How did they calculate that a queen was equivalent to 9 pawns?

Is there a mathematical proof of that?

Also would you rather have 9 pawns or a queen?

@Billmtl said in #1:
> Is there a mathematical proof of that?

The premise is false. There is no universally agreed value.

Different people assign different values to the Queen. Wikipedia has a whole page on relative chess piece values [1], which a section of alternative systems. Values for the Queen range from 7.9 (by Jacob Sarrat, early 19th century) to 13.5 (computer, 1992). Other values include 9.94 (Philidor, Staunton), 9-10 (Lasker), 10 (Euwe) and 9.5 (AlphaZero). Note also that values may change during a game (that is, some pieces may be valued more (or less) in the endgame than in the opening).

[1] en.wikipedia.org/wiki/Chess_piece_relative_value#Alternative_valuations

A rook trapped in the corner is worth 0.

I think they were guessing originally, I'm not sure what the relative piece values are not. depends a lot on the position.

It was not "calculated"; it was "arrived at." :)

I have it on good authority that they put 9 numbered pieces of paper into a hat and drew one out. Voila!

Ask any GM or Dr Google for absolute verification of this little known nugget of chess history.

It's just a rough guideline, but I think it has less to do with its strength against pawn masses, and instead indicates (1) that two rooks are often better than a queen, and equal to a queen and pawn, because the rooks can force the pawn off the board, (2) a queen is normally worth more than a rook and minor piece, because the queen can usually put the pieces into zugzwang, and (3) three minor pieces can match or even defeat a queen if they are coordinated. There are lots of exceptions, however, such as the fact that two knights can usually draw against a queen.

9 is a heuristic.
Mathematically, every position can only be a win, draw or loss. Your queen doesn't matter if your opponent can checkmate you on the next move.
Finding good heuristics is a challenge when making chess engines.

My guess is that over a long period of time they played lots and lots of games. They figured out that a bishop and a knight are about as good as 3 pawns but not as good as 4 (all else being equal). Then they figured out that a queen is about as good as 2 bishops and a knight or 2 knights and a bishop.