Definition of the Attack-Force Metric AF(x,y)
Once upon a time there was a Macintosh freeware program for MacOS 7 which displayed all attacked squares using red, indicating the "force" of the attack using stronger hues of that color. Based on this idea, we set forth to create a Maple program which may perhaps aid in chess analysis, using the notion of attack force, AF.
For a chessboard square (x,y), we define the attack-force metric AF(x,y), as:
Whenever the following are aligned, their forces are summed: R, R-Q, Q-B, B-P, Q-B-P. Note that when a P is on top, the forces are summed only up to the P attacking square and no further. When there is no P, the forces are summed all the way to the boundary of the chessboard or until an obstacle is met.
This definition can be used to calculate AF on every square on the chessboard on any specific configuration. If the calculation is augmented graphically, it can (in principle) be used to analyze some of the elaborate strategic aspects on a given chess position[1].
In order to have a graphical representation of AF however, one needs a color gradation scale, and hence one needs to have a maximum value for the definition. Now, ask yourself: What is the maximum possible AF on a given square for the above metric? Let's assume first that pawn promotions are not allowed. Then look at the following configuration:
The position above, implies a total AF of: F(K)+2*F(P)+2*F(N)+F(B)+2*F(R)+F(Q)=65/18.
Can you find another configuration which tops the AF of the above configuration with no promotions? Now extend the definition to allow promotions and look at the diagram, below:
The position above, implies a total force of: 8*F(N)+2*F(B)+2*F(R)+F(Q)+F(K)+2*F(P)=107/18.
Can you top that? Below are chess configurations generated with a Maple 9 program to graphically display the AF metric, using color gradations of red and green. Green is white. Red is black. Because the general maximum of the metric is difficult to determine, we use the upper bound 65/18 for pure green and pure red. Hence, values below are graded according to color purity.
Note that one can get a good sense of the territorial ownership of the squares of the chessboard using this metric. Study the position above and convince yourself that black has more territory than white, hence is in a much better position (black is actually close to check-mating white).
You can download the Maple 9 worksheet which displays the above analysis, here. Enter the desired board configuration in L and run the entire sheet to get a board colored according to the AF metric, as above.