## Defensive Bishop

#### Problem

In a game of chess, the Bishop can take any piece that lies on a diagonal.

By placing five Bishop pieces on a 4 by 4 grid every square is protected in such a way that no Bishop threatens any other Bishop.

However, it is possible to achieve the same with just four bishops.

What is the least number of Bishop pieces required to protect every square on a standard 8 by 8 chessboard?

#### Solution

It is possible to protect every square on a chessboard by using just eight Bishop pieces; this is one way it can be done.

How many different ways can this be achieved?
What about an $n$ by $n$ grid?
What is the maximum number of non-threatening Bishops pieces that can be placed on a four by four grid?