Call a set of $n$ lines [i]good[/i] if no $3$ lines are concurrent. These $n$ lines divide the Euclidean plane into regions (possible unbounded). A [i]coloring[/i] is an assignment of two colors to each region, one from the set $\{A_1, A_2\}$ and the other from $\{B_1, B_2, B_3\}$, such that no two adjacent regions (adjacent meaning sharing an edge) have the same $A_i$ color or the same $B_i$ color, and there is a region colored $A_i, B_j$ for any combination of $A_i, B_j$.

A number $n$ is [i]colourable[/i] if there is a coloring for any set of $n$ good lines. Find all colourable $n$.
1. **Define the Problem and Initial Conditions:**
   We need to determine for which values of \( n \) there exists a coloring of the regions formed by \( n \) good lines such that no two adjacent regions share the same color from the sets \(\{A_1, A_2\}\) and \(\{B_1, B_2, B_3\}\), and every combination of \(A_i\) and \(B_j\) appears in at least one region.

2. **Case for \( n \leq 4 \):**
   - If \( n \leq 4 \), consider the case where all \( n \) lines are parallel. This configuration divides the plane into \( n+1 \) regions.
   - For \( n = 1 \), there are 2 regions.
   - For \( n = 2 \), there are 3 regions.
   - For \( n = 3 \), there are 4 regions.
   - For \( n = 4 \), there are 5 regions.
   - In each of these cases, it is impossible to assign the colors such that no two adjacent regions share the same \(A_i\) or \(B_i\) color and every combination of \(A_i\) and \(B_j\) appears.

3. **Case for \( n \geq 5 \):**
   - Consider the graph where each vertex represents a region and an edge exists between two vertices if their corresponding regions share a boundary.
   - Represent each region with a binary number of \( n \) digits, where each digit indicates which side of a line the region is on.

4. **Claim 1: The Graph is Bipartite:**
   - To show the graph is bipartite, we need to prove that all cycles in the graph have even length.
   - Two vertices are adjacent if their binary numbers differ by exactly one digit, meaning they share a boundary.
   - Any cycle in this graph must involve an even number of changes in the binary representation, ensuring all cycles are of even length.
   - Therefore, the graph is bipartite.

5. **Claim 2: The Graph has Vertices with Distance \(\geq 5\):**
   - For \( n \geq 5 \), consider the binary representation of regions. The maximum distance between two vertices (regions) in this graph is \( n \), as the Hamming distance between two binary numbers can be at most \( n \).
   - Since \( n \geq 5 \), there exist vertices with a distance of at least 5.

6. **Claim 3: Coloring Exists if Distance \(\geq 5\):**
   - Suppose \( \text{dist}(u, v) \geq 5 \). Define \( S_j = \{ w \mid \text{dist}(u, w) = j \} \) for \( j \in \mathbb{N} \).
   - Since the graph is bipartite, the induced subgraph of \( S_j \) is an anticlique (no edges between vertices in \( S_j \)).
   - There are no edges between \( S_i \) and \( S_j \) if \( |i - j| \geq 2 \).
   - Color all vertices in \( S_j \) with color \( A_{(j \bmod 2)}, B_{(j \bmod 3)} \). This ensures no two adjacent regions share the same \( A_i \) or \( B_i \) color.

7. **Conclusion:**
   - For \( n \geq 5 \), the graph of regions formed by \( n \) good lines can be colored such that no two adjacent regions share the same \( A_i \) or \( B_i \) color, and every combination of \( A_i \) and \( B_j \) appears in at least one region.

The final answer is \( \boxed{ n \geq 5 } \).