Example 3 Fill the numbers $1 \sim 8$ into the 8 squares surrounding the four edges of a $3 \times 3$ chessboard (as shown in Figure 1), so that the sum of the differences (the larger number minus the smaller number) of the adjacent (sharing a common edge) two numbers in these 8 squares is minimized. Find this minimum value.
To make the sum of the 8 differences as small as possible, we hope that each difference is as small as possible. For example, the largest number 8 should have 7 and 6 on its sides, making the differences 1 and 2, which are the smallest (as shown in Figure 6). Therefore, 5 should be placed next to 6 to minimize the difference. Similarly, 4 should be placed next to 5, 3 next to 4, 2 next to 3, and 1 next to 2. Thus, the sum of the 8 differences is
$$
\begin{array}{l}
(8-7)+(8-6)+(6-5)+(5-4)+ \\
(4-3)+(3-2)+(2-1)+(7-1) \\
=(8+8+7+6+5+4+3+2)- \\
\quad(7+6+5+4+3+2+1+1) \\
=14 .
\end{array}
$$