Proposition 1 Fill the $2n$ numbers from $1 \sim 2n$ into the $2n$ cells of the circular ring in Figure 3, so that the sum of the differences (the larger number minus the smaller number) of the numbers in adjacent cells (cells sharing a common edge) is maximized. This maximum value is $2n^2$
Prove: By alternately arranging the larger $n$ numbers $n+1, n+2, \cdots$, $2n$ and the smaller $n$ numbers $1, 2, \cdots, n$ in the $2n$ cells of the circular arrangement in Figure 3, the total sum of the $2n$ differences can be maximized (for the same reason as in the solution of Example 1), and its maximum value is
$$
\begin{array}{l}
{[(n+1)+(n+2)+\cdots+(2 n)] \times 2-} \\
(1+2+\cdots+n) \times 2 \\
=2 n^{2}
\end{array}
$$