Example 9 As shown in Figure 3, a brass plate is equipped with three diamond fine columns, one of which is fitted with 64 rings of different sizes, with the larger ones at the bottom and the smaller ones on top. These rings can be transferred from one column to another one at a time, but it is not allowed to place a larger ring on top of a smaller one. If all 64 rings are to be transferred from one column to another, what is the minimum number of moves required?
Explanation: This problem is about the Tower of Hanoi.
Let $a_{n}$ represent the minimum number of moves required for $n$ disks, with $a_{1}=1$.

For $n>1$, we can first move the $n-1$ disks from column $A$ to column $C$ as required, which takes $a_{n-1}$ moves; then move the largest disk from column $A$ to column $B$, which takes 1 move; finally, move the disks from column $C$ to column $B$ as required, which takes another $a_{n-1}$ moves. Therefore, we have
$$
a_{n}=2 a_{n-1}+1 \text {. }
$$

By transforming and recursively solving, we get
$$
\begin{array}{l}
a_{n}+1=2\left(a_{n-1}+1\right)=2^{2}\left(a_{n-2}+1\right) \\
=\cdots=2^{n-1}\left(a_{1}+1\right)=2^{n} .
\end{array}
$$

Thus, $a_{n}=2^{n}-1$.
Returning to the original problem, we have $a_{64}=2^{64}-1$.