$($d$)$ A solved $8-$puzzle game looks like this:

Given a puzzle state like the one above, where the numbers are in the wrong places, we want to search for a series of moves that ends in the solution below. The number of moves required could be calculated as a path cost.

$\backslash$table$[[7,2,4],[5,,],[8,3,1]]$

Start State

$\backslash$table$[[,1,2],[3,4,5],[6,7,8]]$

Goal State

$($i$)$ Write down a heuristic function, $h,$ for this game $($remember that this estimates the path cost from a particular state to the solution$).$

$($ii$)$ What is the value of $h$ for the above board state?

$($iii$)$ In a greedy search, what move would be chosen next? $($i$.$e$.,$ move the $3,$ the $4$ or the $6$ into the gap?$)$

$($iv$)$ Is this heuristic admissible?

$($v$)$ Can you suggest a better heuristic measure?