<First of all i completely disagree with your previous post. You tried to prove or anyway to give some explanations about: "why a larger board tends to "dilute" the strength of the pieces" and that "pieces on a smaller board are stronger". These two statements CAN NEVER be proven as true or false since they are not defined properly. The reason for that is simple:

These two statements have the expressions: "dilute the strength of the pieces" meaning "reduce the strength of pieces" AND "are stronger".
These 2 expressions are not real and have no meaning since words like "stronger" WITHOUT the "in comparison to" OR the "than that of" are pointless.

Now if you meant: "pieces on a smaller board are stronger than a bigger board", we have again something not very logical since we compare how strong the pieces are in 2 differents boards-worlds. This has no meaning.


What we have to do is simple: If with one way we have that in one game (G-1) (with a board size LxK) we have the values for the pieces:
Piece-X1 = E1
Piece-X2 = E2
.............
Piece-Xn = En
(the values E1,E2,...,En has been at a increasing order.)

and in another game (G-2) (with a board size (L+H)x(K+G)) the values are:
Piece-X1 = R1
Piece-X2 = R2
.............
Piece-Xn = Rn
(the values R1,R2,...,Rn has been at a increasing order.)

Then to have a valid comparison of the strength of every piece (suppose the Piece-Xz) at G-1 in comparison with the same piece in the game G-2, we should compare the Ez/E1 and Rz/R1.

What this means is that in order to compare the strength of every piece at G-1 in comparison with the same piece at the G-2, we compare the relative value of the piece in relation with another piece(Piece-1 for example) at game G-1, with the relative value of the piece in relation with the same piece(Piece-1) as before, at game G-2.


But although not very logical, even if you meant "pieces on a smaller board are stronger than a bigger board" the below procedure you used is wrong.


>You can see on the 100x100 board, with 10,000 squares, there is no way it is going to reach >6,400 (64%) of these squares. It will reach (100-1) x 4 = 396. You can see 396/10,000 is a very >small fraction.

I disagree to your example as a proof for that. And in fact i can find reasons at your example that contradict to your conclusion-statement.
I disagree as a proof because while the Queen on a bigger board covers less percentage of the board, the same exists for the other pieces also.

And altough with a first thought we can say that the Queen covers a smaller percantage of board, so it's weaker, when we compare her power to that of Pawns at 8x8 and 100x100 we can imagine it's much more powerful since with one move it goes from one size to another at every board, while the Pawns at the first case will do 8 centuries, but at the second will do 100 centuries. Also the same exists for the Knights. At a 8x8 board are cats compared to the Ferrari-Queen but on a 100x100 are just turtles.


>In this sense, pieces on a smaller board are stronger since they have a greater "density".