Ain't it funny how the knight moves
Garry Kasparov and Anatoly Karpov (with a tentacle of Garry's octopus knight embracing Anatoly)

Ain't it funny how the knight moves


At times it seems like no one knows how the knight moves.

Even masters find it hard to explain.

Howard Staunton, the man who gave us the English opening, said that the knight's "move is one square in a straight line, and one in an oblique direction..."

According to Wilhelm Steinitz, the first world champion, the knight "moves or captures from the square where he stands to any third square of an opposite colour to the one from which he started, by skipping one diagonal square and then landing on the next square of the same line or row, or vice versa."

"The move can be said to consist of a double move, first moving one square in the manner of a rook and then one square in that of a bishop, but always away from the starting-point" is what Savielly Tartakower offered.

The U.S. chess champion for three decades, Frank James Marshall, said, "Either he goes forward one square and then one square diagonally to the right or left, or he goes immediately to a diagonally right or left square and then forward one square."

British master Hugh Alexander, a man so bright that he cracked the German Enigma machine at Bletchley Park during World War II, maintained that the knight moves "in a direction bisecting the angle between the rook’s and bishop’s move."

FIDE's laws of chess are no clearer: "The knight may move to one of the squares nearest to that on which it stands but not on the same rank, file or diagonal."

The U.S. Chess Federation rules are even murkier. 

The knight’s move is composed of two different steps. First, it makes one step of one single square along the rank or file on which it stands. It does not land on that square, as its move is not complete. Then, still moving away from the square of departure, it moves one step of one single square on a diagonal. It does not matter if the square of the first step is occupied.

This move is sometimes called an L move, as it is equivalent to moving the knight two squares vertically, then one square horizontally (or two squares horizontally, then one square vertically). Note that the knight always moves to a square different in color than that of its starting square. A knight has a maximum of eight possible moves.

Even if these descriptions might be literally accurate (and I have my doubts about at least one of them), it seems to me that none of them are of any use whatsoever in helping us to visualize the knight's move as we're playing a game. Given how poor traditional definitions are in conveying any helpful information, I've been looking for a better way to visualize how the knight moves. 

J.H. Donner

J.H. Donner — The knight moves along a circle

This post is an extended reflection on one way of visualizing how the knight moves, inspired by GM J.H. Donner's observation that "[t]he movement of the knight on the chess board is ... along a circle."

If you've been reading my blog, you might have noticed that this quote by Donner made a cameo appearance in two recent posts, A queen and her knight and Guarini's problem — the puzzle of the four knights.

In fact, it was Donner's quote that led me to blog about Guarini's problem. While the puzzle of the four knights struck me as interesting and while it invites us to explore the history of chess and how mathematics informs our understanding of the game, I wouldn't have blogged about it if I hadn't come across the puzzle soon after I had read a 1972 essay by Donner dedicated to how the knight moves.

Here's Donner's essay, which I've edited down for brevity and which is published in The King: Chess Pieces, a collection of Donner's best articles selected from the thousands that he wrote over more than three decades as a chess journalist.


...The question is whether the knight 'goes' or 'stands'. 'The bishop moves along the diagonal' and 'the rook moves along the horizontal' are utterances that [Tim] Krabbé accepts, but to say that 'the knight moves along the circle' is absurd, according to him.

I will first demonstrate the mathematical incorrectness of his objection and subsequently render it its relative truth, in the hope that my little disquisition may contribute to a deepening of the love for this most paradoxical of the chess pieces.

Strictly speaking, it is incorrect to say of any of the chess pieces that they 'go'. The bishop on b2 doesn't 'go' to the f6-square. 'Going' is covering a route, an itinerary, in which all segments of that route are 'gone over' in chronological order. This chronological element is absolutely absent in chess pieces when seen from the aspect of their geometrical pattern.

Any real 'going' requires a number of moments, several moves. What can therefore be said at most is: the b2-bishop goes to d8 by way of f6.... The bishop moving from b2 to f6 doesn't do so by way of c3, d4 and e5. It jumps from the one square to the other; there is nothing in between the two points....

...[T]he fact that lines such as the diagonal, the vertical and the horizontal can be shortened or extended has its deepest ground in the finiteness of the line as such. The bishop controls a number of squares, a string, in which no 'nearer' or 'further away' can be distinguished (since f6 is just as far away from b2 as a3 - i.e. one move) but which does have a beginning and an end.

[N.b., in graph theory, if two nodes are connected by an edge, then the distance between them equals 1. Since a bishop on b2 can move to any square on the (assumed otherwise empty) a1—h8 diagonal, a graph of the bishop's moves would show that all the squares are connected. So from the bishop's viewpoint, the distance from b2 to a1 is the same as the distance from b2 to f6, i.e., a single move.]

This essential finiteness is the deep tragedy of all chess pieces. Except for the knight! For the string of squares controlled by the knight doesn't constitute a line but a circle, in which neither a beginning nor an end can be discerned. From a purely geometrical point of view therefore, the knight can be said to 'go' along a circle with just as much - or as little - right as the bishop can be said to 'go' along a diagonal and the rook ... etc.

This must be the conclusion when the chess pieces are seen from a geometrical point of view....
Tim Krabbé wasn't the only one to reject the circular course of the knight on emotional grounds. There were many others who had the greatest difficulty imagining what I meant. And there is an obvious reason.

For the peculiar thing about a chess player's thinking is this: he sees movement where there is none. Where the material eye - and the geometrical eye - sees absolute rest, the chess player's mental eye sees a tremendous bustle. To the chess player's eye, the chess pieces are in constant movement, they go ways and paths and have arrived in two moves. It sees the same piece at three different places at the same time. As Nimzowitsch expressed it so eminently: 'To me, the chess pieces have a soul. They have wishes and expectations, which slumber in them unconsciously, and which I must make clear to them.'

The motionlessness of geometry founders in this great turmoil. The board becomes full of pits and holes, and hills and peaks. The diagonals bend and the board is no longer a square either. Ra1-d1 is clearly shorter than Rh1-h4 and I have seen a diagonal bending all of a sudden from b2 to g8, where the enemy king was standing, when a pawn disappeared from d4.

It is the writhing world of geometry in a carnival mirror, where one of the first things to disappear is the circle. And yet, the aura of the circle remains noticeable around the knight in the chess player's groping, naively practical, typically prelogical thinking. In the direct experience of chess-thinking, the dynamics of the knight differs from that of the other pieces.

The bishop has to clear a way for itself, as a result of its essential finiteness. It appears to us as movement par excellence - as its Dutch name of loper (walker) indicates - precisely because it is constantly hindered in its movement. It 'wants' to go from b2 to f6, but there is a pawn on d4 in the way. The bishop feels hindered, because it already sees itself on f6.

With the knight there is no such hindering, because of its essential infiniteness. It is the paradox of this piece that while it is the most jumpy, it is by nature also the most static. We all know those games in which a knight on d4 or e5 takes part in the battle, immovable on its post from the opening through the middle game until far into the endgame, while a destructive war rages around it with great and terrible annihilation.

That is a knight best deployed. Every other piece not played for twenty moves or more is a poor thing, but the opposite goes for the knight: a knight often played cannot find the place where it belongs....

This is the basic paradox of the knight: it 'goes' because it 'stands'. This standing reflects the steadfastness of liberty, deep silence in Absolute Turmoil. It is the image of Divine Quintessentia itself.

Schaakbulletin 60, November 1972

Frankly the first few times I read this essay, I wasn't sure what to make of it. Donner can be an outrageously provocative polemicist. At times it can be tough to tell whether his iconoclasm is serious or sardonic. In his excellent memoir, Smart Chip from St. Petersburg: and other tales from a bygone chess era — the same book that profiles Genrikh Chepukaitis, whom I wrote about in The legendary Genrikh Chepukaitis — Part 1, the welder from Saint Petersburg — Genna Sosonko recalls Donner as a man "attracted by paradoxes and extraordinary, often contradictory opinions" that were original even when they were rubbish.

Even if Donner was serious, it's not entirely clear whether we should take his opinions seriously. Bobby Fischer reputedly once remarked that Donner was the weakest grandmaster in the world. (Then again, Donner beat Fischer at the 1962 Olympiad.) Even Donner's friend Tim Krabbé (who collected the pieces in The King) published a collection of Donner's miniatures — not games Donner had won but the long list of his many defeats in 25 or fewer moves. 

Still the essence of Donner's argument — that the knight moves along a circle — struck me as an interesting way to visualize how to use the knight in a game. To me, his way of looking at the knight is a heck of a lot easier to understand and apply than saying that the knight makes some crazy double move or that its move bisects that of the rook and the bishop. Donner's argument was intriguing enough that I started looking deeper into his idea.

Emanuel Lasker

Emanuel Lasker

Donner isn't alone in viewing the knight as projecting a circle of force.

Long before Donner, Emanuel Lasker, the longest reigning world chess champion, viewed the knight's power as forming a circle, as he wrote in Common Sense in Chess:

  • "As a general rule, it is not good policy to exchange in the early stages of a game the long reaching Bishop against the Knight, whose power does not extend beyond a certain circle."
  • The Knight's "reach never exceeds eight points, situated in a circle."
Jonathan Rowson

Jonathan Rowson

More recently, in The Seven Deadly Chess Sins (a book that @Silman says is "a MUST own!"), GM Jonathan Rowson builds on Donner's analysis. In his chapter devoted to the chess sin of materialism, here is Rowson's discussion of the knight:

I once asked GM Paul Motwani, "If you were a chess piece, which would you be?" Paul replied that he'd be a knight, because it can get everywhere, albeit slowly. This is perhaps why the knight, which controls far fewer squares than a bishop in the centre of the board (8 compared to 13) is considered to be of similar value, because it is limited only by its relative mobility, which is slow, rather than its ability, which is essentially unlimited. Of course it may also be related to the knight's ability to 'jump', especially over pawns which can block much mightier pieces.

The most important feature of a knight from a tactical point of view is that the way it moves is not related at all to any of the other pieces and so it can attack as many as eight squares without being attacked by any piece on those squares in return. It's also worth remembering that a knight attacks squares of an opposite colour to that on which it sits.

That said, to see the unique value of the knight we need a geometrical perspective. If we try to imagine chess without knights, we find an impoverished game with lines, squares, files, ranks and diagonals, but no curves. We should be thankful to the knights, for they are the curvy pieces that bring a circular aspect to an essentially linear game.

Judging by Donner's account in The King, there was a rather heated dispute in Dutch chess around the early 1970s concerning the geometry of the knight. Some saw the knight as the bisector of the bishop's diagonal and the rook's line, but this, according to Donner, overlooks the fact that the knight makes such a short jump. The correct appraisal of the knight in Donner's eyes is that it "moves along a circle".

The circle can be seen, with a sympathetic eye, on the diagram above. The sense in which it "moves" is related to its "essential infiniteness"... "For it is the paradox of this piece that while it is the most jumpy, it is by nature also the most static" ... "Every other piece not played for
twenty moves or more is a poor thing, but the opposite goes for the knight: a knight often cannot find the place where it belongs."

With this in mind, I tend to think of the knight as a lazy cowboy on top of a horse which can move, but does so only in short bursts, and usually with some coercion. The cowboy stands in the middle of a field with a lasso, and is capable of controlling the circle around it by virtue of the threat of reining in any of the opponent's pieces that would dare step into that circle. Thus to my mind the knight is a fascinating piece with an intriguing personality.

Once I discovered that a world champion like Lasker and a modern GM like Rowson each talked about the knight controlling a circle around itself, I gained greater confidence that Donner's concept that the knight moves in a circle might prove useful.

A sculpture by Leigh Dyer of an octopus capturing a rook at Chess Square in Hastings, England

The octopus knight

What really persuaded me that Donner was onto something was when I realized that Donner was describing exactly the same device that GM Raymond Keene has christened as the octopus knight.

Donner didn't merely say that the knight moves along a circle. He went further, arguing that the knight "goes" along a circle because it "stands" in the center of a circle. Once again, here is the conclusion of Donner's 1972 essay, which took me far too long to finally understand:

It is the paradox of this piece that while it is the most jumpy, it is by nature also the most static. We all know those games in which a knight on d4 or e5 takes part in the battle, immovable on its post from the opening through the middle game until far into the endgame, while a destructive war rages around it with great and terrible annihilation.

That is a knight best deployed. Every other piece not played for twenty moves or more is a poor thing, but the opposite goes for the knight: a knight often played cannot find the place where it belongs....

This is the basic paradox of the knight: it 'goes' because it 'stands'. This standing reflects the steadfastness of liberty, deep silence in Absolute Turmoil. It is the image of Divine Quintessentia itself.

Perhaps the best demonstration of Donner's idea that the knight "goes" because it "stands" is a game played 13 years after Donner's 1972 essay, namely, game 16 of the 1985 KasparovKarpov world championship match, the game that led Keene to coin the phrase, the octopus knight.

Since I'll be doing a deep dive into this game in a future post, for now let's admire the game without annotations. As you play through the moves, pay particular attention to Kasparov's queenside knight that ends up on d3. 

Just as Donner described, Kasparov's octopus knight remained motionless on d3 for 17 moves, calm in the center of the storm raging around it. 

As Keene wrote, "[t]his piece starts out as a knight, but shortly transforms into a monstrous centralized octopus, tentacles grasping out in all directions, hovering over the key squares in White's position." 

The five classical elements — fire, air, water, earth and ether

The quintessential knight

Understanding the connection between Donner's "the knight 'goes' because it 'stands'" and Keene's octopus knight enabled me to finally get why Donner calls the knight "the image of Divine Quintessentia itself."

The ancient Greek philosopher Empedocles proposed that the world consisted of four elements: fire, air, water and earth. 

In his work On the Heavens, Aristotle argued that the heavens consisted of a new element, which he called ether. In contrast to the four corruptible earthly elements that were subject to decay, Aristotle's ether was pure and unchanging. 

In the Middle Ages, when European scholars translated Aristotle's works into Latin (often from versions preserved in Arabic translation), they coined the term Quintessentia (from quinta essentia, the fifth element) for Aristotle's new element ether because it was the fifth element after the four earthly elements. 

We now know that Aristotle's ether doesn't exist. That was established by the Michelson–Morley experiment. (A few decades ago, when I first started rock climbing, I visited the plaque outside of Idyllwild, California marking the spot, which climbers call the Relativity Boulders, where Albert Michelson, America's first Nobel laureate in the sciences, conducted this experiment.) 

While ether doesn't exist, Quintessentia survives as a concept. The history of Aristotle's ether has given us the English word quintessential, meaning something that perfectly represents the best of its kind.

And, in this context, we now know why Donner called the knight the image of Divine Quintessentia itself. Aside from pawns, which are not pieces, chess consists of five pieces: the king, the queen, the bishop, the rook and the knight. The first four pieces are alike in that they all move linearly. 

The knight is unique. However we might describe its crazy zigzag move, it, unlike the four other pieces, does not move in a straight line. Not only does the knight move along a circle, rather than along straight lines, but, when its fulfills its potential and self-actualizes its soul's hidden desires, it stands unchanging and motionless at the center of its own circle while the other pieces battle around it. It truly is the fifth element, unlike the four other more common pieces.

That, I believe, is what Donner was trying to convey when he called the knight the image of Divine Quintessentia. 




I'd like to tip my hat tip to chess historian Edward Winter for collecting the many different descriptions of how the knight moves, svarog989 on for photoshopping that octopus tentacle into the photo of Kasparov and Karpov and Bob Seger for inspiring the title of this post.

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If you liked this post, be sure to check out my related post on the knight, Guarini's problem — the puzzle of the four knights.