## Task 2.

A crazy bishop is a piece that can face one of the four diagonally adjacent squares and attacks all squares directly in front of it as well as directly to the left and right (like a chess bishop that cannot see behind itself). Two squares of the game board are said to be diagonally adjacent if they share exactly one common vertex.

Determine the largest natural number $N$ for which it is possible to place $N$ crazy bishops on an $8 \times 8$ game board such that none of them attack each other.
## Solution.

We can imagine the figure of the mad hunter as a point from which 3 arrows emanate, determining the fields that the mad hunter attacks. By diagonal, we mean all diagonals on the game board, not just the main ones. Also, the 4 fields in the corners are also considered diagonals (consisting of only that one field).

On each diagonal, we have two possible directions of movement. Note that we cannot have more than one arrow pointing in the same direction, as then one of the two points to the other, i.e., one mad hunter is attacked. Thus, on each diagonal, we can have at most 2 arrows. The number of diagonals is 30, which means we can have at most 60 arrows. Since each mad hunter contributes with 3 arrows, we conclude that $N \leqslant 20$.

The image below shows an example that it is possible to place 20 mad hunters such that none of them attack any of the others.
![](https://cdn.mathpix.com/cropped/2024_05_30_a077c8100ccb3eae0685g-19.jpg?height=394&width=1176&top_left_y=1949&top_left_x=446)