A short answer to your question is, it depends
(I know, that helps a lot). A lot has to do with the opening. What H&M-S are referring to in chapter one is how control of the center can happen in different openings. This is not H&M-S clearest point. I think it would be easiest to explain it in COA terms of Center Domination. This is probably the easiest way to define it, and generally think about it.
Having said that, the 4 central squares of the Center are the most important, followed by the squares c4,c5,f4,f5, followed by the other 8 squares. Putting a Knight on the 4th or 5th rank in the center that cannot be dislodged by a pawn, and can be dislodged by a piece only with a compensating disadvantage can be crucial in a game. The same goes for a Bishop either in the center or protecting the center.
There are 3 general types of centers.
- Open center, where one or both center files is completely open (no pawn of either side on the file). The battle throughout the game takes place generally in the middle, with minor skirmishes on the flanks. Control of the center, then, in these positions is control of the files that are open.
- Semi-open or half open center, where at least one side has a half-opened file on the d or e file. Control here generally means can one side prevent an outpost, or block a diagonal for the opposing Bishop or rank or file for the Rook and Queen. and
- Closed center, where the pawns are lock with two phalanx (like White pawns on d4, e5 and Black pawns on d5, e6). Control of the center is not as important in closed centers, as the battle generally goes to the flanks.
A much better explanation of the center is found in Kotov's books, Play Like a Grandmaster
and Think Like a Grand Master
, but you aren't there yet. So you should think of COA Center Domination would be a countable point, and as H&M-S say on p. 28 in the summary, 4th Rank v. 3rd is a countable point (in the center), as is two center pawns v. one. All three of these directly affect the center, as will outposts, half and open files, and open diagonals through the center. As you can see, COA is a simple yet effective definition, as it boils down a number of elements of the position, yet chess is dynamic, not static, so it depends
. If a player controls, let's say, two of these 3 items (Center Domination, 4 v. 3, 2 v. 1), I would be comfortable saying he has control, most of the time.
In most games, everything revolves around the center. I hope that my explanation provides more light than fog.