Maximum Non-Attacking Pieces

[!IMPORTANT] Maximum non-attacking piece problems are extremal problems: prove a universal upper bound and then realize that bound with a construction.

Important properties

• Rooks and queens are constrained by row-column occupancy; bishops by diagonal occupancy.
• Kings are often controlled by partitioning the board into $2 \times 2$ blocks.
• Knights interact differently from line-moving pieces, so colour classes and local neighbourhoods become useful.
• A construction is not decoration: it is logically necessary to prove that the upper bound is sharp.

Standard workflow

1. Identify the resource being used up: rows, columns, diagonals, colours, or local neighbourhoods.
2. Convert that resource into an upper bound.
3. Build a symmetric placement that meets the bound exactly.

This is why the family is mathematically valuable: the proof is never just search. It is always counting plus witness.