Number of regions

Question

What is the maximum number of closed regions that can be formed with drawing four lines inside a circle?

Correct answer is $11$ but I'm getting $8$.

Answer 1

HINT

Can you use $3$ lines to create $7$ regions inside the circle? Then use it to add another line splitting $4$ regions...

Answer 2

You can use Euler formula for planar graphs. $$ \nu-\varepsilon +\phi =2$$

If every line cuts every line we have 3 vertices on each line (determined with other lines) and 2 vertices in the intersection with circle. So we have overall $\nu = 14$ vertices.

Now every consecutive vertices on circle determine an edge, so we have 8 edges on circle (since we have 8 vertices on circle). Also each line determine 4 edges (since on each one we have 5 vertices). So we have overall $\varepsilon = 24$ edges.

So we have $$\phi = \varepsilon -\nu +2 = 12$$ faces and one of them is boundless. So we have $11$ regions in the circle.

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You can now easly generalise this for arbitrary lines.