Question 156
Let $n$ be a positive integer. A set of $n$ distinct lines divides the plane into various (possibly unbounded) regions. The set of lines is called "nice" if no three lines intersect at a single point. A "colouring" is an assignment of two colours to each region such that the first colour is from the set $\{A_{1}, A_{2}\}$, and the second colour is from the set $\{B_{1}, B_{2}, B_{3}\}$. Given a nice set of lines, we call it "colourable" if there exists a colouring such that
1. no colour is assigned to two regions that share an edge;
2. for each $i \in \{1,2\}$ and $j \in \{1,2,3\}$ there is at least one region that is assigned with both $A_{i}$ and $B_{j}$.
Determine all $n$ such that every nice configuration of $n$ lines is colourable.
1. no colour is assigned to two regions that share an edge;
2. for each $i \in \{1,2\}$ and $j \in \{1,2,3\}$ there is at least one region that is assigned with both $A_{i}$ and $B_{j}$.
Determine all $n$ such that every nice configuration of $n$ lines is colourable.