CONO-SUR 2014 Problem 6

Let $F$ be a family of subsets of $S = \left \{ 1,2,...,n \right \}$ ( $n \geq 2$ ). A valid play is to choose two disjoint sets $A$ and $B$ from $F$ and add $A \cup B$ to $F$ (without removing $A$ and $B$ ).

Initially, $F$ has all the subsets that contain only one element of $S$ . The goal is to have all subsets of $n - 1$ elements of $S$ in $F$ using valid plays.

Determine the lowest number of plays required in order to achieve the goal.