Four rooks cover the board

A rook attacks every square along its rank and file: $14$ squares from any board square. Colours alternate along each line, so of those $14$ attacked squares, exactly half are light and half are dark when the rook itself sits on a dark square. Count the light ones on the rank, then those on the file, and add.

There are $32$ light squares. If each dark-square rook attacks the same number of them, four such rooks produce exactly $32$ light-square attacks only when that shared per-rook count fits the total with no leftover and no overlap. Coverage of every light square is then possible only when no light square is attacked twice, which forces the rooks far enough apart that their lines do not share a light square. Covering is a counting argument backed by geometry.