A jeweller makes a chain consisting of $N>3$ numbered links.  A querulous customer then asks him to change the order of the links, in such a way that the number of links the jeweller must open is maximized.  What is the maximum number?
To determine the maximum number of links a jeweler must open to reorder a chain of $N$ links, we need to consider the structure of the chain and the constraints given.

1. **Labeling the Links:**
   - Label the part between link $i$ and $i+1$ with the number $i$ for all $i$ such that $1 \le i \le N-1$.
   - Label the part between link $N$ and link $1$ as $N$.

2. **Case I: $N$ is even.**
   - Suppose $N$ is even. Consider the customer asks for part $j$ to be placed in part $j+2$'s spot where $j$ is odd, and the numbers are taken modulo $N$.
   - This means that each link $j$ (where $j$ is odd) will be moved to the position of $j+2$.
   - For example, if $N = 4$, the initial order is $1, 2, 3, 4$. The customer asks for the order $1, 3, 2, 4$.
   - To achieve this, the jeweler must break all $N$ links because each link needs to be repositioned.
   - Therefore, the jeweler has to break all $N$ links.

3. **Case II: $N$ is odd.**
   - Suppose $N$ is odd. Consider the customer asks for the chain to be arranged as $1, 3, 5, \dots$ and then $2, 4, \dots$.
   - This means that the odd-numbered links are placed first, followed by the even-numbered links.
   - For example, if $N = 5$, the initial order is $1, 2, 3, 4, 5$. The customer asks for the order $1, 3, 5, 2, 4$.
   - To achieve this, the jeweler must break all $N$ links because each link needs to be repositioned.
   - Therefore, the jeweler has to break all $N$ links.

In both cases, whether $N$ is even or odd, the jeweler must break all $N$ links to achieve the desired reordering.

The final answer is $\boxed{N}$.