Question $4.$ Min$-$conflicts.

The Five Queens Puzzle is a classic problem in combinatorial mathematics. It involves

placing five queens on a $55$ chessboard in such a way that no two queens threaten each

other. In other words, no two queens can be in the same row, column, or diagonal. The

challenge is to find all possible arrangements of the five queens on the board where they are

safe from each other's attacks. Given the following chessboard:

Figure $4$: The $5-$Queens problem's table

$(1)$ Solve the $5-$Queens problem using Backtracking.

$(2)$ Solve the $5-$Queens problem using the Min Conflicts algorithm.

$(3)$ Compare the differences between the Backtracking and Min Conflicts algorithms.