Let $n \geq 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$-gon by $2n + 1$ distinct integers is called [i]memorable[/i] if the following conditions hold:
(a) Each side has a label that is the arithmetic mean of the labels of its endpoints.
(b) The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices.
Determine all integers $n \geq 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n + 1$ consecutive integers.
1. **Initial Setup and Definitions:**
   - We are given an \( n \)-gon with \( n \geq 3 \).
   - We need to label the \( n \) vertices, \( n \) sides, and the interior of the \( n \)-gon with \( 2n + 1 \) distinct integers such that:
     - Each side's label is the arithmetic mean of its endpoints.
     - The interior's label is the arithmetic mean of all the vertices' labels.
   - We are to determine for which \( n \) there exists such a memorable labeling using \( 2n + 1 \) consecutive integers.

2. **Reduction to Consecutive Integers:**
   - Without loss of generality, we can consider the \( 2n + 1 \) consecutive integers to be \( \{0, 1, 2, \ldots, 2n\} \).

3. **Vertex Labels and Parity:**
   - The minimum \( 0 \) and the maximum \( 2n \) must be used as vertex labels.
   - All vertex labels must have the same parity (either all even or all odd) because otherwise, the arithmetic mean of two vertices (which labels a side) would not be an integer.

4. **Even Vertex Labels:**
   - Since we are using \( 2n + 1 \) consecutive integers, there are \( n + 1 \) even numbers in the set \( \{0, 2, 4, \ldots, 2n\} \).
   - The \( n \) vertices must be labeled with \( n \) of these \( n + 1 \) even numbers.

5. **Sum of Vertex Labels:**
   - The sum of all \( n \) vertex labels must be divisible by \( n \) because the interior label (which is the arithmetic mean of the vertex labels) must be an integer.
   - The sum of all \( n + 1 \) even numbers is \( \sum_{i=0}^n 2i = n(n + 1) \), which is divisible by \( n \).

6. **Exclusion of One Even Number:**
   - The only even number that is not used as a vertex label must be \( n \) because \( n(n + 1) - n = n^2 \), which is divisible by \( n \).
   - Therefore, the sum of the vertex labels is \( n^2 \), and the average (interior label) is \( n \).

7. **Case Analysis:**
   - **Case 1: \( n \) is odd.**
     - If \( n \) is odd, then \( n^2 \) is odd, which contradicts the fact that the sum of the vertex labels (all even) must be even.
     - Hence, \( n \) cannot be odd.
   - **Case 2: \( n \equiv 2 \pmod{4} \).**
     - If \( n \equiv 2 \pmod{4} \), then the vertex labels include numbers \( \{0, 2, 4, \ldots, n-2, n+2, \ldots, 2n\} \).
     - This set has \( \frac{n}{2} + 1 \) numbers congruent to \( 0 \pmod{4} \) and \( \frac{n}{2} - 1 \) numbers congruent to \( 2 \pmod{4} \).
     - This leads to a contradiction because two neighboring vertices both congruent to \( 0 \pmod{4} \) would result in an even side label.
   - **Case 3: \( n \equiv 0 \pmod{4} \).**
     - Let \( n = 4k \).
     - Label the vertices with \( 0, 2, 4, \ldots, 4k-2, 4k+4, 4k+2, 4k+8, \ldots, 8k-2 \).
     - The sides are labeled with \( 1, 3, 5, \ldots, 4k-1, 4k+1, 4k+3, \ldots, 8k-1 \).
     - This labeling satisfies all conditions, and the interior label is \( 4k \).

Conclusion:
- The necessary and sufficient condition for a memorable labeling is \( n \equiv 0 \pmod{4} \).

The final answer is \( \boxed{ n \equiv 0 \pmod{4} } \).