King and a Lot of Queens

Note that wherever the King is placed, no Queen can be placed in its same row, column or diagonals, but a Queen can be placed on any of the other empty squares. If we place place the King on and edge (including the corners), there will be $7$ diagonal squares. This is the most profitable position to place the King because if we place the King closer to the center, the diagonals become bigger.

For a $m\times n$ , with $m\le n$ , there are $mn-(m+n+m)+2=mn-2m-n+2$ squares where a Queen can be placed, as each row consists of $m$ squares, each column consists of $n$ squares and there are a total of $m$ diagonal squares, and we're counting the King square three times.

When $m=8$ and $n=8$ , $64-16-8+2=42$ Queens can be placed on the board. This is biggest number because if we place one more Queen, it will be placed on the King row, column or diagonal, thus, will be attacking the King.

Answer is 42. what was the question again?

Its pretty simple when you draw it out. I'm not sure how to add a picture but draw an 8x8 board and place the king in one of the corners. If you know chess then you know that queens attack horizontally, vertically and diagonally. Cross out these squares and then count them. There should be 7 across, 7 down and 7 on the diagonal as well as the square the king is on. Thats 22 total, subtract from 64 (8x8 (the total number of squares)) and you get 42.

For the generalization I observed what happened with the 8x8 board.

Let m = squares across and n = squares down

The generalization I came up with is

(m-1)+(n-1)+((m+n)/2)-1)+1 which simplifies to 3/2m+3/2n-2