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Good and bad squares

cigoL
Sep 29, 2011, 9:52 AM 5

We all know it's important to control the center. But do we know why? For a long time, I didn't. I just read: control the center, this is important. So, I tried to do so. Lately, I realized why, and it's actually rather obvious. Still, I could imagine that other beginners might not know either. So, for them (and for my own reference), I've written this little article showing the good and bad squares on the board, since it's what center control is very much about. The reason why we have to control the center, is because our pieces have a lot more mobility from there, than they do from the edge of the board. 

So, with that said, let's have a look at each and every piece - besides the pawns - to see how much mobility they have at various squares around the board. Let's being with the King

The King - just not the edge, please 

The King is one of the two pieces who have a lot of good squares on the board. Actually, any square on the board except for the 28 squares around the edges are equally good for the King - from a pure mobility perspective. On all other squares the King has 8 squares to move to. On the edge, the King only have 5 squares to go to, in the corners only 3. So from a mobility perspective the 36 good King squares are 1.7 times better for the King, than the bad squares (corners and edges). The figure 1.7 is the ratio between 8 and the weighted average of the mobility values for all non-ideal squares (in the case of the King, these being the corners and edges (3 and 5)). So, the King is more mobile when it's not on the edge, but it's mobility is as good on b2 as on e5

The Rooks - any square will do 

Now what about the Rooks? The Rooks differs from all the other pieces, by having no favorite squares (again from a purely mobility perspective). The Rooks have 14 squares to move to, 7 in each direction, no matter where on the board they are. So, when it comes to mobility the Rooks doesn't care so much about the center. However, since the center is important for the minor pieces and the Queen, it makes sense to defend the center with the Rooks, e.g. by placing them on the d- or e-files. Since the Rooks are indifferent to where on the board they are placed, their good square/bad square ratio is 1. From now on I will refer to this ratio as GsBs, like this this:

  • King, GsBs(1.7:1
  • Rook, GsBs(1:1

The Queen - somewhat indifferent 

The Queen is, as we all know, the most mobile piece on the board. While the Rooks can go to 14 squares from any other square, the Queen can go to 21 (1½ times the number of squares of the Rook) from the worst of her squares! And where are those worst Queen squares? On the edge, of course. It's not just the corners. Every one of the 28 squares along the border gives the Queen 21 other squares to go to. That's pretty impressive. However, the closer to the center the Queen gets, the more mobile she becomes. From 21 on the edge, to no less than 27 on one of the 4 center squares. So, the Queen does get more mobile the closer to the center she gets, but not be all that much. Her GsBs is 1:1.2

The Bishops - we love the center squares! 

With the Bishops it really becomes clear why the center is so important in chess. From the edge (and the corners) the Bishops only have 7 squares to go to. That's less than the King, when he's not on the edge. On the 4 center squares, the mobility of the Bishops almost doubles, compared to when being on the edge, with 13 good squares. That's just one good square less than the Rooks. So, from a mobility point of view, the value of a Bishop almost doubles when in the center, rather than on the edge (including it's staring squares). On one of the the 12 squares bordering the 4 center squares, the Bishops are doing pretty fine as well, with 11 good squares. So, if it's not possible to place the Bishops on one of the 4 center squares, then at least getting them away from the edge makes a lot of sense. The GsBs for the Bishops is 1:1.5

The Knights - 16 squares of fame 

If you think you are winning the game, and want to challenge yourself, start by placing your Knights in the corners. From here, these funny creatures, are as good as useless - with only 2 squares to jump to. Compare that with any of the 16 central squares, and it becomes obvious why our Knights better be right in the middle of the game. From one of these 16 squares, the Knights can leap to 8 other squares. That's 4(!) times as much mobility as they have while standing in a corner. So, while the Knights don't care if they are on one of the 4 center squares, or one of the 12 squares bordering the center squares, they certainly do not want to be in the corners, or on the edges for that matter. This is naturally reflected in the Knights GsBs ratio of 1:1.8

The GsBs conclusion - or why the center is important 

As we can see now, the center is not just important because someone said so. There are quantitative ways of expressing the value of a square in relation to another square. I call this the GsBs ratio. In short, the higher the GsBs ratio, the more happy the piece will be when moved closer to the center. Th lower GsBs ratio, and the more indifferent to placement (from a mobility point of view) the piece is. So, looking at our list: 

Knight, GsBs(1.8:1) 

King, GsBs(1.7:1) 

Bishop, GsBs(1.5:1) 

Queen, GsBs(1.2:1) 

Rook, GsBs(1.0:1) 

- we can see that the Knight is the most "sensitive to placement", while the Rooks are utterly stoic. Please notice that the ratio is between the ideal and the non-ideal squares for the given piece. So, although the ratio for the Knights and the King is almost even, the King has more ideal squares (36), namely all but the corners and edges. The Knight, on the other hand, only have 16 ideal squares, all in close proximity to the center. 

So, controlling the center is a good idea. And now we know why. 

If you have any comments or question, please feel free to post them below. Thank you!

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