1. For positive integers $n$, let $S_{n}$ be the set of integers $x$ such that $n$ distinct lines, no three concurrent, can divide a plane into $x$ regions (for example, $S_{2}=\{3,4\}$, because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum $i$ such that $S_{i}$ contains at least 4 elements?
Answer: $\square$
For $S_{3}$, either all three lines are parallel (4 regions), exactly two are parallel (6 regions), or none are parallel ( 6 or seven regions, depending on whether they all meet at one point), so $\left|S_{3}\right|=3$. Then, for $S_{4}$, either all lines are parallel (5 regions), exactly three are parallel ( 8 regions), there are two sets of parallel pairs ( 9 regions), exactly two are parallel ( 9 or 10 regions), or none are parallel $(8,9,10$, or 11 regions), so $\left|S_{4}\right|=4$.