In the mathematical chess problem of

**domination**,
you are tasked with finding an arrangement of several pieces
of the same type so that every square on the board is reachable in a
single move by at least one of the pieces.
In the problem of

**independence**, your goal is to place the maximum
number of pieces of the same type so that none of the pieces lie in the
path of another. For example, try to find:

- for each board size, a smallest arrangement of knights that dominates the board;
- a way to place 5 independent queens on a 5x5 board, and a way to place 8 independent queens on an 8x8 board.