How many pawns, bishops, rooks or kings can be put on a $n \times n$ chessboard such that they don't threaten each other? A friend of mine asked me this question and I know this is not easy to solve. I found some informations similar to this question here: < First of all, I'm interested in the pawns-problem. If possible, an algorithm giving the maximum number of pawns (bishops, rooks and so forth) is very appreciated! Maximum of pawns on a $n \times n$ chessboard: book recommendations or PDFs are also appreciated! There are 4 different problems. The first problem, how many pawns can be placed on a chessboard ($n \times n$) such that they don't threaten each other? For the other 3, replace "pawn" with bishop etc.

For bishops choose a colour - in one direction you will find seven diagonals. So at most one on each diagonal, two colours, fourteen bishops - realised by eight on the first rank and six on the eighth.

For rooks, one on each rank - eight along the diagonal will do.

For knights - thirty two all on the same colour.

Not sure what you mean about pawns.

This on a standard chessboard, but generalisable. If the side of the chessboard is odd, put the knights on the colour with most squares.