Single Comment

To continue messing with piece values, let's look at different boards. The piece values given are based on a more or less standard 2D chessboard, rectangular in shape. Change that in any significant way, and you've changed the piece values. Take a 1D board. There are 2 ways you can 'cut' a 1D board from a standard 2D board, orthogonally or diagonally. Consider the value of a rook and a bishop on each board for the slider rook and the slider bishop - not the jumping bishops of One Ring Chess [LLSmith] for example. One one board, the bishop can't move, and the rook has unlimited [except for blocking pieces] movement. So on average in these 2 systems, the rook and bishop are exactly equal. This is carried over into some games played with standard pieces on a diamond-shaped board, where the rooks and bishops trade the number of squares they can move to without trading any other aspect of their respective moves. Clearly this increases the value of the bishop and decreases the value of the rook. 

Now consider the knight's move in 2, 3, and 4 dimensions. That move, the only 2D move in standard chess, explodes in higher dimensions - attacking 8 squares in 2D, 24 in 3D, and 48 in 4D, if I did my numbers right. Beyond that, it takes a Dan Troyka, a Larry Smith, or a Vernon Parton to go. But it's easy to see the value of the knight rises considerably in comparison to, say, the rook. 

So, to sum this up, we seem to have established that the value of a piece depends on both the kinds of other pieces on the board with it, not just the number, and it also depends on the shape of the board it's on. 

Does any piece have an intrinsic value? :)