Why is the Queen the Strongest Piece? Answering two silly chess questions.

When you were first learning to play chess, you probably were told that each piece was worth a certain amount. A pawn is worth 1 point; bishops and knights 3 apiece; rooks 5; and the queen a full 9 points. Then you might have been told that one should develop his knights first, followed by his bishops and queen, and leave the development of the rooks until last.

If you were anything like me, you didn't question what you were told. (Although, to be fair, I think I was six years old when I learned the game.) For the first years of my chess-playing life, I followed those rules as if my life depended on it, playing the Four Knights and only losing material that could be immediately regained. But why those points? Why that order? In this post I look at one way to understand those rules.

In my post about move generation I made a big fuss about how there are only 64 positions a given piece can have, and therefore there are only 64 unblocked move-sets for each type of piece. Each square in the following board shows how many knight moves are available from that square:

Knights obviously suffer when they're in their starting position (a knight on the rim is dim!). Contrast that with a bishop, which begins with some function but must reach the four central squares to be maximized:

That's probably a silly graphic: It's only one color! But it illustrates the fact that an unblocked rook has the same number of moves in every square. That's one reason rooks come out last- they don't necessarily need to be developed at all, since they can be effective from the corners.

Using a very simplistic analysis, we can use these move-sets to tell us which piece to develop first. This table allows one unblocked move for each piece:

What does this mean? Develop your knights first, because that will give you the largest number of additional moves (5). Then come the bishops, gaining 4 moves each. Third is the queen; although it also can gain 4 moves by moving to c3, that square is better used on a knight (so the queen will go somewhere else for 2 additional moves). Rooks should be saved for last!

Explaining why the pieces have the value they do can be roughly done with another table:

The astute reader (or decent chess player) will have a number of objections here. The most important is that these are unblocked move-sets, which completely ignores the fact that rooks are extremely easy (and knights impossible) to block. A similar argument can be made for why one should develop one's rooks- their moves are more diversified in the center than in the corners.

What I've written is probably most appropriate for an endgame, when there are many fewer blockers and move counts are closer to reality. As pieces come off the board, knights will struggle to keep up with bishops- so their value falls to 2.06 accordingly. But even in an endgame, piece values can vary wildly. 

There are many situations- many, many situations- where these piece values have nothing to do with reality. Even rooks and queens can be rendered useless.