This problem might be related to some of the ones suggested above, I did not check carefully.

Assume that you have to prepare the calendar of a basketball league to which $2n$ teams take part, where $n$ is an integer $\geq 1$. The full season consists of $2n-1$ dates where in each date each of the team has to play against another one (different than itself :-)). This has to be done in such a way that, overall, every team meets every other team exactly once (we all know what the calendar of a basketball league look like).

A solution is given in the comment below. The solution I had first suggested was wrong.