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

combinatorics optimization algorithms combinatorial-game-theory

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.

2,045

user302414

Updated on January 04, 2020

Comments

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 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 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 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 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 over 6 years

Kings: at most one in any 2x2 square.
user302414 over 6 years

But this doesn't give the maximum precisely.
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 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 over 6 years

For pawns, only one colour.