The board as an input-output machine

Treat a setup of attacking pieces as a machine. You choose the inputs (which pieces, and where), and it returns an output (how many attacks land on a target square). To reach a wanted number you adjust the inputs and watch the output change.

Tabulating input against output turns guessing into a rule. Often the output moves in steps as you add a piece, so you can predict which input lands on the target instead of stumbling onto it. This is the function idea run backwards: from a wanted output to the input that produces it.