C1. Let $n$ be a positive integer. There are $2018 n+1$ 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 $1 \leq i \leq 2018$, there are exactly $n$ cities that are a distance $i$ away from $C$. (The distance 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?
(Michael Ren)
The answer is $n$ even.
To see that $n$ odd fails, note that by taking $i=1$ we see the graph is $n$-regular; since it has an odd number of vertices we need $n$ to be even.

On the other hand, if $n$ is even, then consider the graph formed by taking the vertices of a regular $(2018 n+1)$-gon and drawing edges between vertices which are at most $n / 2$ apart. Then this works.