How many pawns, bishops, rooks or kings can be put on a $n \times n$ chessboard such that they don't threaten each other?

2,045

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.

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Updated on January 04, 2020

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  • user302414
    user302414 over 2 years

    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: https://en.wikipedia.org/wiki/Eight_queens_puzzle; 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.

    • Git Gud
      Git Gud over 6 years
      Do you mean to maximize $\left|\{x\colon x\text{ is a bishop or } x\text{ is a rook or }x\text{ is a king}\}\right|$?
    • hardmath
      hardmath over 6 years
      With pawns the color of a pawn affects its direction of movement, and so its attack patterns (as does the rule for capture en passant). If you are especially interested in this, please be more specific.
    • user302414
      user302414 over 6 years
      Edited the post. I want to find the maximum number of pawns which can be placed on a chessboard such that no one can "attack" others.
    • Mark Bennet
      Mark Bennet over 6 years
      Pawns on alternate rows - each row complete - looks best compared with pawns on alternate columns - columns can't be complete. If the pawns are white on odd numbered rows, starting with row 3, you can fit a row of black pawns on row 2.So thinking about that you can do pairs of white pawn rows and black pawn rows two rows then a blank row, which does better on larger boards.
  • Empy2
    Empy2 over 6 years
    Kings: at most one in any 2x2 square.
  • user302414
    user302414 over 6 years
    But this doesn't give the maximum precisely.
  • Mark Bennet
    Mark Bennet over 6 years
    @user302414 What do you mean. You can compute the maximums very easily from these configurations. e.g. Bishops $2n-2$, Rooks $n$
  • user302414
    user302414 over 6 years
    Ah, so for kings we have obviously $n^2/4$ for $n$ even and $(n+1)^2/4$ for $n$ odd.
  • user302414
    user302414 over 6 years
    For pawns, only one colour.