persuaded myself to check into this ending. 



So, After 2. Qc5+ Qxc5 3.Kxc5 Black can play

a)      3…. f6

b)      3….Kc7

c)       3….Kd7

 a)3…f6  4.Kd6 g5 5.Ke6 gxh4 6.gxh4 Kc7 7.Kxf6 Kb6 8.Kg5 Kxb5 9.Kxh5 Kc6 10.Kg6 Kd7 11.Kf7 and white wins +- 

 b) 3...Kc7 4.Kd5 Kd7 5.Ke5 Ke7 6.b6 f6+ 7.Kd5 Kd7 8.b7 Kc7 9.Ke6 g5 10.Kxf6 gxh4 11.gxh4 Kxb7 12.Kg6 Kc7 13.Kxh5 Kd7 14.Kg6 Ke8 15.Kg7 And white is just in time. +- 

c)3…Kd7 4.Kd5 4...f6 5.b6 g5 6.b7 Kc7 7.Ke6 gxh4 8.gxh4 Kxb7 9.Kxf6 Kc7 10.Kg6 Kd7 11.Kxh5 Ke7 12.Kg6 Kf8 And Black gets a Draw !! 

After looking at line (b),  the human mind kind of  wavers to reply 3…Kd7 with 4. Kd5. When I gave one of my pupils this position, it took him about 20 minutes to find that when Black plays 3…Kd7, white’s King goes in other direction with 4.Kb6 ! Kc8 5. Ka7 ! +- and white Queens his b-Pawn . I am not saying that he was slow, but this is phenomena that happens frequently in tournament play. We see that sometimes even the elite players miss winning moves when their thinking patterns are 'set' in a particular direction.

So, here another lesson for us to learn is that, if we get stuck in our calculations, we should try to think in a ‘Different  Direction’ . Since this in itself is a big topic, I would like to elaborate on this particular theme in my next blog. 

To conclude, I was able to draw the position from my memory which helped me in making a key decision during a critical moment of the game.  The decision making algorithm that ran through my mind was very analogous to the numerous mathematical problems that I solved during my undergrad years. The formulae here would substitute for the positions that were burnt in my memory and the problem at hand was the decision – Whether to play 1.Ke3 or 1.Kc4. After that, it was merely some mechanical work (Calculations) and evaluating the outcome of these calculations. 

As, I now progress through my Graduation studies in Computer Science, I sense that this particular process is very helpful in writing programs and algorithms. My work has now reduced down to finding what the problem is and how a particular problem was solved using a code snippet from memory. If it is a new problem, I use different snippets and mesh them together as suitable to get to my desired goal. 

I think this is enough for now, and I would like to take leave from you until my next blog. Stay Tuned.

GM Akshayraj Kore

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