Basic Endgames Teach How To Tie Together Mathematics And Logic

In the game of chess, each lowly Pawn has the potential to promote to a powerful Queen by advancing all the way to the 8th rank. Also, there’s a remarkable rule that if one side cannot make any legal moves, the game is actually a draw, rather than a loss for the paralyzed side. These two facts create the phase of a chess game called the endgame, where a player has the opportunity to out-think and out-trick the opponent.


Chess has a well-deserved reputation for being a game of logic. Indeed, fundamentally the game really is a matter of logic, in the sense that everything is about managing the fact that everything boils down to “if I do this, then she can do that, but then I can do this other thing”, and therefore a decision tree of immense breadth and depth. Nowhere is this more true than in the endgame, where being one move ahead of the other side may mean the difference between a win and a draw: and in fact, being one move ahead does not always win, but sometimes even loses (in situations called Zugzwang where getting somewhere first means the other side can make a waiting move and then pounce).

For example, a basic endgame position everyone must learn is the following King and Pawn versus Pawn position. Black to move, there is only one move that draws; the other two moves lose.

This is a perfect position to use to teach children how to think logically, even if they don’t otherwise play chess. They don’t even need to know how to checkmate with a Queen against King. You can just teach them how the King and Pawn work, and set the goal for White as being to get the Pawn to the 8th rank without its being captured. In fact, I think chess would be much more useful in teaching logic if play was arranged starting from simplified positions in endgames, skipping the much more complex phases of the opening and middlegame.


Once a chess player begins applying logical reasoning, an observant player will observe that she is reusing certain patterns in reasoning again and again. This is where reasoning about reasoning, or meta-reasoning, comes in. The concept of “taking the opposition” in chess is one of the simplest examples. In the position above, Black draws by arranging it so that if White’s King advances, Black’s King is in position to “take the opposition” and prevent further progress. So the principle of opposition is not a part of the game of chess, but part of how we can reasoning about the game of chess. A chess player could in theory just apply the “rule” of opposition to play chess well, but without actually understanding why it works, would be missing a huge part of what chess is about: discovering patterns, proving facts about them (this is the “meta-reasoning”), and applying the patterns as building blocks.


This leads to the topic of mathematics in chess. I take the point of view that certain ways of effectively making decisions in chess amount to doing mathematics, going beyond just logic: arithmetic, algebra, geometry. There are many connections to be made here that, when made explicit, can greatly aid in transferring skills out of chess itself.

For today, I’ll just mention a connection with arithmetic and geometry. In the position below, White to move can win, but only by very precise play. The aim is to prevent Black from taking the opposition, and then for White to take the opposition and reduce the problem to the previously mentioned position. The concept of reducing to a previously proved fact is fundamental to logical reasoning, of course. So where does the mathematics come in?

First of all, it must be understood that there is a race between the two Kings to get to one of the critical squares in front of White’s Pawn that will determine whether White can win: White must get the King to d6, e6, or f6. So there may be some kind of counting implicit in whatever logical reasoning is used.

From a geometrical point of view, what is important to understand is that because Kings have to move either horizontally, vertically, or diagonally, “distance” on the chess board is not the same as the “bird’s eye view” visual spatial distance: chess operates on a more abstract geometrical space where, for example, all things being equal, diagonal moves can get a King somewhere much faster than just horizontal or vertical moves.

Arithmetic comes in to tie in this geometric insight with the logic-based goal-setting and reduction: the simple way to determine whether this position is a win for White is to count how many moves it takes to reach a desired square, and to count whether Black can stop this. Arithmetic is basically a meta-reasoning shortcut for otherwise engaging in low-level “if this, then that” logical reasoning. Here, we see that White can, in 3 moves, reach d5 unimpeded, because in 2 moves, Black can at most reach f6. Then we tie up the reasoning with one bit of logic/geometry: after White’s King is on d5 and Black’s King on f6, Black’s King must go to e7 to prevent White from getting to d6. But then this allows White to get to e5, taking the opposition and winning the game.

I believe that this endgame position is very instructive for showing how to apply multiple levels and styles of logical and mathematical understanding to be able to guarantee a desired result. Any student who can master (as tested by playing out as either side to the optimal result) and be able to explain the evaluation of each position in which the Pawn is on e4 and the other Kings are on any other squares on the board will have demonstrated a real understanding of logical reasoning.

Franklin Chen

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About Franklin Chen

Franklin Chen is a United States Chess Federation National Master. Outside his work as a software developer, he also teaches chess and is a member of the Pittsburgh Chess Club in Pennsylvania, USA. He began playing in chess tournaments at age 10 when his father started playing in them himself but retired after five years, taking two decades off until returning to chess as an adult at age 35 in order to continue improving where he left off. He won his first adult chess tournaments including the 2006 PA State Game/29 and Action Chess Championships, and finally achieved the US National Master title at age 45. He is dedicated to the process of continual improvement, and is fascinated by the practical psychology and philosophy of human competition and personal self-mastery. Franklin has a blog about software development, The Conscientious Programmer and a personal blog where he writes about everything else, including his recent journey as an adult improver in playing music.