Question $1$

State Spaces

$[16]$

In the Hungarian Forint and Mexican Peso puzzle, there is a container with five cells. In the initial state of the case the two leftmost cells are occupied by a Forint coin each, while two Peso coins occupy the two rightmost cells. The middle cell is empty. This is illustrated in Figure $1$ as follows:

The puzzle involves rearranging the coins one at a time to reach the goal state. In the goal state, the Peso and Forint are interchanged in Figure $2$ as follows:

Figure $2$: Puzzle Set PPFF

There are only four permissible moves according to the following expressions:

$($i$)$ A Peso Slide $-$ a Peso coin slides one position to the left into an empty cell.

$($ii$)$ A Peso Hop $-$ a Peso coin jumps left across one cell containing a coin into an emptycell.

$($iii$)$ A Forint Slide $-$ a Forint coin slides one position to the right into an empty cell

$($iv$)$ A Forint Hop $-$ a Forint coin jumps right across one cell containing a coin into an empty cell

A cell may never contain more than one coin, and all coins must be in the container after every move. Therefore

$($a$)$ Design a state representation for the problem.

$($b$)$ Using your representation, define the start state, and the goal state.

$($c$)$ Explain why your representation adequately defines any state that could occur in the problem

$(2)$

$($d$)$ Define the operators for this problem.

$($e$)$ How many possible states are there for this problem?