BALTIC-WAY 2015 Problem 9

Let $n>2$ be an integer. A deck contains $\frac{n(n-1)}{2}$ cards, numbered

$$1,2,3, \ldots, \frac{n(n-1)}{2}$$

Two cards form a magic pair if their numbers are consecutive, or if their numbers are 1 and $\frac{n(n-1)}{2}$.

For which $n$ is it possible to distribute the cards into $n$ stacks in such a manner that, among the cards in any two stacks, there is exactly one magic pair?