Maximum Non-Attacking Pieces

These problems ask for the largest possible number of a given piece that can be placed on the board so that no two attack each other.

That makes them classic extremal problems: you must prove both that a number cannot be exceeded and that it can actually be achieved.

The Two-Part Method

Every maximum-placement proof has two halves:

1. Upper bound: show that any arrangement has at most some number of pieces.
2. Construction: exhibit an arrangement that reaches that number.

When those match, the maximum is proved.

Typical Counting Ideas

Different pieces suggest different counting arguments:

• Rooks: at most one per row and one per column.
• Kings: partition the board into $2 \times 2$ blocks.
• Bishops: track occupied diagonals on both colours.
• Knights: colouring and local neighbourhoods become useful.
• Queens: combine row, column, and diagonal restrictions at once.

The board geometry changes from piece to piece, but the strategy stays the same.

What To Look For

• a natural partition of the board into blocks;
• a row or column count that caps the total;
• a colouring that separates safe squares;
• a symmetric construction that reaches the bound cleanly.

Return to the Mathematics in Chess overview for the full collection.