On the Complementarity of Queens and Knights (Guest Blogger)
Hank, a frequent commenter at this blog, and someone I've played in real life (and usually lost) , gave me permission to publish the following excellent piece he wrote. Beautiful, deep, yet simple ideas. Thanks, Hank!
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I've seen a number of different articulations about piece coordination and the complementarity of certain pieces, that explain why the relative value of individual pieces can vary depending on what other pieces they are teamed up with, and especially how redundancy of function can account for the lower combined piece values of certain configurations of pieces (due to a lesser degree of coordination)...
And in fact it is the lack of redundancy that accounts for the higher combined "scores" of other combinations of pieces. For example, strength of the famous "bishop pair" lies precisely in the perfect non-redundancy of the 2 bishops, which will never overlap one iota in the squares that they cover/guard/control/attack.
Another noteworthy example is the queen+knight combo, in which the 2 pieces are said to work particularly well together.
So I was just idly daydreaming about queen moves when the image below popped into my head:
Basically I was realizing that a queen at the center of a 5x5 grid covers or influences a high percentage of the squares in that grid. It turns out that the queen influences 16 of the 24 other squares in the grid (not counting the center square on which it stands), which is 2/3.
And then it struck me that those "other" 8 squares - the remaining 1/3, which are not influenced/guarded/attacked by the queen - are precisely the 8 squares that a knight would influence from the same center square.
So those 16 squares that the queen controls are either a rook's move or a bishop's move (or a king's move) away - betraying a certain redundancy of influence with those other linear pieces, but the elusive in-between squares that remain inaccessible to the queen are all a knight's move away. And in fact the sets of squares influenced by a centralized queen and knight in that 5x5 grid are 100% complementary.
Anyway, I thought there was something particularly tidy and aesthetically pleasing about this image. It conveys somewhat how a "knight's-eye point of view" would tend to see the world somewhat myopically in terms of "local" 5x5 (or smaller) grids, whereas the "vantage" of a bishop/rook/queen can "take in" up to a full 8-square span or expanse. Or you could say that their "sphere of influence" has a radius of 5, versus (up to) 8 for the stronger pieces. Hence we can also see why knights would be at a disadvantage in endgames with few pieces where action is taking place in different quarters/corners/regions of the board (they might run short of viable targets, or end up in the wrong place).
This kind of rudimentary thinking about the way pieces occupy and interact with space is inspired by my current browsings in Maurice Ashley's new book "The Most Valuable Skills in Chess". It's all pretty basic stuff, but I find it helpful (and entertaining) when a couple of insights can be crystallized in a single image like this.
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I've seen a number of different articulations about piece coordination and the complementarity of certain pieces, that explain why the relative value of individual pieces can vary depending on what other pieces they are teamed up with, and especially how redundancy of function can account for the lower combined piece values of certain configurations of pieces (due to a lesser degree of coordination)...
And in fact it is the lack of redundancy that accounts for the higher combined "scores" of other combinations of pieces. For example, strength of the famous "bishop pair" lies precisely in the perfect non-redundancy of the 2 bishops, which will never overlap one iota in the squares that they cover/guard/control/attack.
Another noteworthy example is the queen+knight combo, in which the 2 pieces are said to work particularly well together.
So I was just idly daydreaming about queen moves when the image below popped into my head:
Basically I was realizing that a queen at the center of a 5x5 grid covers or influences a high percentage of the squares in that grid. It turns out that the queen influences 16 of the 24 other squares in the grid (not counting the center square on which it stands), which is 2/3.
And then it struck me that those "other" 8 squares - the remaining 1/3, which are not influenced/guarded/attacked by the queen - are precisely the 8 squares that a knight would influence from the same center square.
So those 16 squares that the queen controls are either a rook's move or a bishop's move (or a king's move) away - betraying a certain redundancy of influence with those other linear pieces, but the elusive in-between squares that remain inaccessible to the queen are all a knight's move away. And in fact the sets of squares influenced by a centralized queen and knight in that 5x5 grid are 100% complementary.
Anyway, I thought there was something particularly tidy and aesthetically pleasing about this image. It conveys somewhat how a "knight's-eye point of view" would tend to see the world somewhat myopically in terms of "local" 5x5 (or smaller) grids, whereas the "vantage" of a bishop/rook/queen can "take in" up to a full 8-square span or expanse. Or you could say that their "sphere of influence" has a radius of 5, versus (up to) 8 for the stronger pieces. Hence we can also see why knights would be at a disadvantage in endgames with few pieces where action is taking place in different quarters/corners/regions of the board (they might run short of viable targets, or end up in the wrong place).
This kind of rudimentary thinking about the way pieces occupy and interact with space is inspired by my current browsings in Maurice Ashley's new book "The Most Valuable Skills in Chess". It's all pretty basic stuff, but I find it helpful (and entertaining) when a couple of insights can be crystallized in a single image like this.
posted by Blue Devil Knight at 11/18/2009 10:47:00 AM
11 Comments:
In the words of Blue Peter "He's one I made earlier"...
http://chesspodcasts.blogspot.com/2009/11/chess-podcast-70-discussion-1.html
Mike Quigley
"So those 16 squares that the queen controls are either a rook's move or a bishop's move (or a king's move) away"
Is logical if one knows that a queen moves like a bishop and rook combined.
"which are not influenced/guarded/attacked by the queen - are precisely the 8 squares that a knight would influence from the same center square."
This is a new insight for me but i wonder if it has any meaning/ influence chess play wise?
Or it must be that it can help in understanding how knight and queen work together but that i am to stupid to see it.
CT: GMs use these ideas all the time, if you believe what they write, to help them decide what trades to make, who has a "qualitative" material advantage. Soltis has a whole book that covers these topics, but not in a fun and clear and simple way like Hank did.
Quigley it would be nice to have a link to the actual podcast at your site. I don't use itunes so it made it a pain to track down. Or maybe it is there and doesn't show up on Firefox?
It's a nice picture to have in mind. And shows the beauty of chess. The knight "L move", which seems arbitrary when you first learn the rules actually fits quite nicely with the movements of the other pieces.
I've long thought of it in the following terms. The rooks move at angles of 0, 90, 180, and 270 degrees. The bishop moves at angles of 45, 135, 225, and 315 degrees. The knight moves at angles of 22.5, 67.5, 112.5, 157.5, etc. Since the knight directions don't cross (the center of) other squares, it passes right through them (like a bishop passing through a pawn chain), as opposed to jumping over them as it is typically described.
Nice observation, it never occurred to me why they work so well together.
Loomis' explanation is all geek to me. I guess my brain isn't wired to think in mathematical terms.
Hank's explanation makes all the sense in the world because it's very linear. I love how knights & queens and knights & rooks work together. CT: The mating possibilities using the knight to guard the checking rook or queen and the escape square for the king are fun. The rook and knight mate is referred to as the Arabian Mate. That's one of the important relationships between the two pieces.
When I teach the knight move to beginners one of the exercises I do with the group is place the knight on e5 and have the kids identify all the squares the knight goes to. I place a pawn on each of the squares they identify. Once all the squares have been marked I ask the kids if they see a pattern. Most see the circle, may will see the pawns on white squares, but few make the connection at first that the knight attacks the opposite color.
I wouldn't dare show them how the queen would cover all the other squares. :-) At least not in that lesson.
I'll be sure to share this with my husband. Great article.
Mrs Chessloser
Absolutely brilliant!
Fischer points this out in "Bobby Fischer Teaches Chess" while describing how the knight moves:
"The Knight moves to the eight nearby squares which are *not* in a horizontal, vertical, or a diagonal straight line from the square it occupies."
Hope all is well for you. No news is good news, i think. Let us know when you relapse. :)
Katar: yes, I've given up on the chess improvement for the time being, focusing on chess enjoyment mostly in the form of a few relaxing games of blitz before bed. Things at lab are really busy, I'm writing a book proposal (about neuroscience, not chess). That said, I did install Bookup on my computer last night. :)
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