$($a$)$ The representation for the n$-$queens problem in the genetic algorithm $($GA$)$ is defined as n$-$tuple

having n unique integers between $1$ and n inclusive. The position within the tuple denotes the

grid column, and the number stored at that position is the grid row. For the $8-$queens problem,

the maximum fitness value is $28.$ Derive the maximum fitness value for n$-$queens. Support your

deduction with a proper explanation in one line.

$($b$)$ Given an initial state of an $8-$puzzle problem and the goal state to be reached in figure $4.$

Find the most cost$-$effective path to reach the goal state from the initial state using the A$*$

Initial State

Goal State

Figure $4$: $8-$puzzle of Question $5($b$).$

Algorithm. Consider $g(n)=$ Depth of node and $h(n)=$ Number of misplaced tiles.

$(2+4=6)$