A short proof of a variant of the round-robin scheduling problem

Suppose 2n people participate in a sports tournament that consists of multiple rounds. In each round, two teams of n people are formed to play against each other. We require that every two players play at least once in opposing teams and once in the same team. For this variant of the round-robin scheduling problem, an explicit formula for the minimum number of rounds needed to satisfy both conditions has recently been published. In this short note, an alternative and short proof of this is given.