At the $i$th row and $j$th column of the seat. For the new semester, the seats are to be rearranged. If a student's original seat is $(i, j)$, and the new seat is $(m, n)$, then the student has made a move $[a, b]=[i-m, j-n]$, and $a+b$ is called the position number of the student. The sum of all students' position numbers is denoted as $S$. Find the difference between the maximum possible value of $S$ and the minimum possible value of $S$.
Add a virtual student $A$, at this point the sum of the position numbers is denoted as $S^{\prime}$. Note that swapping the positions of two adjacent students does not change the value of $S^{\prime}$. Through a finite number of swaps of adjacent students, the previous semester's positions can be restored, therefore $S^{\prime}=0$. Thus, $S$ plus the position number of $A$ is 0. When $A$ is at the top right corner, the position number of $A$ is the largest; when $A$ is at the bottom left corner, the position number of $A$ is the smallest. Therefore, the difference between the maximum and minimum values of $S$ is $5+$ $7=12$.