Q$3.$ Consider the $2$X$2$ Rubik's cube shown below:

$($a$)$ Develop a list$-$based state representation for the $22$ Rubik's cube so that a search algorithm

can be applied to solving it$.$ Keep it simple! Explain your state representation.

$($b$)$ Any state in which all $4$ pieces on each of the $6$ sides all have the same number as each other

is a goal state $($e$.$g$.$ as in the picture above$).$ Can your representation represent more than one

such goal state or is it unique? Show the goal state or states and explain why or why not.

$($c$)$ Any state that is not a goal state is a potential start state for the search. Show and explain two

different start states.

$($d$)$ A move for the $22$ rubric cube is a rotation of one of the $6$ faces by either $90$ degrees left, $90$

degrees right or $180$ degrees $($direction doesn't matter$).$ Since there are $6$ faces, that makes $6$

times $3$ possible moves. Present the transition model, mapping the state before the move to the

state after the move, for all the moves but just for one side.

$($e$)$ Assuming you have moves for all $6$ sides, what is the branching factor for this search?

$($f$)$ Can you propose a modification of the action model to reduce this complexity? Be sure to

explain how it reduces the complexity.

$($g$)$ To apply A$*$ search to this problem we need to have an admissible heuristic. Explain in detail

whether the following is or is not an admissible heuristic:

$h(s)=$ sum $of$ the number $of$ mismatched pieces each side

E$.$g$.$ if all the numbers on a side are different, then $3$ are mismatched; if $2$ are the same, then $2$ are

mismatched; if $3$ are the same then $1$ is mismatched.