Example 1 Place the numbers $1,2, \cdots, 8$ in the 8 squares around the perimeter of a $3 \times 3$ chessboard (as shown in Figure 1), such that the sum of the absolute values of the differences between adjacent numbers (numbers in squares that share a common edge) is maximized. Find this maximum value.
To make the description convenient, we define the difference between two adjacent numbers as the larger number minus the smaller number, thus avoiding the concept of absolute value.

To maximize the sum of the 8 differences, we want each difference to be as large as possible. For example, the largest number 8 should be placed next to the smallest number 1 and the second smallest number 2, making the differences 7 and 6 as large as possible (as shown in Figure 2). The second largest number 7 is adjacent to 1, and 3 is placed next to 7. The third largest number 6 is adjacent to 2, and also adjacent to 4. This arrangement ensures that the differences between adjacent numbers are as large as possible. The remaining number 5 is placed in the last position, adjacent to 3 and 4. Carefully observe Figure 2; it is impossible to adjust the positions of the numbers to further increase the sum of these 8 differences. The sum of these 8 differences is
$$
\begin{array}{l}
(8-1)+(8-2)+(7-1)+(7-3)+ \\
(6-2)+(6-4)+(5-4)+(5-3) \\
=(8+7+6+5) \times 2-(1+2+3+4) \times 2 \\
=32
\end{array}
$$