P$9\mathrm{.}4\mathrm{.}6[10$ pts$]$

In the logic boat puzzle, you are on one side of a river with a cabbage, a goat, and a wolf,

and you want to cross the river with all three items, using a boat which can take only you

and one of the items at a time. There is an additional constraint that you may not leave

the wolf alone with the goat, or the goat alone with the cabbage. Model the state space of

this problem by a graph, with nodes for allowable states and an edge from node $x$ to node

$y$ if there is a legal crossing of the boat that changes the state from $x$ to $y.($Note: You will

need to review the solution in the textbook for Exercise $4\mathrm{.}11\mathrm{.}7.)$

$($a$)$ Write down the graph representing the state space and explain what each node and

edge represents.

$($b$)$ Is there a solution to the puzzle? If there is no solution, prove that there is no solution.

If there is a solution, show a solution, step by step.

$($c$)$ If your answer in part $($b$)$ was that there is no solution, just write $"$no solution" here.

If your answer to part $($b$)$ is that there is a solution now answer "whether there is more

than one solution". If there is only one solution, explain why the answer in part $($b$)$ is

the only solution. If there is more than one solution, describe the second solution.