$1)$ Consider the N$-$Queens problem, as shown below for a standard $8$x$8$ Chess board. The problem is to find a placement of n queens on the board such that no queen is attacking any other $($i$.$e$.,$ in the same row, column, or diagonal$).$ Since there must be one and only one queen in each column, potential solutions can be represented as a vector of length $\backslash ($ n $\backslash )$; for the board below it would be $\backslash ([3,5,8,4,1,7,2,6]\backslash )($indexing from $1$ at the top$).$

Now, consider a $\backslash (5\backslash$times $5\backslash )$ board with the starting vector $[5,4,2,2,4].$ Draw this board, and show how a basic local search using the minimum conflicts heuristic would proceed for this problem. In each step, select the queen with the largest number of current conflicts that was $*$not$*$ moved in the previous step; break any ties by selecting the queen furthest to the left. Then, move that queen to the row the has the lowest number of conflicts $($break ties by selecting the row with the lowest index, i$.$e$.,$ the highest row$).$ Continue this process until you either reach a valid solution or detect a cycle $($repeat a previous state$).$