for the A$*$ algorithm $($best$-$first branch$-$and$-$bound$).$

Define the cost function C$($x$)$ of the search node in the state$-$space tree as follows

C$($x$)=$ g$($x$)+$ h$($x$)$ where

g$($x$)=$ cost of reaching the current node x from the root

h$($X$)=$The Manhattan distance between a board and the goal board

the sum of the Manhattan distances $($sum of the vertical and horizontal distance$)$ from the tiles to their goal positions.

File in the cost of A$,$B$,$ and C for the corresponding nodes in the state$-$space tree.