Can you tell me how many bishops can be on an N by N chess board without them threatening each other, and what the reasoning is behind how these bishops are placed to achieve this?
A bishop can travel in any of the four diagonals. Therefore we can place bishops if it is not in any diagonal of another bishop. The maximum bishops that can be placed on an n * n chessboard will be 2 * (n – 1). 

- Place n bishops in first row
- Place n-2 bishops in last row. We only leave two corners of last row
What is maximum number of queens that could be placed on a N by N board without them threatening each other and how would the queens be placed?
The maximum number of queens that can be placed on an NxN chessboard such that no two queens threaten each other is N. This is known as the N-Queens problem, and has a well-known solution based on backtracking algorithms.

The idea is to place queens in columns, one by one, in such a way that no two queens are in the same row, same column, or same diagonal. To do this, we can start from the first column and place a queen in all rows one by one. Then we move to the next column and repeat the same process. If we reach a column where we can't place a queen in any row, we backtrack and try the next row in the previous column.