$($c$)$ State and justify a non$-$trivial admissible heuristic for this problem which is not the Manhattan distance to the exit.

$($d$)$ If we used an inadmissible heuristic in $A^{**}$ graph search, would the search be complete? Would it be optimal?

$($e$)$ If we used an admissible heuristic in $A^{**}$ graph search, is it guaranteed to return an optimal solution? What if the heuristic was consistent? What if we were using $A^{**}$ tree search instead of $A^{**}$ graph search?

$($f$)$ When might we want to use an inadmissible heuristic over an admissible heuristic?