China Girls' Mathematical Olympiad

Add a virtual student $A$ so that every seat is occupied by exactly one student. Denote $S^{\prime }$ the sum of the position values in this situation. Notice that an exchange of two students occupying the adjacent seats will not change the value of $S^{\prime }$. Every student can return to his/her original seat by a finite number of such exchanges of adjacent students. Then $S^{\prime }=0$. Since $S^{\prime }=S+a_A+b_A$, where $a_A+b_A$ is the position value of student $A$, then we have $S$ is the greatest when student $A$ occupies seat $(1,1)$, and $S$ is the smallest when $A$ occupies seat $(6,8)$. So the difference between the greatest and the smallest possible values of $S$ is $14$.