Five queens dominate the board

What is the fewest queens that together attack or occupy every square? That minimum is the queen domination number of the $8 \times 8$ board. A queen covers its rank, file, and both diagonals, a powerful cross, yet a short cover still needs careful placement, and one queen fewer than the domination number always leaves a gap.

Domination differs from the non-attacking problem: here queens may attack each other, and the goal is coverage, not peace. Finding a minimal cover is a search; proving that fewer pieces cannot suffice needs a counting or case argument about how much their crosses can reach.