Let $n$ be a positive integer. There are $2018n+1$ cities in the Kingdom of Sellke Arabia. King Mark wants to build two-way roads that connect certain pairs of cities such that for each city $C$ and integer $1\le i\le 2018,$ there are exactly $n$ cities that are a distance $i$ away from $C.$ (The [i]distance[/i] between two cities is the least number of roads on any path between the two cities.)
    
For which $n$ is it possible for Mark to achieve this?

[i]Proposed by Michael Ren[/i]
1. **Graph Interpretation and Degree Calculation:**
   - Consider the cities as vertices of a graph \( G \).
   - Each city \( C \) must have exactly \( n \) cities at a distance \( i \) for \( 1 \leq i \leq 2018 \).
   - This implies that each vertex \( v \) in \( G \) has a degree \( \deg(v) = n \).

2. **Handshake Lemma Application:**
   - By the Handshake Lemma, the sum of the degrees of all vertices in a graph is twice the number of edges:
     \[
     2|E| = \sum_{v \in G} \deg(v)
     \]
   - Since there are \( 2018n + 1 \) vertices and each vertex has degree \( n \):
     \[
     \sum_{v \in G} \deg(v) = n(2018n + 1)
     \]
   - Therefore:
     \[
     2|E| = n(2018n + 1)
     \]

3. **Divisibility Condition:**
   - For \( 2|E \) to be an integer, \( n(2018n + 1) \) must be even.
   - Since \( 2018n + 1 \) is always odd (as \( 2018n \) is even and adding 1 makes it odd), \( n \) must be even for the product \( n(2018n + 1) \) to be even.
   - Hence, \( 2 \mid n \).

4. **Construction for Even \( n \):**
   - To show that \( 2 \mid n \) works, consider the vertices of \( G \) arranged in a circle.
   - For each vertex \( v \), connect it to the first \( \frac{n}{2} \) vertices to the right and the first \( \frac{n}{2} \) vertices to the left.
   - This ensures that each vertex \( v \) has \( n \) neighbors, and the distances to these neighbors cover all required distances \( 1 \leq i \leq 2018 \).

5. **Verification:**
   - Each vertex \( v \) will have exactly \( n \) vertices at each distance \( i \) from 1 to 2018, satisfying the problem's conditions.
   - This construction works for any even \( n \).

Conclusion:
- We have shown that \( 2 \mid n \) is a necessary condition.
- We have also provided a valid construction for any even \( n \).

The final answer is \( \boxed{ 2 \mid n } \).