On the Complementarity of Queens and Knights (Guest Blogger)
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.